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

20, 104, 464, 1952, 130304, 522752, 8382464, 134193152, 549754241024, 8796086730752, 140737463189504, 144115187270549504, 196159429230833773869868419445529014560349481041922097152, 3450873173395281893717377931138512601610429881249330192849350210617344

Offset: 1

Author

Vladimir Shevelev, Nov 06 2010

Keywords

• nonn

Comments

This is a subsequence of A181595. [Proof: sigma(m) = (2^t-1)*(2^t-2) leads to an abundance of m which is 2.]

Numbers m such that the sum of the even divisors of m equals the square of the odd divisors of m.

Proof: let s0 the sum of the even divisors and s1 the sum of the odd divisors.

s1 = 2^t-2 because 2^t-3 is prime.

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

Links

• Eric Chen, Table of n, a(n) for n = 1..28
• Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 12.

Crossrefs

Cf. A181595, A181701, A000396, A050414, A050415.

Programs

Maple

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

PARI

for(k=1, 200, if(ispseudoprime(2^k-3), print1(2^(k-1)*(2^k-3), ", "))) \\ Eric Chen, Jun 13 2018

Formula

a(n) = 2^(A050414(n)-1) * (2^A050414(n) - 3). - Max Alekseyev, Jul 31 2025

Extensions

Edited and extended by D. S. McNeil, Nov 18 2010

Definition simplified by R. J. Mathar, Nov 18 2010