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A316094
FDH numbers of strict integer partitions with odd parts.
1
1, 2, 4, 7, 8, 11, 14, 16, 19, 22, 25, 28, 31, 32, 38, 41, 44, 47, 50, 53, 56, 61, 62, 64, 71, 76, 77, 79, 82, 83, 88, 94, 97, 100, 101, 103, 106, 107, 109, 112, 113, 121, 122, 124, 127, 128, 131, 133, 137, 139, 142, 149, 151, 152, 154, 157, 158, 163, 164, 166
OFFSET
1,2
COMMENTS
Also numbers n such that A305829(n) is odd.
Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH number of a strict integer partition (y_1,...,y_k) is f(y_1)*...*f(y_k).
EXAMPLE
Sequence of all integer partitions with distinct odd parts begins (), (1), (3), (5), (3,1), (7), (5,1), (9), (11), (7,1), (13), (5,3), (15), (9,1), (11,1), (17), (7,3), (19), (13,1), (21), (5,3,1), (23), (15,1), (9,3), (25), (11,3), (7,5), (27), (17,1), (29), (7,3,1), (19,1), (31).
MATHEMATICA
nn=100;
FDfactor[n_]:=If[n==1, {}, Sort[Join@@Cases[FactorInteger[n], {p_, k_}:>Power[p, Cases[Position[IntegerDigits[k, 2]//Reverse, 1], {m_}->2^(m-1)]]]]];
FDprimeList=Array[FDfactor, nn, 1, Union]; FDrules=MapIndexed[(#1->#2[[1]])&, FDprimeList];
Select[Range[nn], OddQ[Times@@(FDfactor[#]/.FDrules)]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 24 2018
STATUS
approved