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 A386570 #14 Aug 13 2025 10:22:46
%S A386570 1,15,118,1119,10585,102093,986262,9593611,93752493,918726697,
%T A386570 9024991249
%N A386570 The number of solutions x to d(x) = d(x+1) below 10^n, where d(x) is the number of divisors function (A000005).
%H A386570 Paul Erdős, <a href="https://doi.org/10.1017/S0305004100019277">On a problem of Chowla and some related problems</a>, Proc. Cambridge Philos. Soc., Vol. 32, No. 4 (1936), pp. 530-540; <a href="http://www.renyi.hu/~p_erdos/1936-03.pdf">alternate link</a>.
%H A386570 Paul Erdős, Carl Pomerance, and András Sárközy, <a href="http://www.renyi.hu/~p_erdos/1987-14.pdf">On locally repeated values of certain arithmetic functions, II</a>, Acta Math. Hungarica, Vol. 49 (1987), pp. 251-259; <a href="http://www.math.dartmouth.edu/~carlp/PDF/58.pdf">alternate link</a>.
%H A386570 D. R. Heath-Brown, <a href="https://doi.org/10.1112/S0025579300010743">The divisor function at consecutive integers</a>, Mathematika, Vol. 31 (1984), pp. 141-149.
%H A386570 Adolf Hildebrand, <a href="http://projecteuclid.org/euclid.pjm/1102690578">The divisor function at consecutive integers</a>, Pacific J. Math., Vol. 129, No. 2 (1987), pp. 307-319.
%H A386570 Christopher G. Pinner, <a href="https://doi.org/10.1093/qmath/48.4.499">Repeated values of the divisor function</a>, Quarterly Journal of Mathematics, Vol. 48, No. 192 (1997), pp. 499-502.
%F A386570 a(n) = A074802(10^n).
%e A386570 Below 10 there is one solution, x = 2, hence a(1) = 1.
%e A386570 Below 10^2 there are 15 solutions, x = 2, 14, 21, 26, 33, 34, 38, 44, 57, 75, 85, 86, 93, 94, 98, hence a(2) = 15.
%t A386570 With[{s = Array[DivisorSigma[0, #]&, 10^5]}, Array[Count[Range[10^# - 1], _?(s[[#]] == s[[# + 1]] &)] &, IntegerLength@ Length@ s - 1]] (* after _Michael De Vlieger_ at A300285 *)
%t A386570 cnt = 0; lst = {}; k = 1; n = 1; ds = 1; dt = 2; Do[ While[k < 10^n, cnt += Boole[ds == dt]; k++; ds = dt; dt = DivisorSigma[0, k+1]]; AppendTo[ lst, cnt], {n, 12}]; lst - (* _Robert G. Wilson v_, Jul 31 2025 *)
%Y A386570 Cf. A000005, A005237, A074802, A284783.
%Y A386570 Similar sequences: A300285, A317474.
%K A386570 nonn,more
%O A386570 1,2
%A A386570 _Amiram Eldar_, Jul 26 2025