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Theorem List for Metamath Proof Explorer - 15001-15100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremvdwapf 15001 The arithmetic progression function is a function. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP

Theoremvdwapval 15002* Value of the arithmetic progression function. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP

Theoremvdwapun 15003 Remove the first element of an arithmetic progression. (Contributed by Mario Carneiro, 11-Sep-2014.)
AP AP

Theoremvdwapid1 15004 The first element of an arithmetic progression. (Contributed by Mario Carneiro, 12-Sep-2014.)
AP

Theoremvdwap0 15005 Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP

Theoremvdwap1 15006 Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP

Theoremvdwmc 15007* The predicate " The -coloring contains a monochromatic AP of length ". (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP AP

Theoremvdwmc2 15008* Expand out the definition of an arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP

Theoremvdwpc 15009* The predicate " The coloring contains a polychromatic -tuple of AP's of length ". A polychromatic -tuple of AP's is a set of AP's with the same base point but different step lengths, such that each individual AP is monochromatic, but the AP's all have mutually distinct colors. (The common basepoint is not required to have the same color as any of the AP's.) (Contributed by Mario Carneiro, 18-Aug-2014.)
PolyAP AP

Theoremvdwlem1 15010* Lemma for vdw 15023. (Contributed by Mario Carneiro, 12-Sep-2014.)
AP                      MonoAP

Theoremvdwlem2 15011* Lemma for vdw 15023. (Contributed by Mario Carneiro, 12-Sep-2014.)
MonoAP MonoAP

Theoremvdwlem3 15012 Lemma for vdw 15023. (Contributed by Mario Carneiro, 13-Sep-2014.)

Theoremvdwlem4 15013* Lemma for vdw 15023. (Contributed by Mario Carneiro, 12-Sep-2014.)

Theoremvdwlem5 15014* Lemma for vdw 15023. (Contributed by Mario Carneiro, 12-Sep-2014.)
AP                      AP

Theoremvdwlem6 15015* Lemma for vdw 15023. (Contributed by Mario Carneiro, 13-Sep-2014.)
AP                      AP                                    PolyAP MonoAP

Theoremvdwlem7 15016* Lemma for vdw 15023. (Contributed by Mario Carneiro, 12-Sep-2014.)
AP        PolyAP PolyAP MonoAP

Theoremvdwlem8 15017* Lemma for vdw 15023. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP               PolyAP

Theoremvdwlem9 15018* Lemma for vdw 15023. (Contributed by Mario Carneiro, 12-Sep-2014.)
MonoAP                      PolyAP MonoAP               MonoAP                      PolyAP MonoAP

Theoremvdwlem10 15019* Lemma for vdw 15023. Set up secondary induction on . (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP               PolyAP MonoAP

Theoremvdwlem11 15020* Lemma for vdw 15023. (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP        MonoAP

Theoremvdwlem12 15021 Lemma for vdw 15023. base case of induction. (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP

Theoremvdwlem13 15022* Lemma for vdw 15023. Main induction on ; , base cases. (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP

Theoremvdw 15023* Van der Waerden's theorem. For any finite coloring and integer , there is an such that every coloring function from to contains a monochromatic arithmetic progression (which written out in full means that there is a color and base, increment values such that all the numbers lie in the preimage of , i.e. they are all in and evaluated at each one yields ). (Contributed by Mario Carneiro, 13-Sep-2014.)

Theoremvdwnnlem1 15024* Corollary of vdw 15023, and lemma for vdwnn 15027. If is a coloring of the integers, then there are arbitrarily long monochromatic APs in . (Contributed by Mario Carneiro, 13-Sep-2014.)

Theoremvdwnnlem2 15025* Lemma for vdwnn 15027. The set of all "bad" for the theorem is upwards-closed, because a long AP implies a short AP. (Contributed by Mario Carneiro, 13-Sep-2014.)

Theoremvdwnnlem3 15026* Lemma for vdwnn 15027. (Contributed by Mario Carneiro, 13-Sep-2014.) (Proof shortened by AV, 27-Sep-2020.)

