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A085829
a(n) = least k such that the average number of divisors of {1..k} is >= n.
12
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
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