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

1,1

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

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 *)

Prog

(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));

Crossrefs

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

Subsequence of A335431 provided there are no odd perfect numbers.

Sequence in context: A350277 A350278 A233757 * A139247 A124611 A281900

Adjacent sequences: A341619 A341620 A341621 * A341623 A341624 A341625

Keyword

nonn

Author

Antti Karttunen, Feb 19 2021

Status

approved