A355535 - OEIS

%I #8 Jul 24 2022 14:13:43

%S 9,21,25,27,45,49,57,63,75,81,99,105,115,117,121,125,133,135,147,153,

%T 159,171,175,189,195,207,225,231,243,245,261,273,275,279,285,289,297,

%U 315,325,333,343,345,351,357,361,363,369,371,375,387,393,399,405,423

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

%C 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 The terms together with their prime indices begin:

%e     9: {2,2}

%e    21: {2,4}

%e    25: {3,3}

%e    27: {2,2,2}

%e    45: {2,2,3}

%e    49: {4,4}

%e    57: {2,8}

%e    63: {2,2,4}

%e    75: {2,3,3}

%e    81: {2,2,2,2}

%e    99: {2,2,5}

%e   105: {2,3,4}

%e 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 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],OddQ[#]&&Select[Tuples[primeMS/@primeMS[#]],UnsameQ@@#&]=={}&]

%Y Including evens gives A355529.

%Y The version for all divisors including evens is A355740, zeros of A355739.

%Y Choices of a prime factor of each prime index: A355741, unordered A355744.

%Y A001221 counts distinct prime factors, with sum A001414.

%Y A001222 counts prime factors with multiplicity.

%Y A003963 multiplies together the prime indices of n.

%Y A056239 adds up prime indices, row sums of A112798.

%Y A120383 lists numbers divisible by all of their prime indices.

%Y Cf. A000720, A076610, A289509, A302796, A327486, A355731, A355733, A355742.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jul 22 2022