cp's OEIS Frontend

A343337 Numbers with no prime index divisible by all the other prime indices.

%I A343337 #12 Aug 27 2021 05:58:40
%S A343337 1,15,30,33,35,45,51,55,60,66,69,70,75,77,85,90,91,93,95,99,102,105,
%T A343337 110,119,120,123,132,135,138,140,141,143,145,150,153,154,155,161,165,
%U A343337 170,175,177,180,182,186,187,190,198,201,203,204,205,207,209,210,215
%N A343337 Numbers with no prime index divisible by all the other prime indices.
%C A343337 Alternative name: 1 and numbers whose greatest prime index is not divisible by all the other prime indices.
%C A343337 First differs from A318992 in lacking 195.
%C A343337 First differs from A343343 in lacking 195.
%C A343337 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A343337 Also Heinz numbers of partitions with greatest part not divisible by all the others (counted by A343341). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
%e A343337 The sequence of terms together with their prime indices begins:
%e A343337       1: {}            90: {1,2,2,3}      141: {2,15}
%e A343337      15: {2,3}         91: {4,6}          143: {5,6}
%e A343337      30: {1,2,3}       93: {2,11}         145: {3,10}
%e A343337      33: {2,5}         95: {3,8}          150: {1,2,3,3}
%e A343337      35: {3,4}         99: {2,2,5}        153: {2,2,7}
%e A343337      45: {2,2,3}      102: {1,2,7}        154: {1,4,5}
%e A343337      51: {2,7}        105: {2,3,4}        155: {3,11}
%e A343337      55: {3,5}        110: {1,3,5}        161: {4,9}
%e A343337      60: {1,1,2,3}    119: {4,7}          165: {2,3,5}
%e A343337      66: {1,2,5}      120: {1,1,1,2,3}    170: {1,3,7}
%e A343337      69: {2,9}        123: {2,13}         175: {3,3,4}
%e A343337      70: {1,3,4}      132: {1,1,2,5}      177: {2,17}
%e A343337      75: {2,3,3}      135: {2,2,2,3}      180: {1,1,2,2,3}
%e A343337      77: {4,5}        138: {1,2,9}        182: {1,4,6}
%e A343337      85: {3,7}        140: {1,1,3,4}      186: {1,2,11}
%e A343337 For example, 195 has prime indices {2,3,6}, and 6 is divisible by both 2 and 3, so 195 does not belong to the sequence.
%t A343337 Select[Range[1000],#==1||With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(Max@@p/p)]&]
%Y A343337 The complement is counted by A130689.
%Y A343337 The dual version is A342193.
%Y A343337 The case with smallest prime index not dividing all the others is A343338.
%Y A343337 The case with smallest prime index dividing by all the others is A343340.
%Y A343337 These are the Heinz numbers of the partitions counted by A343341.
%Y A343337 Including the dual version gives A343343.
%Y A343337 A000005 counts divisors.
%Y A343337 A000070 counts partitions with a selected part.
%Y A343337 A006128 counts partitions with a selected position.
%Y A343337 A056239 adds up prime indices, row sums of A112798.
%Y A343337 A067824 counts strict chains of divisors starting with n.
%Y A343337 A253249 counts strict chains of divisors.
%Y A343337 A339564 counts factorizations with a selected factor.
%Y A343337 Cf. A083710, A257993, A338470, A341450, A343342, A343345, A343346, A343377, A343379, A343381, A343382.
%K A343337 nonn
%O A343337 1,2
%A A343337 _Gus Wiseman_, Apr 13 2021