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).
1, 15, 118, 1119, 10585, 102093, 986262, 9593611, 93752493, 918726697, 9024991249
Offset: 1
Examples
Below 10 there is one solution, x = 2, hence a(1) = 1. 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.
Links
- Paul Erdős, On a problem of Chowla and some related problems, Proc. Cambridge Philos. Soc., Vol. 32, No. 4 (1936), pp. 530-540; alternate link.
- Paul Erdős, Carl Pomerance, and András Sárközy, On locally repeated values of certain arithmetic functions, II, Acta Math. Hungarica, Vol. 49 (1987), pp. 251-259; alternate link.
- D. R. Heath-Brown, The divisor function at consecutive integers, Mathematika, Vol. 31 (1984), pp. 141-149.
- Adolf Hildebrand, The divisor function at consecutive integers, Pacific J. Math., Vol. 129, No. 2 (1987), pp. 307-319.
- Christopher G. Pinner, Repeated values of the divisor function, Quarterly Journal of Mathematics, Vol. 48, No. 192 (1997), pp. 499-502.
Programs
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Mathematica
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 *) 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 *)
Formula
a(n) = A074802(10^n).