Theoremvdwnn 15027* Van der Waerden's theorem, infinitary version. For any finite coloring of the positive integers, there is a color that contains arbitrarily long arithmetic progressions. (Contributed by Mario Carneiro, 13-Sep-2014.)

6.2.14  Ramsey's theorem

Syntaxcram 15028 Extend class notation with the Ramsey number function.
Ramsey

Syntaxcramold 15029 Extend class notation with the Ramsey number function (old version).
Ramsey

Theoremramtlecl 15030* The set of numbers with the Ramsey number property is upward-closed. (Contributed by Mario Carneiro, 21-Apr-2015.)

Definitiondf-ram 15031* Define the Ramsey number function. The input is a number for the size of the edges of the hypergraph, and a tuple from the finite color set to lower bounds for each color. The Ramsey number Ramsey is the smallest number such that for any set with Ramsey and any coloring of the set of -element subsets of (with color set ), there is a color and a subset such that and all the hyperedges of (that is, subsets of of size ) have color . (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.)
Ramsey inf

Definitiondf-ramOLD 15032* Define the Ramsey number function. The input is a number for the size of the edges of the hypergraph, and a tuple from the finite color set to lower bounds for each color. The Ramsey number Ramsey is the smallest number such that for any set with Ramsey and any coloring of the set of -element subsets of (with color set ), there is a color and a subset such that and all the hyperedges of (that is, subsets of of size ) have color . (Contributed by Mario Carneiro, 20-Apr-2015.) Obsolete version of df-ram 15031 as of 14-Sep-2020. (New usage is discouraged.)
Ramsey

Theoremhashbcval 15033* Value of the "binomial set", the set of all -element subsets of . (Contributed by Mario Carneiro, 20-Apr-2015.)

Theoremhashbccl 15034* The binomial set is a finite set. (Contributed by Mario Carneiro, 20-Apr-2015.)

Theoremhashbcss 15035* Subset relation for the binomial set. (Contributed by Mario Carneiro, 20-Apr-2015.)

Theoremhashbc0 15036* The set of subsets of size zero is the singleton of the empty set. (Contributed by Mario Carneiro, 22-Apr-2015.)

Theoremhashbc2 15037* The size of the binomial set is the binomial coefficient. (Contributed by Mario Carneiro, 20-Apr-2015.)

Theorem0hashbc 15038* There are no subsets of the empty set with size greater than zero. (Contributed by Mario Carneiro, 22-Apr-2015.)

Theoremramval 15039* The value of the Ramsey number function. (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by AV, 14-Sep-2020.)
Ramsey inf

TheoremramvalOLD 15040* The value of the Ramsey number function. (Contributed by Mario Carneiro, 21-Apr-2015.) Obsolete version of ramval 15039 as of 14-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Ramsey

Theoremramcl2lem 15041* Lemma for extended real closure of the Ramsey number function. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.)
Ramsey inf

Theoremramcl2lemOLD 15042* Lemma for extended real closure of the Ramsey number function. (Contributed by Mario Carneiro, 20-Apr-2015.) Obsolete version of ramcl2lem 15041 as of 14-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Ramsey

Theoremramtcl 15043* The Ramsey number has the Ramsey number property if any number does. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.)
Ramsey

TheoremramtclOLD 15044* The Ramsey number has the Ramsey number property if any number does. (Contributed by Mario Carneiro, 20-Apr-2015.) Obsolete version of ramtcl 15043 as of 14-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Ramsey

Theoremramtcl2 15045* The Ramsey number is an integer iff there is a number with the Ramsey number property. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.)
Ramsey

Theoremramtcl2OLD 15046* The Ramsey number is an integer iff there is a number with the Ramsey number property. (Contributed by Mario Carneiro, 20-Apr-2015.) Obsolete version of ramtcl2 15045 as of 14-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Ramsey

Theoremramtub 15047* The Ramsey number is a lower bound on the set of all numbers with the Ramsey number property. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.)
Ramsey

TheoremramtubOLD 15048* The Ramsey number is a lower bound on the set of all numbers with the Ramsey number property. (Contributed by Mario Carneiro, 20-Apr-2015.) Obsolete version of ramtub 15047 as of 14-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Ramsey

