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A317474 The number of solutions to sigma(x) = sigma(x+1) below 10^n, where sigma(x) is the sum of divisors function (A000203). 0

%I #14 Aug 19 2018 18:12:05

%S 0,1,3,9,24,62,113,232,533,1097,2295,4804,10135

%N The number of solutions to sigma(x) = sigma(x+1) below 10^n, where sigma(x) is the sum of divisors function (A000203).

%C Data extracted from A002961.

%C The terms were calculated by:

%C a(1)-a(4) - Andrzej Makowski (1960)

%C a(5) - Mientka & Vogt (1970), Lal, Eldridge & Gillard (1972)

%C a(6)-a(7) - Hunsucker, Nebb & Stearns (1973)

%C a(8) - Pentti Haukkanen (1993)

%C a(9) - _Jud McCranie_ (1997)

%C a(10) - _T. D. Noe_ (2007)

%C a(11)-a(12) - _T. D. Noe_ (2010)

%C a(13) - _Giovanni Resta_ (2014)

%D John L. Hunsucker, Jack Nebb, and Robert E. Stearns, Jr., Computational results concerning some equations involving sigma(n), The Mathematics Student, Vol. 41 (1973), pp. 285-289.

%H Pentti Haukkanen, <a href="https://www.researchgate.net/publication/282603270_SOME_COMPUTATIONAL_RESULTS_CONCERNING_THE_DIVISOR_FUNCTIONS_dn_AND_SIGMAn">Some computational results concerning the divisor functions d(n) and sigma(n)</a>, The Mathematics Student, Vol. 62 (1993), pp. 166-168.

%H M. Lal, C. Eldridge & P. Gillard, Solutions of sigma(n) = sigma(n+k), 1972, <a href="https://doi.org/10.1090/S0025-5718-73-99700-7">Review</a> in Mathematics of Computation, Vol. 27, No. 123 (1973), p. 676.

%H Andrzej Makowski, <a href="http://www.jstor.org/stable/2310107">On Some Equations Involving Functions phi(n) and sigma(n)</a>, The American Mathematical Monthly, Vol. 67, No. 7 (1960), pp. 668-670; <a href="http://www.jstor.org/stable/10.2307/2311516">Correction</a>, ibid., Volume 68, No. 7 (1961), p. 650.

%H Walter E. Mientka and Richard L. Vogt, <a href="https://eudml.org/doc/258944">Computational results relating to problems concerning sigma(n)</a>, Matematicki Vesnik, Vol. 7, No. 51 (1970), pp. 35-36.

%F Conjecture: Limit_{n->oo} a(n)/A300285(n) = 1.

%e Below 10^3 there are 3 solutions x = 14, 206, 957, hence a(3) = 3.

%t With[{s = Array[DivisorSigma[1,#]&, 10^5]}, Array[Count[Range[10^# - 1], _?(s[[#]] == s[[# + 1]] &)] &, IntegerLength@ Length@ s - 1]] (* after _Michael De Vlieger_ at A300285 *)

%Y Cf. A000203, A002961, A300285.

%K nonn,more

%O 1,3

%A _Amiram Eldar_, Jul 29 2018

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