A363261 The partial sums of the prime indices of n include half the sum of all prime indices of n.

4, 9, 12, 16, 25, 30, 40, 48, 49, 63, 64, 70, 81, 84, 108, 112, 121, 144, 154, 160, 165, 169, 192, 198, 220, 256, 264, 270, 273, 286, 289, 325, 351, 352, 360, 361, 364, 390, 442, 448, 468, 480, 520, 529, 561, 567, 576, 595, 624, 625, 640, 646, 675, 714, 729

Offset

1,1

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..55.

Example

The terms together with their prime indices begin:
   4: {1,1}
   9: {2,2}
  12: {1,1,2}
  16: {1,1,1,1}
  25: {3,3}
  30: {1,2,3}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  49: {4,4}
  63: {2,2,4}
  64: {1,1,1,1,1,1}
  70: {1,3,4}
  81: {2,2,2,2}
  84: {1,1,2,4}

Mathematica

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

Crossrefs

Partitions of this type are counted by A322439.

For parts instead of partial sums we have A344415, counted by A035363.

A025065 counts palindromic partitions, ranked by A265640.

A027746 lists prime factors, A112798 indices, length A001222, sum A056239.

A301987 lists numbers whose sum of prime indices equals their product.

A322109 ranks partitions of n with no part > n/2, counted by A110618.

Cf. A001414, A106529, A327473, A327476, A334201, A340387, A360005.

Sequence in context: A357976 A330879 A357636 * A360953 A348272 A285419

Adjacent sequences: A363258 A363259 A363260 * A363262 A363263 A363264

Keyword

nonn

Author

Gus Wiseman, Jun 06 2023

Status

approved