Theoremramub 15049* The Ramsey number is a lower bound on the set of all numbers with the Ramsey number property. (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theoremramub2 15050* It is sufficient to check the Ramsey property on finite sets of size equal to the upper bound. (Contributed by Mario Carneiro, 23-Apr-2015.)
Ramsey

Theoremrami 15051* The defining property of a Ramsey number. (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey               Ramsey

Theoremramcl2 15052 The Ramsey number is either a nonnegative integer or plus infinity. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.)
Ramsey

Theoremramcl2OLD 15053 The Ramsey number is either a nonnegative integer or plus infinity. (Contributed by Mario Carneiro, 20-Apr-2015.) Obsolete version of ramcl2 15052 as of 14-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Ramsey

Theoremramxrcl 15054 The Ramsey number is an extended real number. (This theorem does not imply Ramsey's theorem, unlike ramcl 15066.) (Contributed by Mario Carneiro, 20-Apr-2015.)
Ramsey

Theoremramubcl 15055 If the Ramsey number is upper bounded, then it is an integer. (Contributed by Mario Carneiro, 20-Apr-2015.)
Ramsey Ramsey

Theoremramlb 15056* Establish a lower bound on a Ramsey number. (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theorem0ram 15057* The Ramsey number when . (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theorem0ram2 15058 The Ramsey number when . (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theoremram0 15059 The Ramsey number when . (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theorem0ramcl 15060 Lemma for ramcl 15066: Existence of the Ramsey number when . (Contributed by Mario Carneiro, 23-Apr-2015.)
Ramsey

Theoremramz2 15061 The Ramsey number when has value zero for some color . (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theoremramz 15062 The Ramsey number when is the zero function. (Contributed by Mario Carneiro, 22-Apr-2015.)
Ramsey

Theoremramub1lem1 15063* Lemma for ramub1 15065. (Contributed by Mario Carneiro, 23-Apr-2015.)
Ramsey               Ramsey                      Ramsey

Theoremramub1lem2 15064* Lemma for ramub1 15065. (Contributed by Mario Carneiro, 23-Apr-2015.)
Ramsey               Ramsey                      Ramsey

Theoremramub1 15065* Inductive step for Ramsey's theorem, in the form of an explicit upper bound. (Contributed by Mario Carneiro, 23-Apr-2015.)
Ramsey               Ramsey        Ramsey Ramsey

Theoremramcl 15066 Ramsey's theorem: the Ramsey number is an integer for every finite coloring and set of upper bounds. (Contributed by Mario Carneiro, 23-Apr-2015.)
Ramsey

Theoremramsey 15067* Ramsey's theorem with the definition Ramsey eliminated. If is an integer, is a specified finite set of colors, and is a set of lower bounds for each color, then there is an such that for every set of size greater than and every coloring of the set of all -element subsets of , there is a color and a subset such that is larger than and the -element subsets of are monochromatic with color . This is the hypergraph version of Ramsey's theorem; the version for simple graphs is the case . This is Metamath 100 proof #31. (Contributed by Mario Carneiro, 23-Apr-2015.)

6.2.15  Primorial function

According to Wikipedia "Primorial", https://en.wikipedia.org/wiki/Primorial (28-Aug-2020): "In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying [all] positive integers [less than or equal to a given number], the function only multiplies [the] prime numbers [less than or equal to the given number]. The name "primorial", coined by Harvey Dubner, draws an analogy to primes similar to the way the name "factorial" relates to factors."

