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Heinz numbers of integer partitions whose only odd part is the smallest.
21

%I #17 Mar 07 2021 19:05:04

%S 2,5,6,11,14,17,18,23,26,31,35,38,41,42,47,54,58,59,65,67,73,74,78,83,

%T 86,95,97,98,103,106,109,114,122,126,127,137,142,143,145,149,157,158,

%U 162,167,174,178,179,182,185,191,197,202,209,211,214,215,222,226

%N Heinz numbers of integer partitions whose only odd part is the smallest.

%C The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers whose only odd prime index (counting multiplicity) is the smallest.

%F Also numbers n > 1 such that A055396(n) is odd and A032742(n) belongs to A066207.

%e The sequence of partitions together with their Heinz numbers begins:

%e 2: (1) 54: (2,2,2,1) 109: (29)

%e 5: (3) 58: (10,1) 114: (8,2,1)

%e 6: (2,1) 59: (17) 122: (18,1)

%e 11: (5) 65: (6,3) 126: (4,2,2,1)

%e 14: (4,1) 67: (19) 127: (31)

%e 17: (7) 73: (21) 137: (33)

%e 18: (2,2,1) 74: (12,1) 142: (20,1)

%e 23: (9) 78: (6,2,1) 143: (6,5)

%e 26: (6,1) 83: (23) 145: (10,3)

%e 31: (11) 86: (14,1) 149: (35)

%e 35: (4,3) 95: (8,3) 157: (37)

%e 38: (8,1) 97: (25) 158: (22,1)

%e 41: (13) 98: (4,4,1) 162: (2,2,2,2,1)

%e 42: (4,2,1) 103: (27) 167: (39)

%e 47: (15) 106: (16,1) 174: (10,2,1)

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

%t Select[Range[2,100],OddQ[First[primeMS[#]]]&&And@@EvenQ[Rest[primeMS[#]]]&]

%Y These partitions are counted by A035363 (shifted left once).

%Y Terms of A340932 can be factored into elements of this sequence.

%Y The even version is A341447.

%Y A001222 counts prime factors.

%Y A005408 lists odd numbers.

%Y A026804 counts partitions whose smallest part is odd.

%Y A027193 counts odd-length partitions, ranked by A026424.

%Y A031368 lists odd-indexed primes.

%Y A032742 selects largest proper divisor.

%Y A055396 selects smallest prime index.

%Y A056239 adds up prime indices.

%Y A058695 counts partitions of odd numbers, ranked by A300063.

%Y A061395 selects largest prime index.

%Y A066207 lists numbers with all even prime indices.

%Y A066208 lists numbers with all odd prime indices.

%Y A112798 lists the prime indices of each positive integer.

%Y A244991 lists numbers whose greatest prime index is odd.

%Y A340932 lists numbers whose smallest prime index is odd.

%Y Cf. A006141, A160786, A257991, A257992, A300272, A340604, A340607, A340854/A340855, A340933.

%K nonn

%O 1,1

%A _Gus Wiseman_, Feb 12 2021