A319826 GCD of the strict integer partition with FDH number n; GCD of the indices (in A050376) of Fermi-Dirac prime factors of n.

0, 1, 2, 3, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 2, 9, 10, 1, 11, 1, 1, 1, 12, 1, 13, 1, 2, 1, 14, 1, 15, 1, 1, 1, 1, 3, 16, 1, 2, 1, 17, 1, 18, 1, 2, 1, 19, 1, 20, 1, 2, 1, 21, 1, 1, 1, 1, 1, 22, 1, 23, 1, 1, 3, 4, 1, 24, 1, 2, 1, 25, 1, 26, 1, 1, 1, 1, 1, 27, 1, 28

Offset: 1

Author

Gus Wiseman, Sep 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

			45 is the FDH number of (6,4), which has GCD 2, so a(45) = 2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

A left inverse of A050376.

Cf. A052331, A056239, A064547, A213925, A289508, A289509, A290103, A299755, A299757, A319825.

Programs

Mathematica

nn=200;
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];
GCD@@@Table[Reverse[FDfactor[n]/.FDrules],{n,nn}]

PARI

A319826(n) = { my(i=1,g=0,x=A052331(n)); while(x,if(x%2,g = gcd(g,i)); x>>=1; i++); (g); }; \\ (Uses the program given in A052331) - Antti Karttunen, Feb 18 2023

Formula

For all n >= 1, a(A050376(n)) = n. - Antti Karttunen, Feb 18 2023

Extensions

Secondary definition added by Antti Karttunen, Feb 18 2023