A366321 Numbers m whose prime indices have even sum k such that k/2 is not a prime index of m.

1, 3, 7, 10, 13, 16, 19, 21, 22, 27, 28, 29, 34, 36, 37, 39, 43, 46, 48, 52, 53, 55, 57, 61, 62, 64, 66, 71, 75, 76, 79, 81, 82, 85, 87, 88, 89, 90, 91, 94, 100, 101, 102, 107, 108, 111, 113, 115, 116, 117, 118, 120, 129, 130, 131, 133, 134, 136, 138, 139, 144

Offset

0,2

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=0..60.

Example

The prime indices of 84 are y = {1,1,2,4}, with even sum 8; but 8/2 = 4 is in y, so 84 is not in the sequence.
The terms together with their prime indices begin:
    1: {}
    3: {2}
    7: {4}
   10: {1,3}
   13: {6}
   16: {1,1,1,1}
   19: {8}
   21: {2,4}
   22: {1,5}
   27: {2,2,2}
   28: {1,1,4}
   29: {10}
   34: {1,7}
   36: {1,1,2,2}

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], EvenQ[Total[prix[#]]]&&FreeQ[prix[#], Total[prix[#]]/2]&]

Crossrefs

Partitions of this type are counted by A182616, strict A365828.

A066207 lists numbers with all even prime indices, odd A066208.

A086543 lists numbers with at least one odd prime index, counted by A366322.

A300063 ranks partitions of odd numbers.

A366319 ranks partitions of n not containing n/2.

A366321 ranks partitions of 2k that do not contain k.

Cf. A000041, A006827, A047967, A320924, A339662, A365825, A365920, A366318, A366528, A366530.

Sequence in context: A103541 A183178 A340604 * A310183 A145004 A310184

Adjacent sequences: A366318 A366319 A366320 * A366322 A366323 A366324

Keyword

nonn

Author

Gus Wiseman, Oct 13 2023

Status

approved