A181701 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.

12, 20, 56, 88, 104, 368, 464, 992, 1504, 1888, 1952, 16256, 24448, 28544, 30592, 32128, 98048, 122624, 128768, 130304, 507392, 521728, 522752, 2087936, 7337984, 8124416, 8353792, 8378368, 8382464, 25161728, 67100672, 125820928, 132112384, 133685248, 134193152

Offset

1,1

Comments

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.

Links

Donovan Johnson, Table of n, a(n) for n = 1..1000

Yanbin Li and Qunying Liao, A class of new near-perfect numbers, J. Korean Math. Soc. 52 (2015), No. 4, pp. 751-763.

Paul Pollack and Vladimir Shevelev, On perfect and near-perfect numbers, J. Number Theory 132 (2012), pp. 3037-3046. arXiv:1011.6160

X.-Z. Ren, Y.-G. Chen, On near-perfect numbers with two distinct prime factors, Bulletin of the Australian Mathematical Society, No 3 (2013) , available on CJO2013. doi:10.1017/S0004972713000178.

M. Tang, X. Z. Ren and M. Li, On near-perfect and deficient-perfect numbers, Colloq. Math. 133 (2013), 221-226.

Mathematica

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 *)

Prog

(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 */

Crossrefs

Cf. A181595, A181596, A000396, A181692.

Sequence in context: A259174 A109788 A341361 * A025104 A163323 A060159

Adjacent sequences: A181698 A181699 A181700 * A181702 A181703 A181704

Keyword

nonn

Author

Vladimir Shevelev, Nov 06 2010

Extensions

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

Status

approved