A140863 Odd numbers k such that sigma(m) = 2m+k has a solution in m, where sigma is the sum-of-divisors function A000203.

3, 7, 17, 19, 31, 39, 41, 51, 59, 65, 71, 89, 115, 119, 127, 161, 185, 199, 215, 243, 251, 259, 265, 269, 299, 309, 353, 363, 399, 401, 455, 459, 467, 499, 519, 593, 635, 713, 737, 815, 831, 845, 899, 921, 923, 965, 967, 983, 1011, 1021, 1025, 1049, 1053, 1055

Offset: 1

Author

Lekraj Beedassy, Jul 20 2008

Keywords

• nonn

Comments

From M. F. Hasler and Farideh Firoozbakht, Nov 26 2009: (Start)

The sequence of Mersenne primes, A000668 is a subsequence of this sequence.

Because if k=2^p-1 is prime then n=2^(p-1)*(2^p-1)^2 is a solution of the equation sigma(x)=2x+k. The proof is easy. (End)

The definition is equivalent to asking for a number m with abundance A033880(m) = k. - M. F. Hasler, Mar 10 2025

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 196, p. 58, Ellipses, Paris 2008.

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..579
• Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.

Crossrefs

Cf. A000668. - M. F. Hasler and Farideh Firoozbakht, Nov 26 2009

Cf. A156903. - Robert G. Wilson v, Dec 09 2018

Cf. A000203 (sigma), A033880 (abundance: sigma(n)-2n).

Cf. A380866 (smallest solutions m to the given equation).

Formula

A033880(A156903), image of A156903 under A033880, or range of A033880 restricted to A156903, where A033880 is the abundance sigma(x)-2x, and A156903 are numbers with odd positive abundance. - M. F. Hasler, Mar 10 2025

Extensions

a(13)-a(54) from Donovan Johnson, Dec 09 2008