A305829 Factor n into distinct Fermi-Dirac primes (A050376), normalize by replacing every instance of the k-th Fermi-Dirac prime with k, then multiply everything together.

1, 1, 2, 3, 4, 2, 5, 3, 6, 4, 7, 6, 8, 5, 8, 9, 10, 6, 11, 12, 10, 7, 12, 6, 13, 8, 12, 15, 14, 8, 15, 9, 14, 10, 20, 18, 16, 11, 16, 12, 17, 10, 18, 21, 24, 12, 19, 18, 20, 13, 20, 24, 21, 12, 28, 15, 22, 14, 22, 24, 23, 15, 30, 27, 32, 14, 24, 30, 24, 20, 25

Offset

1,3

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. Then a(n) = s_1 * ... * s_k.

Multiplicative because for coprime m and n the Fermi-Dirac factorizations of m and n are disjoint and their union is the Fermi-Dirac factorization of m * n. - Andrew Howroyd, Aug 02 2018

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Gus Wiseman, Tree of x -> a(x) for n = 1...75

Mathematica

nn=100;
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];
Table[Times@@(FDfactor[n]/.FDrules), {n, nn}]

Prog

(PARI) \\ here isfd is membership test for A050376.
isfd(n)={my(e=isprimepower(n)); e && e == 1<<valuation(e, 2)}
seq(n)={my(v=select(isfd, [1..n])); vector(n, n, my(f=factor(n)); prod(i=1, #f~, my([p, e]=f[i, ]); prod(j=0, logint(e, 2), if(bittest(e, j), vecsearch(v, p^(1<<j)), 1))))} \\ Andrew Howroyd, Aug 02 2018

Crossrefs

Cf. A003963, A050376, A064547, A213925, A279065, A279614, A299755, A299756, A299757, A305830, A305831, A305832.

Sequence in context: A100798 A302785 A319825 * A121701 A161759 A260643

Adjacent sequences: A305826 A305827 A305828 * A305830 A305831 A305832

Keyword

nonn,mult

Author

Gus Wiseman, Jun 10 2018

Status

approved