Syntaxcprmo 15068 Extend class notation to include the primorial of nonnegative integers.
#p

Definitiondf-prmo 15069* Define the primorial function on nonnegative integers as the product of all prime numbers less than or equal to the integer. For example, #p ; because ; (see prmo6 15179). In the literature, the primorial function is written as a postscript hash: 6# = 30. In contrast to prmorcht 24184, where the primorial function is defined by using the sequence builder ( ), the more specialized definition of a product of a series is used here. (Contributed by AV, 28-Aug-2020.)
#p

Theoremprmoval 15070* Value of the primorial function for nonnegative integers. (Contributed by AV, 28-Aug-2020.)
#p

Theoremprmocl 15071 Closure of the primorial function. (Contributed by AV, 28-Aug-2020.)
#p

Theoremprmone0 15072 The primorial function is nonzero. (Contributed by AV, 28-Aug-2020.)
#p

Theoremprmo0 15073 The primorial of 0. (Contributed by AV, 28-Aug-2020.)
#p

Theoremprmo1 15074 The primorial of 1. (Contributed by AV, 28-Aug-2020.)
#p

Theoremprmop1 15075 The primorial of a successor. (Contributed by AV, 28-Aug-2020.)
#p #p #p

Theoremprmonn2 15076 Value of the primorial function expressed recursively. (Contributed by AV, 28-Aug-2020.)
#p #p #p

Theoremprmo2 15077 The primorial of 2. (Contributed by AV, 28-Aug-2020.)
#p

Theoremprmo3 15078 The primorial of 3. (Contributed by AV, 28-Aug-2020.)
#p

Theoremprmdvdsprmo 15079* The primorial of a number is divisible by each prime less then or equal to the number. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 28-Aug-2020.)
#p

Theoremprmdvdsprmop 15080* The primorial of a number plus an integer greater than 1 and less then or equal to the number is divisible by a prime less then or equal to the number. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 28-Aug-2020.)
#p

Theoremfvprmselelfz 15081* The value of the prime selection function is in a finite sequence of integers if the argument is in this finite sequence of integers. (Contributed by AV, 19-Aug-2020.)

Theoremfvprmselgcd1 15082* The greatest common divisor of two values of the prime selection function for different arguments is 1. (Contributed by AV, 19-Aug-2020.)

Theoremprmolefac 15083 The primorial of a positive integer is less than or equal to the factorial of the integer. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 29-Aug-2020.)
#p

Theoremprmodvdslcmf 15084 The primorial of a nonnegative integer divides the least common multiple of all positive integers less than or equal to the integer. (Contributed by AV, 19-Aug-2020.) (Revised by AV, 29-Aug-2020.)
#p lcm

Theoremprmolelcmf 15085 The primorial of a positive integer is less than or equal to the least common multiple of all positive integers less than or equal to the integer. (Contributed by AV, 19-Aug-2020.) (Revised by AV, 29-Aug-2020.)
#p lcm

6.2.16  Prime gaps

According to Wikipedia "Prime gap", https://en.wikipedia.org/wiki/Prime_gap (16-Aug-2020): "A prime gap is the difference between two successive prime numbers. The n-th prime gap, denoted gn or g(pn) is the difference between the (n+1)-th and the n-th prime numbers, i.e. gn = pn+1 - pn . We have g1 = 1, g2 = g3 = 2, and g4 = 4."

It can be proven that there are arbitrary large gaps, usually by showing that "in the sequence n!+2, n!+3, ..., n!+n the first term is divisible by 2, the second term is divisible by 3, and so on. Thus, this is a sequence of n-1 consecutive composite integers, and it must belong to a gap between primes having length at least n.", see prmgap 15108.

Instead of using the factorial of n (see df-fac 12498), any function can be chosen for which f(n) is not relatively prime to the integers 2, 3, ..., n. For example, the least common multiple of the integers 2, 3, ..., n, see prmgaplcm 15110, or the primorial n# (the product of all prime numbers less than or equal to n), see prmgapprmo 15112, are such functions, which provide smaller values than the factorial function (see lcmflefac 14700 and prmolefac 15083 resp. prmolelcmf 15085): "For instance, the first prime gap of size larger than 14 occurs between the primes 523 and 541, while 15! is the vastly larger number 1307674368000." But the least common multiple of the integers 2, 3, ..., 15 is 360360, and 15# is 30030 (p3248 = 30029 and P3249 = 30047, so g3248 = 18).

Theoremprmgaplem1 15086 Lemma for prmgap 15108: The factorial of a number plus an integer greater than 1 and less then or equal to the number is divisible by that integer. (Contributed by AV, 13-Aug-2020.)

