This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A317474 #30 Jul 14 2025 10:11:24
%S A317474 0,1,3,9,24,62,113,232,533,1097,2295,4804,10135
%N A317474 The number of solutions to sigma(x) = sigma(x+1) below 10^n, where sigma(x) is the sum of divisors function (A000203).
%C A317474 Data extracted from A002961.
%C A317474 The terms were calculated by:
%C A317474 a(1)-a(4) - Andrzej Makowski (1960);
%C A317474 a(5) - Mientka & Vogt (1970), Lal, Eldridge & Gillard (1972);
%C A317474 a(6)-a(7) - Hunsucker, Nebb & Stearns (1973);
%C A317474 a(8) - Pentti Haukkanen (1993);
%C A317474 a(9) - _Jud McCranie_ (1997);
%C A317474 a(10) - _T. D. Noe_ (2007);
%C A317474 a(11)-a(12) - _T. D. Noe_ (2010);
%C A317474 a(13) - _Giovanni Resta_ (2014).
%C A317474 It is not known whether there exist infinitely many natural numbers x for which sigma(x) = sigma(x+1) (see Sierpiński, p. 177). - _Stefano Spezia_, Jul 13 2025
%D A317474 Wacław Sierpiński, Elementary Theory of Numbers, Warszawa, 1988.
%H A317474 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 A317474 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; <a href="https://schoolbooksarchive.azimpremjiuniversity.edu.in/handle/20.500.12497/11788">entire volume</a>.
%H A317474 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 A317474 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 A317474 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 A317474 Conjecture: Limit_{n->oo} a(n)/A300285(n) = 1.
%e A317474 Below 10^3 there are 3 solutions x = 14, 206, 957, hence a(3) = 3.
%t A317474 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 A317474 Cf. A000203, A002961, A300285.
%K A317474 nonn,more
%O A317474 1,3
%A A317474 _Amiram Eldar_, Jul 29 2018