

A305829


Factor n into distinct FermiDirac primes (A050376), normalize by replacing every instance of the kth FermiDirac prime with k, then multiply everything together.


12



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
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OFFSET

1,3


COMMENTS

Let f(n) = A050376(n) be the nth FermiDirac 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 FermiDirac factorizations of m and n are disjoint and their union is the FermiDirac 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^(m1)]]]]];
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



