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

1,3

Comments

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)

References

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.

Links

Table of n, a(n) for n=1..13.

Pentti Haukkanen, Some computational results concerning the divisor functions d(n) and sigma(n), The Mathematics Student, Vol. 62 (1993), pp. 166-168.

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.

Formula

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

Example

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

Mathematica

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 *)

Crossrefs

Cf. A000203, A002961, A300285.

Sequence in context: A189162 A079282 A117585 * A279978 A269601 A006684

Adjacent sequences: A317471 A317472 A317473 * A317475 A317476 A317477

Keyword

nonn,more

Author

Amiram Eldar, Jul 29 2018

Status

approved