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 A283156 Number of preimages of even integers under the sum-of-proper-divisors function. 8
 0, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 2, 1, 2, 0, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 0, 2, 2, 1, 0, 1, 2, 1, 2, 4, 2, 2, 1, 2, 1, 1, 0, 1, 0, 1, 1, 2, 1, 3, 2, 1, 3, 1, 1, 0, 2, 2, 2, 3, 2, 1, 1, 0, 1, 2, 1, 2, 1, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Let sigma(n) denote the sum of divisors function, and s(n):=sigma(n)-n. The k-th element a(k) corresponds to the number of solutions to 2k=s(m) in positive integers, where m is a variable. In 2016, C. Pomerance proved that, for every e > 0, the number of solutions is O_e((2k)^{2/3+e}). Note that for odd numbers n the problem of solving n=s(m) is quite different from the case when n is even. According to a slightly stronger version of Goldbach's conjecture, for every odd number n there exist primes p and q such that n = s(pq) = p + q + 1. This conjecture was verified computationally by Oliveira e Silva to 10^18. Thus the problem is (almost) equivalent to counting the solutions to n=p+q+1 in primes. LINKS Anton Mosunov, Table of n, a(n) for n = 1..10000 R. K. Guy, J. L. Selfridge, What drives an aliquot sequence?, Math. Comp. 29 (129), 1975, 101-107. P. Pollack, C. Pomerance, Some problems of Erdos on the sum-of-divisors function, Trans. Amer. Math. Soc., Ser. B, 3 (2016), 1-26. C. Pomerance, The first function and its iterates, A Celebration of the Work of R. L. Graham, S. Butler, J. Cooper, and G. Hurlbert, eds., Cambridge U. Press, to appear. C. Pomerance, H.-S. Yang, Variant of a theorem of Erdos on the sum-of-proper-divisors function, Math. Comp., 83 (2014), 1903-1913.T. Oliveira e Silva, Goldbach conjecture verification, 2015. FORMULA a(n) = A048138(2*n). - Michel Marcus, Mar 04 2017 EXAMPLE a(1)=0, because 2*1=s(m) has no solutions; a(2)=1, because 2*2=s(9); a(3)=2, because 2*3=s(6)=s(25); a(4)=2, because 2*4=s(10)=s(49); a(5)=1, because 2*5=s(14). PROG (PARI) a(n) = sum(k=1, (2*n-1)^2, (sigma(k) - k) == 2*n); \\ Michel Marcus, Mar 04 2017 CROSSREFS Cf. A048138, A283152. Sequence in context: A055230 A290262 A112050 * A298231 A320473 A194884 Adjacent sequences: A283153 A283154 A283155 * A283157 A283158 A283159 KEYWORD nonn AUTHOR Anton Mosunov, Mar 01 2017 STATUS approved

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