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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. 4
1, 5, 9, 11, 13, 15, 21, 23, 25, 27, 29, 33, 35, 37, 43, 45 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Cohen (1982) shows all odd squares are members. The remaining 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)

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: A080765 A226039 A257292 * A314584 A294277 A043721

Adjacent sequences:  A234282 A234283 A234284 * A234286 A234287 A234288

KEYWORD

nonn,more,hard,nice

AUTHOR

N. J. A. Sloane, Dec 28 2013

STATUS

approved

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Last modified January 29 16:58 EST 2020. Contains 331347 sequences. (Running on oeis4.)