A300285 -id:A300285 - cp's OEIS frontend

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

Offset: 1

Amiram Eldar, Jul 29 2018

Data extracted from A002961.
The terms were calculated by:
a(1)-a(4) - Andrzej Makowski (1960);
a(5) - Mientka & Vogt (1970), Lal, Eldridge & Gillard (1972);
a(6)-a(7) - Hunsucker, Nebb & Stearns (1973);
a(8) - Pentti Haukkanen (1993);
a(9) - Jud McCranie (1997);
a(10) - T. D. Noe (2007);
a(11)-a(12) - T. D. Noe (2010);
a(13) - Giovanni Resta (2014).
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

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

Wacław Sierpiński, Elementary Theory of Numbers, Warszawa, 1988.

Pentti Haukkanen, Some computational results concerning the divisor functions d(n) and sigma(n), The Mathematics Student, Vol. 62 (1993), pp. 166-168.
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; entire volume.
M. Lal, C. Eldridge & P. Gillard, Solutions of sigma(n) = sigma(n+k), 1972, Review in Mathematics of Computation, Vol. 27, No. 123 (1973), p. 676.
Andrzej Makowski, On Some Equations Involving Functions phi(n) and sigma(n), The American Mathematical Monthly, Vol. 67, No. 7 (1960), pp. 668-670; Correction, ibid., Volume 68, No. 7 (1961), p. 650.
Walter E. Mientka and Richard L. Vogt, Computational results relating to problems concerning sigma(n), Matematicki Vesnik, Vol. 7, No. 51 (1970), pp. 35-36.

Cf. A000203, A002961, A300285.

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 *)

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

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

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula