A383014 Numbers whose prime indices can be partitioned into constant blocks with a common sum.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 40, 41, 43, 47, 48, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 108, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169

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.

Links

Table of n, a(n) for n=1..62.

Example

The prime indices of 36 are {1,1,2,2}, and a partition into constant blocks with a common sum is: {{2},{2},{1,1}}, so 36 is in the sequence.
The prime indices of 43200 are {1,1,1,1,1,1,2,2,2,3,3}, and a partition into constant blocks with a common sum is: {{{1,1,1,1,1,1},{2,2,2},{3,3}}}, so 43200 is in the sequence.
The prime indices of 520000 are {1,1,1,1,1,1,3,3,3,3,6} and a partition into constant blocks with a common sum is: {{1,1,1,1,1,1},{3,3},{3,3},{6}}, so 520000 is in the sequence.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  11: {5}
  12: {1,1,2}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  37: {12}
  40: {1,1,1,3}

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
mce[y_]:=Table[ConstantArray[y[[1]], #]&/@ptn, {ptn, IntegerPartitions[Length[y]]}];
Select[Range[100], Select[Join@@@Tuples[mce/@Split[prix[#]]], SameQ@@Total/@#&]!={}&]

Crossrefs

Twice-partitions of this type (constant blocks with a common sum) are counted by A279789.

Includes all elements of A353833.

For distinct sums we have the complement of A381636.

For strict blocks we have the complement of A381719.

For distinct sums and strict blocks we have the complement of A381806.

The complement is A381871, counted by A381993.

These are the positions of positive terms in A381995.

Partitions of this type are counted by A383093.

Constant blocks: A000688, A006171, A279784, A295935, A381453 (lower), A381455 (upper).

A001055 counts factorizations (multiset partitions of prime indices), strict A045778.

A050361 counts factorizations into distinct prime powers.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A317141 counts coarsenings of prime indices, refinements A300383.

Cf. A000720, A000961, A001222, A321469, A381635, A381715, A381716, A381717.

Sequence in context: A326841 A274222 A300273 * A383110 A353844 A353833

Adjacent sequences: A383011 A383012 A383013 * A383015 A383016 A383017

Keyword

nonn

Author

Gus Wiseman, Apr 22 2025

Status

approved