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A319878
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Odd numbers whose product of prime indices (A003963) is a square of a squarefree number (A062503).
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1
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1, 7, 9, 23, 25, 97, 121, 151, 161, 169, 175, 183, 185, 195, 207, 225, 227, 289, 541, 661, 679, 687, 781, 841, 847, 873, 957, 961, 1009, 1089, 1193, 1427, 1563, 1589, 1681, 1819, 1849, 1879, 1895, 2023, 2043, 2167, 2193, 2209, 2231, 2425, 2437, 2585, 2601
(list;
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refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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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 2-regular (all vertex-degrees are 2) multiset partitions (no empty parts).
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LINKS
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EXAMPLE
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The sequence of multiset partitions whose MM-numbers belong to the sequence begins:
1: {}
7: {{1,1}}
9: {{1},{1}}
23: {{2,2}}
25: {{2},{2}}
97: {{3,3}}
121: {{3},{3}}
151: {{1,1,2,2}}
161: {{1,1},{2,2}}
169: {{1,2},{1,2}}
175: {{2},{2},{1,1}}
183: {{1},{1,2,2}}
185: {{2},{1,1,2}}
195: {{1},{2},{1,2}}
207: {{1},{1},{2,2}}
225: {{1},{1},{2},{2}}
227: {{4,4}}
289: {{4},{4}}
541: {{1,1,3,3}}
661: {{5,5}}
679: {{1,1},{3,3}}
687: {{1},{1,3,3}}
781: {{3},{1,1,3}}
841: {{1,3},{1,3}}
847: {{1,1},{3},{3}}
873: {{1},{1},{3,3}}
957: {{1},{3},{1,3}}
961: {{5},{5}}
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[1, 100, 2], Or[#==1, SameQ[##, 2]&@@Last/@FactorInteger[Times@@primeMS[#]]]&]
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CROSSREFS
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Cf. A003963, A005117, A005176, A062503, A064573, A072774, A295193, A302505, A319877, A319899, A320325, A322526, A322527, A322530.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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