Theoremprmgaplem2 15087 Lemma for prmgap 15108: The factorial of a number plus an integer greater than 1 and less then or equal to the number are not coprime. (Contributed by AV, 13-Aug-2020.)

Theoremprmgaplcmlem1 15088 Lemma for prmgaplcm 15110: The least common multiple of all positive integers less than or equal to a number plus an integer greater than 1 and less then or equal to the number is divisible by that integer. (Contributed by AV, 14-Aug-2020.) (Revised by AV, 27-Aug-2020.)
lcm

Theoremprmgaplcmlem2 15089 Lemma for prmgaplcm 15110: The least common multiple of all positive integers less than or equal to a number plus an integer greater than 1 and less then or equal to the number are not coprime. (Contributed by AV, 14-Aug-2020.) (Revised by AV, 27-Aug-2020.)
lcm

TheoremlcmsmapnnOLD 15090* The function mapping a positive integer to the least common multiple of all positive integers less than or equal to the integer is a mapping from the positive integers to the positive integers. (Contributed by AV, 14-Aug-2020.) Obsolete as of 27-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.)

Theoremprmgaplcmlem1OLD 15091* Lemma for prmgaplcmOLD 15109: The least common multiple of all positive integers less than or equal to a number plus an integer greater than 1 and less then or equal to the number is divisible by that integer. (Contributed by AV, 14-Aug-2020.) Obsolete version of prmgaplcmlem1 15088 as of 27-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.)

Theoremprmgaplcmlem2OLD 15092* Lemma for prmgaplcmOLD 15109: The least common multiple of all positive integers less than or equal to a number plus an integer greater than 1 and less then or equal to the number are not coprime. (Contributed by AV, 14-Aug-2020.) Obsolete version of prmgaplcmlem2 15089 as of 27-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.)

TheoremprmormapnnOLD 15093* The primorial function is a mapping from the positive integers to the positive integers. In contrast to prmorcht 24184, where the primorial is defined by using the sequence builder ( ), the more specialized definition of a product of a series is used here. (Contributed by AV, 14-Aug-2020.) Obsolete as of 29-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.)

TheoremprmdvdsprmorOLD 15094* The primorial of a number is divisible by each prime less then or equal to the number. (Contributed by AV, 15-Aug-2020.) Obsolete version of prmdvdsprmo 15079 as of 29-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.)

TheoremprmdvdsprmorpOLD 15095* The primorial of a number plus an integer greater than 1 and less then or equal to the number is divisible by a prime less then or equal to the number. (Contributed by AV, 15-Aug-2020.) Obsolete version of prmdvdsprmop 15080 as of 29-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.)

TheoremprmgapprmorlemOLD 15096* Lemma for prmgapprmorOLD 15113: The primorial of a number plus an integer greater than 1 and less then or equal to the number are not coprime. (Contributed by AV, 15-Aug-2020.) Obsolete version of prmgapprmolem 15111 as of 29-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.)

TheoremprmorlefacOLD 15097* The primorial of a positive integer is less than or equal to the factorial of the integer. (Contributed by AV, 15-Aug-2020.) Obsolete version of prmolefac 15083 as of 29-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.)

TheoremprmordvdslcmfOLD 15098* The primorial of a positive integer divides the least common multiple of all positive integers less than or equal to the integer. (Contributed by AV, 19-Aug-2020.) (Revised by AV, 27-Aug-2020.) Obsolete version of prmodvdslcmf 15084 as of 29-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.)
lcm

TheoremprmorlelcmfOLD 15099* The primorial of a positive integer is less than or equal to the least common multiple of all positive integers less than or equal to the integer. (Contributed by AV, 19-Aug-2020.) (Revised by AV, 27-Aug-2020.) Obsolete version of prmolelcmf 15085 as of 29-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.)
lcm

TheoremprmordvdslcmsOLDOLD 15100* The primorial of a positive integer divides the least common multiple of all positive integers less than or equal to the integer. (Contributed by AV, 19-Aug-2020.) Obsolete version of prmordvdslcmfOLD 15098 as of 27-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.)

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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