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A316265
FDH numbers of strict integer partitions with prime parts.
2
1, 3, 4, 7, 11, 12, 19, 21, 25, 28, 33, 41, 44, 47, 57, 61, 75, 76, 77, 83, 84, 97, 100, 121, 123, 132, 133, 139, 141, 151, 164, 169, 175, 183, 188, 197, 209, 228, 231, 233, 241, 244, 249, 271, 275, 287, 289, 291, 300, 307, 308, 329, 332, 347, 361, 363, 388
OFFSET
1,2
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, preceded by their FDH numbers, begins:
1: ()
3: (2)
4: (3)
7: (5)
11: (7)
12: (3,2)
19: (11)
21: (5,2)
25: (13)
28: (5,3)
33: (7,2)
41: (17)
44: (7,3)
47: (19)
57: (11,2)
61: (23)
75: (13,2)
76: (11,3)
77: (7,5)
83: (29)
84: (5,3,2)
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], And@@PrimeQ/@(FDfactor[#]/.FDrules)&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 28 2018
STATUS
approved