%I #20 Apr 04 2015 10:00:21

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

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

%C 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).

%C Comments from _Farideh Firoozbakht_, Jan 12 2014: (Start)

%C 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).

%C 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.).

%C Examples:

%C 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

%C 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

%C 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.).

%C 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.

%C 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.

%C Special cases:

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

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

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

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

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

%C and so on. (End)

%H Graeme L. Cohen, <a href="http://dx.doi.org/10.1017/S000497270001159X">Generalised quasiperfect numbers</a>, Ph.D. Dissertation, University of New South Wales, Sydney, 1982. Abstract in Bull. Australian Math. Soc., 27 (1983), 153-155.

%H Nichole Davis, Dominic Klyve and Nicole Kraght, <a href="http://dx.doi.org/10.2140/involve.2013.6.493">On the difference between an integer and the sum of its proper divisors</a>, Involve, Vol. 6 (2013), No. 4, 493-504; DOI: 10.2140/involve.2013.6.493

%H Farideh Firoozbakht and M. F. Hasler, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Hasler/hasler2.html">Variations on Euclid's formula for perfect numbers</a>, Journal of Integer Sequences 13 (2010), 18 pp. #10.3.1.

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

%Y For negative values of n see A234286.

%K nonn,more,hard,nice

%O 1,2

%A _N. J. A. Sloane_, Dec 28 2013