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A272895 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 2s>k. If such a natural number does not exist, we set a(n)=0.

%I A272895 #23 Jul 25 2017 06:29:55
%S A272895 0,3,4,5,5,2,6,7,6,4,3,4,7,1,8,9,8,6,3,4,6,5,1,6,8,2,6,3,9,4,10,11,10,
%T A272895 8,4,6,5,5,5,9,6,7,2,3,8,7,3,8,9,4,8,4,4,6,4,9,10,6,7,4,11,3,12,13,12,
%U A272895 10,5,8,8,6,6,6,5,6,6,7,3,5,10,4,8,9,0,4,8,4,5,11,8,8,3,6,10,9,4,10
%N A272895 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 2s>k. If such a natural number does not exist, we set a(n)=0.
%H A272895 Tom Müller, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Mueller/mueller10.html">On the Exponents of Non-Trivial Divisors of Odd Numbers and a Generalization of Proth's Primality Theorem</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.7.
%p A272895 H:=2n+1:
%p A272895 smax:=floor(evalf(log[2](H))):
%p A272895 R:=A272894(n)
%p A272895 for m from 0 to smax do;
%p A272895 for s from m+1 to smax+1 do;
%p A272895 hmax:=floor(evalf(H/2^m)):
%p A272895 for h from 1 to hmax by 2 do;
%p A272895 k:=(2^(s-m)*H-h)/(2^s*h+1);
%p A272895 if k<h then break fi;
%p A272895 if type(k,integer)=true and R<2*s-m then R:=2*s-m fi;
%p A272895 od;
%p A272895 od;
%p A272895 od;
%p A272895 a(n):=R;
%Y A272895 Cf. A272894.
%K A272895 nonn,easy
%O A272895 0,2
%A A272895 _Tom Mueller_, May 09 2016