cp's OEIS Frontend

A135546 Let p be the n-th prime and let g be the order of 2 mod p (see A014664). Then if g is even, a(n) = p*(2^(g/2) - 1), otherwise a(n) = 2^g - 1.

%I A135546 #9 Apr 16 2025 10:57:50
%S A135546 3,15,7,341,819,255,9709,2047,475107,31,9699291,41943,5461,8388607,
%T A135546 3556769739,31675383749,65498251203,575525617597,34359738367,511,
%U A135546 549755813887,182518930210733,2047,1627389855,113715890591104923,2251799813685247,963770320257286037
%N A135546 Let p be the n-th prime and let g be the order of 2 mod p (see A014664). Then if g is even, a(n) = p*(2^(g/2) - 1), otherwise a(n) = 2^g - 1.
%C A135546 Karpenkov asks how often is it the case that if p is the n-th prime (n >= 2) then A038553(p) = a(n)? The first failure is at p = 37. Is it true that a(n) is always divisible by A038553(p)?
%H A135546 N. J. A. Sloane, <a href="/A135546/b135546.txt">Table of n, a(n) for n = 2..1000</a>
%H A135546 O. N. Karpenkov, <a href="http://dx.doi.org/10.1007/s11853-007-0010-z">On examples of difference operators for {0,1}-valued functions over finite sets</a>, Funct. Anal. Other Math. 1 (2006), 175-180. [The function q(n)]
%H A135546 O. N. Karpenkov, <a href="http://arxiv.org/abs/math/0611940">On examples of difference operators for {0,1}-valued functions over finite sets</a>, arXiv:math/0611940 [math.CO], 2006.
%p A135546 (First load the b-file for A014664 as the array b1.)
%p A135546 a := proc(i) local p,g; p:=ithprime(i); g:=b1[i-1]; if g mod 2 = 0 then p*(2^(g/2)-1) else 2^g-1; fi; end;
%t A135546 g[n_]:=MultiplicativeOrder[2, Prime[n]];a[n_]:=If[EvenQ[g[n]],Prime[n]*(2^(g[n]/2)-1),2^g[n]-1];Table[a[n],{n,2,28}] (* _James C. McMahon_, Apr 16 2025 *)
%Y A135546 Cf. A038553, A014664.
%K A135546 nonn
%O A135546 2,1
%A A135546 _N. J. A. Sloane_, Feb 24 2008