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.

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

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;
od;
od;
R;
end:
seq(a(n), n=0..100);

Crossrefs

Cf. A272895.

Sequence in context: A100995 A319273 A329615 * A268387 A136566 A330235

Adjacent sequences: A272891 A272892 A272893 * A272895 A272896 A272897

Keyword

nonn,easy

Author

Tom Mueller, May 09 2016

Status

approved