A299757 Weight of the strict integer partition with FDH number n.

0, 1, 2, 3, 4, 3, 5, 4, 6, 5, 7, 5, 8, 6, 6, 9, 10, 7, 11, 7, 7, 8, 12, 6, 13, 9, 8, 8, 14, 7, 15, 10, 9, 11, 9, 9, 16, 12, 10, 8, 17, 8, 18, 10, 10, 13, 19, 11, 20, 14, 12, 11, 21, 9, 11, 9, 13, 15, 22, 9, 23, 16, 11, 12, 12, 10, 24, 13, 14, 10, 25, 10, 26, 17

Offset: 1

Author

Gus Wiseman, Feb 18 2018

Keywords

• nonn

Comments

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. Every positive integer n has a unique factorization of the form n = f(s_1)*...*f(s_k) where the s_i are strictly increasing positive integers. This determines a unique strict integer partition (s_k...s_1) whose FDH number is then defined to be n.

In analogy with the Heinz number correspondence between integer partitions and positive integers (see A056239), FDH numbers give a correspondence between strict integer partitions and positive integers.

Examples

			Sequence of strict integer partitions begins: () (1) (2) (3) (4) (2,1) (5) (3,1) (6) (4,1) (7) (3,2) (8) (5,1) (4,2) (9).
		

Crossrefs

Cf. A004111, A050376, A056239, A061775, A064547, A106400, A213925, A215366, A246867, A279065, A279614, A299090, A299755, A299756, A299758, A299759.

Programs

Mathematica

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)]]]]];
nn=200;FDprimeList=Array[FDfactor,nn,1,Union];
FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Table[Total[FDfactor[n]/.FDrules],{n,nn}]