A103292 -id:A103292 - cp's OEIS frontend

A103291 Numbers k such that sigma(2^k-1) >= 2*(2^k-1)-1, i.e., the number 2^k-1 is perfect, abundant, or least deficient.

1, 12, 24, 36, 40, 48, 60, 72, 80, 84, 90, 96, 108, 120, 132, 140, 144, 156, 160, 168, 180, 192, 200, 204, 210, 216, 220, 228, 240, 252, 264, 270, 276, 280, 288, 300, 312, 320, 324, 330, 336, 348, 360, 372, 384, 396, 400, 408, 420, 432, 440, 444, 450, 456, 468

Offset: 1

Max Alekseyev, Jan 28 2005

Is there an odd term besides 1? Numbers 2^a(i)-1 form set difference of sequences A103289 and A096399.

Odd terms > 1 exist, but there are none < 10^7. If k > 1 is an odd term, then 2^k-1 must have more than 900000 distinct prime factors and all of them must be members of A014663. - David Wasserman, Apr 15 2008

Amiram Eldar, Table of n, a(n) for n = 1..136

Cf. A000203, A075708, A103288, A103289, A103292, A023196.

for(i=1,1000,n=2^i-1;if(sigma(n)>=2*n-1,print1(i, ", ")));

Numbers k such that 2^k-1 is in A103288.

More terms from David Wasserman, Apr 15 2008

A359084 Numbers k such that A246601(k) > 2*k.

Original entry on oeis.org

4095, 8190, 16380, 32760, 65520, 131040, 262080, 524160, 1048320, 2096640, 4193280, 8386560, 16773120, 16777215, 33546240, 33550335, 33554430, 67092480, 67096575, 67100670, 67108860, 134184960, 134189055, 134193150, 134201340, 134217720, 268369920, 268374015

Offset: 1

Amiram Eldar, Dec 15 2022

An analog of abundant numbers k (A005101), in which the divisor sum is restricted to divisors d whose 1-bits in their binary expansions are common with those of k.
If k is a term then 2*k is also a term. Therefore all the terms can be generated from the primitive set of the odd terms (A359085).
The least term that is not divisible by 4095 is a(208) = 1099511627775 = 2^40 - 1.
Since A246601(2^k-1) = sigma(2^k-1), 2^k-1 is a term for all k in A103292, unless 2^k-1 is an odd perfect number (A000396).

Cf. A000203 (sigma), A000396, A103292, A246601.
Subsequence of A005101.
A359085 is a subsequence.

s[n_] := DivisorSum[n, # &, BitAnd[n, #] == # &]; Select[Range[10^6], s[#] > 2*# &]

is(n) = sumdiv(n, d, d * (bitor(n, d) == n)) > 2*n;

cp's OEIS Frontend

A103291 Numbers k such that sigma(2^k-1) >= 2*(2^k-1)-1, i.e., the number 2^k-1 is perfect, abundant, or least deficient.

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