A181703 Numbers of the form m = 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

1,1

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

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

Prog

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

Crossrefs

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

Sequence in context: A045768 A088831 A063785 * A334419 A187756 A248087

Adjacent sequences: A181700 A181701 A181702 * A181704 A181705 A181706

Keyword

nonn

Author

Vladimir Shevelev, Nov 06 2010

Extensions

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

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

Status

approved