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).
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
REFERENCES
Wacław Sierpiński, Elementary Theory of Numbers, Warszawa, 1988.
LINKS
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.
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
KEYWORD
nonn,more
AUTHOR
Amiram Eldar, Jul 29 2018
STATUS
approved