|
%I #24 Apr 26 2018 03:33:26
%S 12,20,56,88,104,368,464,992,1504,1888,1952,16256,24448,28544,30592,
%T 32128,98048,122624,128768,130304,507392,521728,522752,2087936,
%U 7337984,8124416,8353792,8378368,8382464,25161728,67100672,125820928,132112384,133685248,134193152
%N Near-perfect numbers (A181595) of the form 2^(t-1)*(2^t-2^k-1), where 2^t-2^k-1 is prime, k>=1, t>k.
%C There exist near-perfect numbers of the form 2^r*p, where p is prime, which are not in the sequence (e.g., 24,40,224). For given k, the smallest value of t gives sequence A181692.
%H Donovan Johnson, <a href="/A181701/b181701.txt">Table of n, a(n) for n = 1..1000</a>
%H Yanbin Li and Qunying Liao, <a href="http://www.mathnet.or.kr/mathnet/kms_content.php?no=413359">A class of new near-perfect numbers</a>, J. Korean Math. Soc. 52 (2015), No. 4, pp. 751-763.
%H Paul Pollack and Vladimir Shevelev, <a href="https://doi.org/10.1016/j.jnt.2012.06.008">On perfect and near-perfect numbers</a>, J. Number Theory 132 (2012), pp. 3037-3046. <a href="http://arxiv.org/abs/1011.6160">arXiv:1011.6160</a>
%H X.-Z. Ren, Y.-G. Chen, <a href="http://dx.doi.org/10.1017/S0004972713000178">On near-perfect numbers with two distinct prime factors</a>, Bulletin of the Australian Mathematical Society, No 3 (2013) , available on CJO2013. doi:10.1017/S0004972713000178.
%H M. Tang, X. Z. Ren and M. Li, <a href="http://dx.doi.org/10.4064/cm133-2-8">On near-perfect and deficient-perfect numbers</a>, Colloq. Math. 133 (2013), 221-226.
%t s = Sort@ Flatten@ Table[p = (2^t - 2^k - 1); If[PrimeQ@ p, 2^(t - 1) p, Nothing], {t, 2, 14}, {k, t - 1}]; Select[Select[s, DivisorSigma[1, #] > 2 # &], MemberQ[Divisors@ #, DivisorSigma[1, #] - 2 #] &] (* _Michael De Vlieger_, Sep 23 2015, after _Alonso del Arte_ at A181595 *)
%o (PARI) mx=2^269*(2^270-2^122-1); v=vector(1000); n=0; for(k=1, 269, for(t=k+1, 270, p=2^t-2^k-1; m=2^(t-1)*p; if(m>mx, next(2)); if(isprime(p), n++; v[n]=m))); v=vecsort(v); for(n=1, 1000, write("b181701.txt", n " " v[n])) /* _Donovan Johnson_, May 24 2013 */
%Y Cf. A181595, A181596, A000396, A181692.
%K nonn
%O 1,1
%A _Vladimir Shevelev_, Nov 06 2010
%E Edited, corrected, and extended by _D. S. McNeil_, Nov 18 2010
|
