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A343343 Numbers with either no prime index dividing, or no prime index divisible by all the other prime indices.

%I A343343 #11 Apr 22 2021 22:01:48
%S A343343 1,15,30,33,35,45,51,55,60,66,69,70,75,77,85,90,91,93,95,99,102,105,
%T A343343 110,119,120,123,132,135,138,140,141,143,145,150,153,154,155,161,165,
%U A343343 170,175,177,180,182,186,187,190,195,198,201,203,204,205,207,209,210
%N A343343 Numbers with either no prime index dividing, or no prime index divisible by all the other prime indices.
%C A343343 After 1, first differs from A318992 in lacking 390, with prime indices {1,2,3,6}.
%C A343343 First differs from A343337 in having 195, with prime indices {2,3,6}.
%C A343343 Alternative name: 1 and numbers where either the smallest prime index is not a divisor of all the other prime indices, or the greatest prime index is not divisible by all the other prime indices.
%C A343343 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 A343343 Also Heinz numbers of partitions that either empty, have smallest part not dividing all the others, or have greatest part not divisible by all the others (counted by A343346). 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.
%F A343343 Equals the union of A342193 and A343337.
%e A343343 The sequence of terms together with their prime indices begins:
%e A343343       1: {}            90: {1,2,2,3}      141: {2,15}
%e A343343      15: {2,3}         91: {4,6}          143: {5,6}
%e A343343      30: {1,2,3}       93: {2,11}         145: {3,10}
%e A343343      33: {2,5}         95: {3,8}          150: {1,2,3,3}
%e A343343      35: {3,4}         99: {2,2,5}        153: {2,2,7}
%e A343343      45: {2,2,3}      102: {1,2,7}        154: {1,4,5}
%e A343343      51: {2,7}        105: {2,3,4}        155: {3,11}
%e A343343      55: {3,5}        110: {1,3,5}        161: {4,9}
%e A343343      60: {1,1,2,3}    119: {4,7}          165: {2,3,5}
%e A343343      66: {1,2,5}      120: {1,1,1,2,3}    170: {1,3,7}
%e A343343      69: {2,9}        123: {2,13}         175: {3,3,4}
%e A343343      70: {1,3,4}      132: {1,1,2,5}      177: {2,17}
%e A343343      75: {2,3,3}      135: {2,2,2,3}      180: {1,1,2,2,3}
%e A343343      77: {4,5}        138: {1,2,9}        182: {1,4,6}
%e A343343      85: {3,7}        140: {1,1,3,4}      186: {1,2,11}
%e A343343 For example, the prime indices of 90 are {1,2,2,3}, and, because 1 divides all the other parts, 90 is in the sequence, even though 3 is not divisible by all the other parts.
%t A343343 Select[Range[100],#==1||With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(Max@@p/p)||!And@@IntegerQ/@(p/Min@@p)]&]
%Y A343343 The partitions without these Heinz numbers are counted by A130714.
%Y A343343 The first condition alone gives A342193.
%Y A343343 The second condition alone gives A343337.
%Y A343343 The "and" instead of "or" version is A343338.
%Y A343343 The partitions with these Heinz numbers are counted by A343346.
%Y A343343 A000005 counts divisors.
%Y A343343 A000070 counts partitions with a selected part.
%Y A343343 A006128 counts partitions with a selected position.
%Y A343343 A015723 counts strict partitions with a selected part.
%Y A343343 A018818 counts partitions into divisors (strict: A033630).
%Y A343343 A056239 adds up prime indices, row sums of A112798.
%Y A343343 A067824 counts strict chains of divisors starting with n.
%Y A343343 A253249 counts strict chains of divisors.
%Y A343343 A339564 counts factorizations with a selected factor.
%Y A343343 Cf. A083710, A257993, A338470, A339562, A341450, A343339, A343340, A343341, A343342, A343378, A343379, A343382.
%K A343343 nonn
%O A343343 1,2
%A A343343 _Gus Wiseman_, Apr 15 2021