A272895 -id:A272895

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A272894

a(n) is the largest natural number k such that the composite number (2n+1) 2^k+1 has a nontrivial divisor of the form h2^s+1 (h odd) with s>k. If such a natural number does not exist, we set a(n)=0.

+10
1

0, 0, 0, 1, 0, 2, 1, 0, 1, 3, 2, 0, 1, 1, 2, 2, 0, 4, 3, 0, 2, 1, 1, 2, 1, 2, 3, 3, 1, 1, 2, 0, 2, 5, 4, 2, 3, 0, 1, 1, 2, 2, 1, 3, 2, 3, 1, 0, 1, 0, 4, 4, 0, 2, 3, 2, 0, 1, 1, 2, 3, 2, 0, 3, 1, 6, 5, 0, 4, 1, 2, 1, 3, 1, 1, 2, 3, 3, 2, 2, 2, 3, 0, 4, 3, 4, 2, 1, 1, 0, 3, 0, 2, 1, 3, 2, 1, 1, 5, 5, 2

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OFFSET

0,6

LINKS

Table of n, a(n) for n=0..100.

Tom Müller, On the Exponents of Non-Trivial Divisors of Odd Numbers and a Generalization of Proth's Primality Theorem, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.7.

EXAMPLE

We always have 2^k + 1 < h2^s + 1 if k < s. Thus a(1)=0.

MAPLE

a:= proc(n)

H:=2*n+1:

smax:=floor(evalf(log[2](H))):

R:=0:

for r from 1 to smax-1 do;

for s from r+1 to smax do;

kmax:=floor(evalf(H/2^s)):

for k from 1 to kmax by 2 do;

h:=(H-2^(s-r)*k)/(2^s*k+1):

if h<1 then break fi;

if type(h, integer) and R<r then R:=r fi;

od;

end:

seq(a(n), n=0..100);

CROSSREFS

Cf. A272895.

KEYWORD

nonn,easy

AUTHOR

Tom Mueller, May 09 2016

STATUS

approved

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