cp's OEIS Frontend

A355535 Odd numbers of which it is not possible to choose a different prime factor of each prime index.

%I A355535 #8 Jul 24 2022 14:13:43
%S A355535 9,21,25,27,45,49,57,63,75,81,99,105,115,117,121,125,133,135,147,153,
%T A355535 159,171,175,189,195,207,225,231,243,245,261,273,275,279,285,289,297,
%U A355535 315,325,333,343,345,351,357,361,363,369,371,375,387,393,399,405,423
%N A355535 Odd numbers of which it is not possible to choose a different prime factor of each prime index.
%C A355535 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.
%e A355535 The terms together with their prime indices begin:
%e A355535     9: {2,2}
%e A355535    21: {2,4}
%e A355535    25: {3,3}
%e A355535    27: {2,2,2}
%e A355535    45: {2,2,3}
%e A355535    49: {4,4}
%e A355535    57: {2,8}
%e A355535    63: {2,2,4}
%e A355535    75: {2,3,3}
%e A355535    81: {2,2,2,2}
%e A355535    99: {2,2,5}
%e A355535   105: {2,3,4}
%e A355535 For example, the prime indices of 897 are {2,6,9}, of which we can choose prime factors in two ways: (2,2,3) or (2,3,3); but neither of these has all distinct elements, so 897 is in the sequence.
%t A355535 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A355535 Select[Range[100],OddQ[#]&&Select[Tuples[primeMS/@primeMS[#]],UnsameQ@@#&]=={}&]
%Y A355535 Including evens gives A355529.
%Y A355535 The version for all divisors including evens is A355740, zeros of A355739.
%Y A355535 Choices of a prime factor of each prime index: A355741, unordered A355744.
%Y A355535 A001221 counts distinct prime factors, with sum A001414.
%Y A355535 A001222 counts prime factors with multiplicity.
%Y A355535 A003963 multiplies together the prime indices of n.
%Y A355535 A056239 adds up prime indices, row sums of A112798.
%Y A355535 A120383 lists numbers divisible by all of their prime indices.
%Y A355535 Cf. A000720, A076610, A289509, A302796, A327486, A355731, A355733, A355742.
%K A355535 nonn
%O A355535 1,1
%A A355535 _Gus Wiseman_, Jul 22 2022