A357707 Numbers whose prime indices have equal number of parts congruent to each of 1 and 3 (mod 4).

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

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

Table of n, a(n) for n=1..59.

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 partitions are ranked by A357636, reverse A357632.

The conjugate reverse version is A357640 (aerated).

A056239 adds up prime indices, row sums of A112798.

A316524 gives alternating sum of prime indices, reverse A344616.

A344651 counts partitions by alternating sum, ordered A097805.

A357705 counts reversed partitions by skew-alternating sum, half A357704.

Cf. A035363, A035550, A035594, A053251, A298311, A357486, A357623, A357638.

Sequence in context: A284526 A344413 A339618 * A294571 A083214 A091210

Adjacent sequences: A357704 A357705 A357706 * A357708 A357709 A357710

Keyword

nonn

Author

Gus Wiseman, Oct 12 2022

Status

approved