A316266 FDH numbers of strict integer partitions with prime parts and prime length.

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

Author

Gus Wiseman, Jun 28 2018

Keywords

• nonn

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).

Examples

			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)
		

Crossrefs

Cf. A000586, A045450, A050376, A064547, A213925, A299755, A299757, A316185, A316265, A316267.

Programs

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)]&]