A341622 Numbers that are either already perfect, or a perfect number is eventually reached if we start doubling them.

3, 6, 7, 14, 28, 31, 62, 124, 127, 248, 254, 496, 508, 1016, 2032, 4064, 8128, 8191, 16382, 32764, 65528, 131056, 131071, 262112, 262142, 524224, 524284, 524287, 1048448, 1048568, 1048574, 2096896, 2097136, 2097148, 4193792, 4194272, 4194296, 8387584, 8388544, 8388592, 16775168, 16777088, 16777184, 33550336, 33554176, 33554368

Offset: 1

Author

Antti Karttunen, Feb 19 2021

Keywords

• nonn

Comments

Numbers whose closure under map x -> 2x contains a perfect number (one of the terms of A000396).

Numbers k such that A341621(k) > A336915(k). No powers of 2 are included because they stay deficient forever.

Sequence is the union of odd perfect numbers (whose existence is contested, see e.g., A326051), and the numbers of the form (2^p - 1) * 2^e, where p is one of the primes in A000043, and e < p.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..5153
• Index entries for sequences where odd perfect numbers must occur, if they exist at all

Crossrefs

Cf. A000043, A000396, A005101, A326051, A336915, A336921.

Subsequence of A335431 provided there are no odd perfect numbers.

Programs

Mathematica

m = MersennePrimeExponent[Range[8]]; f[p_] := 2^Range[0, p - 1]*(2^p - 1); Select[Sort @ Flatten[f /@ m], # <= 2^m[[-1]] - 1 &] (* Amiram Eldar, Feb 20 2021, for calculating terms below 10^1500, the current lower bound for odd perfect numbers *)

PARI

isA341622(n) = if(!bitand(n,n-1), 0, for(i=0,oo,my(n2 = n+n); if(sigma(n) >= n2, return(sigma(n)==n2)); n = n2));