A382215 MM-numbers of multiset partitions into constant blocks with a common sum.

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 23, 25, 27, 31, 32, 35, 41, 49, 53, 59, 64, 67, 81, 83, 97, 103, 109, 121, 125, 127, 128, 131, 157, 175, 179, 191, 209, 211, 227, 241, 243, 245, 256, 277, 283, 289, 311, 331, 343, 353, 361, 367, 391, 401, 419, 431, 461

Offset: 1

Author

Gus Wiseman, Mar 21 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, sum A056239. The multiset of multisets with MM-number n is 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 terms together with their prime indices of prime indices begin:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  11: {{3}}
  16: {{},{},{},{}}
  17: {{4}}
  19: {{1,1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  31: {{5}}
  32: {{},{},{},{},{}}
  35: {{2},{1,1}}
  41: {{6}}
  49: {{1,1},{1,1}}
  53: {{1,1,1,1}}
  59: {{7}}
		

Crossrefs

Twice-partitions of this type are counted by A279789.

For just constant blocks we have A302492, counted by A000688.

For sets of constant multisets we have A302496, counted by A050361.

For just common sums we have A326534, counted by A321455.

Factorizations of this type are counted by A381995.

For strict blocks and distinct sums we have A382201, counted by A381633.

Normal multiset partitions of this type are counted by A382204.

For strict instead of constant blocks we have A382304, counted by A382080.

For sets of constant multisets with distinct sums A382426, counted by A381635.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A124010 gives prime signature, sorted A118914, length A001221, sum A001222.

Cf. A000720, A000961, A001055, A302242, A302601, A368100, A381715, A381716, A381636, A381871.

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@@SameQ@@@prix/@prix[#]&]

PARI

is(k) = my(f=factor(k)[, 1]~, k, p, v=vector(#f, i, primepi(f[i]))); for(i=1, #v, k=isprimepower(v[i], &p); if(k||v[i]==1, v[i]=k*primepi(p), return(0))); #Set(v)<2; \\ Jinyuan Wang, Apr 02 2025

Formula

Equals A326534 /\ A302492.