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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
0, 1, 3, 9, 24, 62, 113, 232, 533, 1097, 2295, 4804, 10135 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 29 12:24 EST 2021. Contains 349416 sequences. (Running on oeis4.)