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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: A111622 A356065 A082418 * A085493 A343621 A139252 Adjacent sequences: A319896 A319897 A319898 * A319900 A319901 A319902 KEYWORD nonn AUTHOR Gus Wiseman, Dec 17 2018 STATUS approved

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Last modified February 24 05:54 EST 2024. Contains 370294 sequences. (Running on oeis4.)