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

1,3

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).

Links

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

Formula

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

Example

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

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}]

Prog

(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

Crossrefs

A left inverse of A050376.

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

Sequence in context: A240185 A292032 A097150 * A278108 A302786 A322026

Adjacent sequences: A319823 A319824 A319825 * A319827 A319828 A319829

Keyword

nonn

Author

Gus Wiseman, Sep 28 2018

Extensions

Secondary definition added by Antti Karttunen, Feb 18 2023

Status

approved