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

1,2

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.

Links

Table of n, a(n) for n=1..58.

Example

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}}

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[#]]&]

Prog

(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

Crossrefs

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

Sequence in context: A201644 A342949 A322400 * A302601 A064076 A050842

Adjacent sequences: A322550 A322551 A322552 * A322554 A322555 A322556

Keyword

nonn

Author

Gus Wiseman, Dec 15 2018

Status

approved