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A317474
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The number of solutions to sigma(x) = sigma(x+1) below 10^n, where sigma(x) is the sum of divisors function (A000203).
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0
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0, 1, 3, 9, 24, 62, 113, 232, 533, 1097, 2295, 4804, 10135
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OFFSET
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1,3
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COMMENTS
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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)
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REFERENCES
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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.
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LINKS
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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.
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FORMULA
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Conjecture: Limit_{n->oo} a(n)/A300285(n) = 1.
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EXAMPLE
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Below 10^3 there are 3 solutions x = 14, 206, 957, hence a(3) = 3.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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