0,1

It's not obvious that a(n) exists for all n; I'd like to see a proof. - David Wasserman (wasserma(AT)spawar.navy.mil), 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 (noe(AT)sspectra.com), Jan 21 2006

According to Sierpinski, H. J. Kanold proved that there is a k such that sigma(x)=k has n or more solutions. Sierpinski states that Erdos 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 (noe(AT)sspectra.com), Oct 18 2006

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.

T. D. Noe, Table of n, a(n) for n=0..429

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

W. Sierpi\'{n}ski, Elementary Theory of Numbers, Warszawa 1964, page 166.

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

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}] (Noe)

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).

Adjacent sequences: A007365 A007366 A007367 this_sequence A007369 A007370 A007371

Sequence in context: A045873 A110060 A061081 this_sequence A054677 A012929 A013161

nonn

njas, Mira Bernstein, Robert G. Wilson v (rgwv(AT)rgwv.com)

More terms from David W. Wilson (davidwwilson(AT) comcast.net).