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A319899 Numbers whose number of prime factors with multiplicity (A001222) is the number of distinct prime factors (A001221) in the product of the prime indices (A003963). 2
1, 3, 5, 7, 11, 15, 17, 19, 23, 26, 31, 33, 35, 39, 41, 51, 53, 55, 58, 59, 65, 67, 69, 74, 77, 83, 85, 86, 87, 91, 93, 94, 95, 97, 103, 109, 111, 119, 122, 123, 127, 129, 131, 142, 146, 155, 157, 158, 161, 165, 169, 177, 178, 179, 183, 185, 187, 191, 201, 202 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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. 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 square multiset multisystems, meaning the number of edges is equal to the number of distinct vertices.

LINKS

Table of n, a(n) for n=1..60.

EXAMPLE

The sequence of multiset multisystems whose MM-numbers belong to the sequence begins:

   1: {}

   3: {{1}}

   5: {{2}}

   7: {{1,1}}

  11: {{3}}

  15: {{1},{2}}

  17: {{4}}

  19: {{1,1,1}}

  23: {{2,2}}

  26: {{},{1,2}}

  31: {{5}}

  33: {{1},{3}}

  35: {{2},{1,1}}

  39: {{1},{1,2}}

  41: {{6}}

  51: {{1},{4}}

  53: {{1,1,1,1}}

  55: {{2},{3}}

  58: {{},{1,3}}

  59: {{7}}

  65: {{2},{1,2}}

  67: {{8}}

  69: {{1},{2,2}}

  74: {{},{1,1,2}}

MATHEMATICA

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

Select[Range[100], PrimeOmega[#]==PrimeNu[Times@@primeMS[#]]&]

CROSSREFS

Cf. A003963, A057151, A064573, A120732, A319616, A319877, A320325, A322527, A322530.

Sequence in context: A029626 A111622 A082418 * A085493 A139252 A076245

Adjacent sequences:  A319896 A319897 A319898 * A319900 A319901 A319902

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 17 2018

STATUS

approved

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Last modified November 11 22:31 EST 2019. Contains 329046 sequences. (Running on oeis4.)