A070172 Smallest k such that sigma(k) >= n.

1, 2, 2, 3, 4, 4, 4, 6, 6, 6, 6, 6, 8, 8, 8, 10, 10, 10, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 16, 16, 16, 18, 18, 18, 18, 18, 18, 18, 18, 20, 20, 20, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 36, 36

Offset: 1

Author

Benoit Cloitre, May 06 2002

Keywords

• easy
• nonn

Comments

Also smallest m to partition n into distinct divisors of m; highly abundant numbers are record values: a(i) < A002093(n) for 1<=i < A085443(n), A002093(n) = a(A085443(n)). - Reinhard Zumkeller, Jun 30 2003

1 followed by A002093(k) appearing A034885(k+1)-A034885(k) times, for k >= 2. - Amiram Eldar, Apr 04 2025

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Amiram Eldar, Plot of a(n)*log(log(n))/n for a sample of values of n <= 5.33*10^28.

Crossrefs

Cf. A002093, A085443.

Programs

Mathematica

nn=80;With[{s=Table[{n,DivisorSigma[1,n]},{n,nn}]},Transpose[ Flatten[ Table[ Select[s,#[[2]]>=i&,1],{i,nn}],1]][[1]]] (* Harvey P. Dale, Dec 28 2013 *)
seq[lim_] := Module[{han = Cases[Import["https://oeis.org/A002093/b002093.txt", "Table"], {, }][[;; , 2]], hmax, sigma, d}, hmax = han[[-1]]; If[lim > hmax, Print["Error: lim is too large"]; {}, han = Select[han, # <= lim &]; sigma = DivisorSigma[1, han]; d = Join[{1}, Differences[sigma]]; Flatten[Table[han[[i]], {i, 1, Length[han]}, {d[[i]]}]]]]; seq[100] (* Amiram Eldar, Apr 04 2025 *)

PARI

for(n=1,150,s=1; while(sigma(s)
				

Formula

It seems that lim_{n -> oo} a(n)*log(log(n))/n = C = 0.6...