cp's OEIS Frontend

A181703 Numbers of the form 2^(t-1)*(2^t-3), where 2^t-3 is prime.

%I A181703 #35 Aug 01 2025 01:59:31
%S A181703 20,104,464,1952,130304,522752,8382464,134193152,549754241024,
%T A181703 8796086730752,140737463189504,144115187270549504,
%U A181703 196159429230833773869868419445529014560349481041922097152,3450873173395281893717377931138512601610429881249330192849350210617344
%N A181703 Numbers of the form 2^(t-1)*(2^t-3), where 2^t-3 is prime.
%C A181703 This is a subsequence of A181595. [Proof: sigma(m) = (2^t-1)*(2^t-2) leads to an abundance of m which is 2.]
%C A181703 Numbers m such that the sum of the even divisors of m equals the square of the odd divisors of m.
%C A181703 Proof: let s0 the sum of the even divisors and s1 the sum of the odd divisors.
%C A181703 s1 = 2^t-2 because 2^t-3 is prime.
%C A181703 s0 = 2 + 4 + 8 + ... + 2^(t-1) + (2^t - 3)(2 + 4 + 8 + ... + 2^(t-1)) = (2^t - 2)^2 => s0 = s1^2. - _Michel Lagneau_, Apr 17 2013
%H A181703 Eric Chen, <a href="/A181703/b181703.txt">Table of n, a(n) for n = 1..28</a>
%H A181703 Christian Kassel and Christophe Reutenauer, <a href="https://arxiv.org/abs/2507.15780">Pairs of intertwined integer sequences</a>, arXiv:2507.15780 [math.NT], 2025. See p. 12.
%F A181703 a(n) = 2^(A050414(n)-1) * (2^A050414(n) - 3). - _Max Alekseyev_, Jul 31 2025
%p A181703 with(numtheory):for n from 1 to 600000 do:x:=divisors(n):n0:=nops(x):s0:=0:s1:=0:for k from 1 to n0 do:if irem(x[k],2)=0 then s0:=s0+ x[k]:else s1:=s1+ x[k]:fi:od:if s0=s1^2 then print(n):else fi:od: # _Michel Lagneau_, Apr 17 2013
%o A181703 (PARI) for(k=1, 200, if(ispseudoprime(2^k-3), print1(2^(k-1)*(2^k-3), ", "))) \\ _Eric Chen_, Jun 13 2018
%Y A181703 Cf. A181595, A181701, A000396, A050414, A050415.
%K A181703 nonn
%O A181703 1,1
%A A181703 _Vladimir Shevelev_, Nov 06 2010
%E A181703 Edited and extended by _D. S. McNeil_, Nov 18 2010
%E A181703 Definition simplified by _R. J. Mathar_, Nov 18 2010