A322553 Odd numbers whose product of prime indices is a prime power.

1, 3, 5, 7, 9, 11, 17, 19, 21, 23, 25, 27, 31, 41, 49, 53, 57, 59, 63, 67, 81, 83, 97, 103, 109, 115, 121, 125, 127, 131, 133, 147, 157, 159, 171, 179, 189, 191, 211, 227, 241, 243, 277, 283, 289, 311, 331, 343, 353, 361, 367, 371, 393, 399, 401, 419, 431, 441

Offset: 1

Author

Gus Wiseman, Dec 15 2018

Keywords

• nonn

Comments

Differs from A322400 in having 1 and lacking 377, the MM-number of {{1,2},{1,3}}.

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.

Examples

			The multiset multisystem 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 multisystem with MM-number 78 is {{},{1},{1,2}}. The sequence of multiset partitions whose MM-numbers belong to this sequence begins:
   1: {}
   3: {{1}}
   5: {{2}}
   7: {{1,1}}
   9: {{1},{1}}
  11: {{3}}
  17: {{4}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  31: {{5}}
  41: {{6}}
  49: {{1,1},{1,1}}
  53: {{1,1,1,1}}
  57: {{1},{1,1,1}}
  59: {{7}}
  63: {{1},{1},{1,1}}
  67: {{8}}
  81: {{1},{1},{1},{1}}
  83: {{9}}
  97: {{3,3}}
		

Crossrefs

Cf. A003963, A056239, A112798, A290103, A302242, A320325, A320698.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1,100,2],#==1||PrimePowerQ[Times@@primeMS[#]]&]

PARI

isok(n) = {if (n % 2, my(f = factor(n), pk = prod(k=1, #f~, primepi(f[k,1]))); (pk == 1) || isprimepower(pk););} \\ Michel Marcus, Dec 16 2018