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A316266
FDH numbers of strict integer partitions with prime parts and prime length.
2
12, 21, 28, 33, 44, 57, 75, 76, 77, 84, 100, 123, 132, 133, 141, 164, 175, 183, 188, 209, 228, 231, 244, 249, 275, 287, 291, 300, 308, 329, 332, 363, 388, 399, 417, 427, 451, 453, 475, 484, 492, 507, 517, 525, 532, 556, 564, 581, 591, 604, 627, 671, 676, 679
OFFSET
1,1
COMMENTS
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 strict integer partitions with prime parts and prime length, preceded by their FDH numbers, begins:
12: (3,2)
21: (5,2)
28: (5,3)
33: (7,2)
44: (7,3)
57: (11,2)
75: (13,2)
76: (11,3)
77: (7,5)
84: (5,3,2)
MATHEMATICA
nn=1000;
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], And[PrimeQ[Length[FDfactor[#]]], And@@PrimeQ/@(FDfactor[#]/.FDrules)]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 28 2018
STATUS
approved