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A319826
GCD of the strict integer partition with FDH number n; GCD of the indices (in A050376) of Fermi-Dirac prime factors of n.
7
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
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 28 2018
EXTENSIONS
Secondary definition added by Antti Karttunen, Feb 18 2023
STATUS
approved