A085829 a(n) = least k such that the average number of divisors of {1..k} is >= n.

1, 4, 15, 42, 120, 336, 930, 2548, 6930, 18870, 51300, 139440, 379080, 1030484, 2801202, 7614530, 20698132, 56264040, 152941824, 415739030, 1130096128, 3071920000, 8350344420, 22698590508, 61701166395, 167721158286, 455913379324, 1239301050624, 3368769533514

Offset

1,2

Comments

Does a(n+1)/a(n) converge to e?

Since the total number of divisors of {1..k} (see A006218) is k * (log(k) + 2*gamma - 1) + O(sqrt(k)), the average number of divisors of {1..k} approaches (log(k) + 2*gamma - 1). Since log(a(n)) + 2*gamma - 1 approaches n, a(n+1)/a(n) approaches e. - Jon E. Schoenfield, Aug 13 2007

References

Julian Havil, "Gamma: Exploring Euler's Constant", Princeton University Press, Princeton and Oxford, pp. 112-113, 2003.

Links

Donovan Johnson, Table of n, a(n) for n = 1..40 (first 36 terms from Jon E. Schoenfield)

Example

a(20) = 415739030 because the average number of divisors of {1..415739030} is >= 20.

Mathematica

s = 0; k = 1; Do[ While[s = s + DivisorSigma[0, k]; s < k*n, k++ ]; Print[k]; k++, {n, 1, 20}]

Prog

(PARI) A085829(n) = {local(s, k); s=1; k=1; while(s<k*n, k++; s=s+numdiv(k)); k} \\ Michael B. Porter, Oct 23 2009

Crossrefs

Cf. A050226, A057494, A085567, A006218.

Sequence in context: A011844 A075468 A100503 * A085567 A213498 A294259

Adjacent sequences: A085826 A085827 A085828 * A085830 A085831 A085832

Keyword

nonn,nice

Author

Robert G. Wilson v, Jul 07 2003

Extensions

Edited by Don Reble, Nov 06 2005

More terms from Jon E. Schoenfield, Aug 13 2007

Status

approved