

A357707


Numbers whose prime indices have equal number of parts congruent to each of 1 and 3 (mod 4).


1



1, 3, 7, 9, 10, 13, 19, 21, 27, 29, 30, 34, 37, 39, 43, 49, 53, 55, 57, 61, 62, 63, 70, 71, 79, 81, 87, 89, 90, 91, 94, 100, 101, 102, 107, 111, 113, 115, 117, 129, 130, 131, 133, 134, 139, 147, 151, 159, 163, 165, 166, 169, 171, 173, 181, 183, 186, 187, 189
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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.


LINKS



EXAMPLE

The terms together with their prime indices begin:
1: {}
3: {2}
7: {4}
9: {2,2}
10: {1,3}
13: {6}
19: {8}
21: {2,4}
27: {2,2,2}
29: {10}
30: {1,2,3}


MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Count[primeMS[#], _?(Mod[#, 4]==1&)]==Count[primeMS[#], _?(Mod[#, 4]==3&)]&]


CROSSREFS

These partitions are counted by A035544.
Includes A066207 = products of primes of even index.
The conjugate reverse version is A357640 (aerated).
A357705 counts reversed partitions by skewalternating sum, half A357704.


KEYWORD

nonn


AUTHOR



STATUS

approved



