A234285 Positive odd numbers n such that sigma(m) - 2m is never equal to n, where sigma(.) is the sum of divisors function A000203. Conjectural.

1, 5, 9, 11, 13, 15, 21, 23, 25, 27, 29, 33, 35, 37, 43, 45

Offset

1,2

Comments

The terms shown here are conjectural, based on a search up to 10^20 made by Davis et al. (2013).

Comments from Farideh Firoozbakht, Jan 12 2014: (Start)

1. Mersenne primes are not in this sequence. Because if M=2^p-1 is prime then M=sigma(m)-2m, where m=2^(p-1)*(2^p-1)^2=(1/2)*(M+1)*M^2 (please see Proposition 2.1 of Firoozbakht-Hasler, 2010).

2. If M = 2^p - 1 is a Mersenne prime then M^2 + 3M + 1 = 4^p + 2^p - 1 is not in the sequence. Because M^2 + 3M + 1 = sigma(m) - 2m where m = M^3 + M^2 = 2^p(2^p-1)^2 (please see Proposition 2.5, op. cit.).

Examples:

p = 2, M = 3, 4^p + 2^p - 1 = 19, m = M^3 + M^2 = 2^p(2^p-1)^2 = 36; sigma(m) - 2m = 19

p = 3, M = 7, 4^p + 2^p - 1 = 71, m = M^3 + M^2 = 2^p(2^p-1)^2 = 392; sigma(m) - 2m = 71

3. Note that if r is an even number and if for a number k p = 2^k - r - 1 is an odd prime then r = sigma(m) - 2m where m = 2^(k-1)*p. Namely r is not in the sequence (see Theorem 1.1, op. cit.).

It seems that for each even number r, there exists at least one odd prime of the form 2^k - r - 1. This means there is no even term in the sequence.

Moreover I conjecture that, for each even number r, there exist infinitely many primes p of the form 2^k - r - 1, or equivalently, I conjecture that: For each odd number s, there exists infinitely many primes p of the form 2^k - s.

Special cases:

(i): s = 1, there exist infinitely many Mersenne primes.

(ii): s = -1, there exist infinitely many Fermat primes.

(iii): s = 3, sequence A050414 is infinite.

(iv): s = -3, sequence A057732 is infinite.

(v): s = -5, sequence A059242 is infinite.

and so on. (End)

Cohen (1983) showed that 203^2 is not a term since sigma(m) - 2*m = 203^2 has a solution m = 742^2. - Max Alekseyev, Aug 29 2025

Links

Table of n, a(n) for n=1..16.

Graeme L. Cohen, Generalised quasiperfect numbers, Ph.D. Dissertation, University of New South Wales, Sydney, 1982. Abstract in Bull. Australian Math. Soc., 27 (1983), 153-155.

Nichole Davis, Dominic Klyve and Nicole Kraght, On the difference between an integer and the sum of its proper divisors, Involve, Vol. 6 (2013), No. 4, 493-504; DOI: 10.2140/involve.2013.6.493

Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for perfect numbers, Journal of Integer Sequences 13 (2010), 18 pp. #10.3.1.

Crossrefs

Cf. A000203, A033879 (2n - sigma(n)).

For negative values of n see A234286.

Sequence in context: A226039 A386982 A257292 * A314584 A335486 A347771

Adjacent sequences: A234282 A234283 A234284 * A234286 A234287 A234288

Keyword

nonn,more,hard,nice

Author

N. J. A. Sloane, Dec 28 2013

Extensions

Edited by Max Alekseyev, Aug 29 2025

Status

approved