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A322554 Numbers whose product of prime indices is a power of a squarefree number (A072774). 6
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 66, 67, 68, 72, 73, 76, 79, 80 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The complement is {35, 37, 39, 45, 61, 65, ...}.

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. 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}}. This sequence lists all MM-numbers of regular multiset multisystems, where regularity means all vertex-degrees are equal.

LINKS

Table of n, a(n) for n=1..67.

EXAMPLE

Most small numbers belong to this sequence. However, the sequence of multiset multisystems whose MM-numbers do not belong to this sequence begins:

  35: {{2},{1,1}}

  37: {{1,1,2}}

  39: {{1},{1,2}}

  45: {{1},{1},{2}}

  61: {{1,2,2}}

  65: {{2},{1,2}}

  69: {{1},{2,2}}

  70: {{},{2},{1,1}}

  71: {{1,1,3}}

  74: {{},{1,1,2}}

  75: {{1},{2},{2}}

  77: {{1,1},{3}}

  78: {{},{1},{1,2}}

  87: {{1},{1,3}}

  89: {{1,1,1,2}}

  90: {{},{1},{1},{2}}

  91: {{1,1},{1,2}}

  95: {{2},{1,1,1}}

  99: {{1},{1},{3}}

MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Select[Range[100], SameQ@@Last/@FactorInteger[Times@@primeMS[#]]&]

CROSSREFS

Cf. A003963, A056239, A062503, A072774, A112798, A302242, A302491, A320325, A320698, A322553.

Sequence in context: A273883 A132145 A272284 * A330232 A273882 A277112

Adjacent sequences:  A322551 A322552 A322553 * A322555 A322556 A322557

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 15 2018

STATUS

approved

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Last modified March 29 18:31 EDT 2020. Contains 333117 sequences. (Running on oeis4.)