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

%I A341622 #25 Feb 20 2021 10:37:09
%S A341622 3,6,7,14,28,31,62,124,127,248,254,496,508,1016,2032,4064,8128,8191,
%T A341622 16382,32764,65528,131056,131071,262112,262142,524224,524284,524287,
%U A341622 1048448,1048568,1048574,2096896,2097136,2097148,4193792,4194272,4194296,8387584,8388544,8388592,16775168,16777088,16777184,33550336,33554176,33554368
%N A341622 Numbers that are either already perfect, or a perfect number is eventually reached if we start doubling them.
%C A341622 Numbers whose closure under map x -> 2x contains a perfect number (one of the terms of A000396).
%C A341622 Numbers k such that A341621(k) > A336915(k). No powers of 2 are included because they stay deficient forever.
%C A341622 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.
%H A341622 Amiram Eldar, <a href="/A341622/b341622.txt">Table of n, a(n) for n = 1..5153</a>
%H A341622 <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>
%t A341622 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 *)
%o A341622 (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));
%Y A341622 Cf. A000043, A000396, A005101, A326051, A336915, A336921.
%Y A341622 Subsequence of A335431 provided there are no odd perfect numbers.
%K A341622 nonn
%O A341622 1,1
%A A341622 _Antti Karttunen_, Feb 19 2021