OFFSET
1,2
COMMENTS
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, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
These are the Heinz numbers of integer partitions whose parts and conjugate parts are all odd. They are counted by A053253.
EXAMPLE
The terms together with their prime indices begin:
1: {}
2: {1}
5: {3}
8: {1,1,1}
11: {5}
17: {7}
20: {1,1,3}
23: {9}
31: {11}
32: {1,1,1,1,1}
41: {13}
44: {1,1,5}
47: {15}
59: {17}
67: {19}
68: {1,1,7}
73: {21}
80: {1,1,1,1,3}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Select[Range[100], And@@OddQ/@primeMS[#]&&And@@OddQ/@conj[primeMS[#]]&]
CROSSREFS
The restriction to primes is A031368.
These partitions appear to be counted by A053253.
The even version is A066207^2.
For even instead of odd conjugate parts we get A066208^2.
A238745 gives the Heinz number of the conjugate prime signature.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 18 2022
STATUS
approved