A181703 - OEIS

%I #28 Jun 14 2018 03:17:30

%S 20,104,464,1952,130304,522752,8382464,134193152,549754241024,

%T 8796086730752,140737463189504,144115187270549504,

%U 196159429230833773869868419445529014560349481041922097152,3450873173395281893717377931138512601610429881249330192849350210617344

%N Numbers of the form m = 2^(t-1)*(2^t-3), where 2^t-3 is prime.

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

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

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

%C 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 Eric Chen, <a href="/A181703/b181703.txt">Table of n, a(n) for n = 1..28</a>

%p 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 (PARI) for(k=1, 200, if(ispseudoprime(2^k-3), print1(2^(k-1)*(2^k-3), ", "))) \\ _Eric Chen_, Jun 13 2018

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

%K nonn

%O 1,1

%A _Vladimir Shevelev_, Nov 06 2010

%E Edited and extended by _D. S. McNeil_, Nov 18 2010

%E Definition simplified by _R. J. Mathar_, Nov 18 2010