A284783 -id:A284783 - cp's OEIS frontend

A385430 Least number k such that k and k + n! have the same number of divisors.

2, 3, 5, 5, 7, 7, 11, 23, 17, 11, 17, 29, 46, 19, 43, 23, 31, 37, 89, 29, 31, 31, 97, 62, 41, 59, 47, 67, 159, 107, 127, 79, 37, 97, 61, 131, 86, 43, 97, 53, 61, 97, 71, 47, 94, 101, 233, 53, 83, 61, 249, 53, 71, 158, 71, 149, 107, 134, 254, 206, 166, 131, 271

Offset: 1

Robert G. Wilson v, Jul 31 2025

Inspired by A284783.

First differs from A037153 at n=13 (and when they differ a(n) is a composite < A037153(n)).

			a(1) = 2 since d(2) = d(3) = 2;
a(5) = 7 since d(7) = d(7+5!) = 2;
a(13) = 46 since d(46) = d(46+13!) = 4; etc.

Sean A. Irvine, Table of n, a(n) for n = 1..82

Cf. A000005, A000142, A006881, A037153, A284783.

a[n_] := Block[{k = 2}, While[ DivisorSigma[0, k] != DivisorSigma[0, k + n!], k++]; k]; Array[ a, 51]

a(n) = my(k=1); while (numdiv(k) != numdiv(k+n!), k++); k; \\ Michel Marcus, Aug 02 2025

More terms from Sean A. Irvine, Aug 08 2025

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).

Original entry on oeis.org

1, 15, 118, 1119, 10585, 102093, 986262, 9593611, 93752493, 918726697, 9024991249

Offset: 1

Amiram Eldar, Jul 26 2025

			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.

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.

Cf. A000005, A005237, A074802, A284783.

Similar sequences: A300285, A317474.

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 *)

a(n) = A074802(10^n).

cp's OEIS Frontend

A385430 Least number k such that k and k + n! have the same number of divisors.

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