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A322553 Odd numbers whose product of prime indices is a prime power. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified June 20 00:16 EDT 2021. Contains 345154 sequences. (Running on oeis4.)