A007368 Smallest k such that sigma(x) = k has exactly n solutions. (Formerly M4829)

2, 1, 12, 24, 96, 72, 168, 240, 336, 360, 504, 576, 1512, 1080, 1008, 720, 2304, 3600, 5376, 2520, 2160, 1440, 10416, 13392, 3360, 4032, 3024, 7056, 6720, 2880, 6480, 10800, 13104, 5040, 6048, 4320, 13440, 5760, 18720, 20736, 19152, 22680, 43680

Offset

0,1

Comments

It's not obvious that a(n) exists for all n; I'd like to see a proof. - David Wasserman, Jun 07 2002

Note that k-1 is frequently prime. See A115374 for the least prime. For each n, it appears that there are an infinite number of k such that sigma(x)=k has exactly n solutions. - T. D. Noe, Jan 21 2006

According to Sierpiński, H. J. Kanold proved that there is a k such that sigma(x)=k has n or more solutions. Sierpiński states that Erdős proved that if, for some k, sigma(x)=k has exactly n solutions, then there are an infinite number of such k. - T. D. Noe, Oct 18 2006

Index of the first occurrence of n in A054973. - Jaroslav Krizek, Apr 25 2009

References

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe and Donovan Johnson, Table of n, a(n) for n = 0..5000 (terms up to a(429) from T. D. Noe)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Kevin Ford, Sergei Konyagin, On two conjectures of Sierpiński concerning the arithmetic functions σ and ϕ, arXiv:1910.08452 [math.NT], 2019.

W. Sierpiński, Elementary Theory of Numbers, Warszawa 1964, page 166.

R. G. Wilson, V, Letter to N. J. A. Sloane, Jul. 1992

Example

a(10) = 504; {204, 220, 224, 246, 284, 286, 334, 415, 451, 503} is the set of x such that sigma(x) = 504.

Mathematica

Needs["Statistics`DataManipulation`"]; s=DivisorSigma[1, Range[10^5]]; f=Frequencies[s]; fs=Sort[f]; tfs=Transpose[fs][[1]]; utfs=Union[tfs]; firstMissing=First[Complement[Range[Last[utfs]], utfs]]; pos=1; Table[While[tfs[[pos]]<n, pos++ ]; fs[[pos, 2]], {n, firstMissing-1}] (* T. D. Noe *)
terms = 100; cnt = DivisorSigma[1, Range[terms^3]] // Tally // Sort; a[0] = 2; a[n_] := SelectFirst[cnt, #[[2]] == n&][[1]]; Table[a[n], {n, 0, terms - 1}] (* Jean-François Alcover, Jul 18 2017 *)

Crossrefs

Cf. A000203, A054973, A002191, A007609.

Cf. A115374 (least prime p such that sigma(x)=sigma(p) has exactly n solutions).

Cf. A007369, A007370, A007371, A007372 (n such that sigma(x)=k has 0, 1, 2 and 3 solutions).

Cf. A184393, A184394, A201915 (smallest solution, largest solution, triangle of solutions for sigma(x)=a(n)).

Sequence in context: A265022 A110060 A061081 * A054677 A290533 A012929

Adjacent sequences: A007365 A007366 A007367 * A007369 A007370 A007371

Keyword

nonn

Author

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

Extensions

More terms from David W. Wilson

Status

approved