A383309 Numbers whose prime indices are prime powers > 1 with a common sum of prime indices.

1, 3, 5, 7, 9, 11, 17, 19, 23, 25, 27, 31, 35, 41, 49, 53, 59, 67, 81, 83, 97, 103, 109, 121, 125, 127, 131, 157, 175, 179, 191, 209, 211, 227, 241, 243, 245, 277, 283, 289, 311, 331, 343, 353, 361, 367, 391, 401, 419, 431, 461, 509, 529, 547, 563, 587, 599

Offset: 1

Author

Gus Wiseman, Apr 25 2025

Keywords

• nonn

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. We define the multiset of multisets with MM-number n to be formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

Examples

			The systems with these MM-numbers begin:
   1: {}
   3: {{1}}
   5: {{2}}
   7: {{1,1}}
   9: {{1},{1}}
  11: {{3}}
  17: {{4}}
  19: {{1,1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  31: {{5}}
  35: {{2},{1,1}}
  41: {{6}}
  49: {{1,1},{1,1}}
  53: {{1,1,1,1}}
  59: {{7}}
  67: {{8}}
  81: {{1},{1},{1},{1}}
  83: {{9}}
  97: {{3,3}}
		

Crossrefs

Twice-partitions of this type are counted by A279789.

For just a common sum we have A326534.

For just constant blocks we have A355743.

Numbers without a factorization of this type are listed by A381871, counted by A381993.

The multiplicative version is A381995.

This is the odd case of A382215.

For strict instead of constant blocks we have A382304.

A001055 counts factorizations, strict A045778.

A023894 counts partitions into prime-powers.

A034699 gives maximal prime-power divisor.

A050361 counts factorizations into distinct prime powers.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A246655 lists the prime-powers (A000961 includes 1), towers A164336.

A317141 counts coarsenings of prime indices, refinements A300383.

A353864 counts rucksack partitions, ranked by A353866.

A355742 chooses a prime-power divisor of each prime index.

Cf. A000688, A000720, A001222, A006171, A038041, A279784, A302242, A302493, A321455, A326518, A381719.

Programs

Mathematica

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

Formula

Equals A326534 /\ A355743.