0,2

It appears that a(n) = A102469(n) (largest prime factor of the same numerator) except when n = 3. The smallest factorial divisible by the corresponding denominator is n!. Omitting the 0-th term in the sum, it appears that the Smarandache number and the largest prime factor, of the numerator of Sum_{k=1...n} 1/k! are both equal to A096058(n).

The Mathematica program given below was used to generate the sequence. If the numerator of Sum_{k=0...n}(1/k!) is squarefree, the program prints the value of the numerator's largest prime factor, which must equal a(n). Otherwise, the program prints the complete factorization of the numerator so a(n) can be determined by inspection. - Ryan Propper (rpropper(AT)stanford.edu), Jul 31 2005

A. J. Kempner, Miscellanea, Amer. Math. Monthly, 25 (1918), 201-210. See Section II, "Concerning the smallest integer m! divisible by a given integer n."

J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.

Eric Weisstein's World of Mathematics, SmarandacheFunction.

Index entries for sequences related to factorial numbers.

A002034(A061354(n)).

Sum_{k=0...3} 1/k! = 8/3 and 4! is the smallest factorial divisible by 8, so a(3) = 4.

Do[l = FactorInteger[Numerator[Sum[1/k!, {k, 0, n}]]]; If[Length[l] == Plus @@ Last /@ l, Print[Max[First /@ l]], Print[l]], {n, 1, 30}] (Propper)

Cf. A102469, A000522, A002034, A061354, A096058, A007917.

Sequence in context: A080067 A117824 A122212 this_sequence A079053 A002518 A093727

Adjacent sequences: A102465 A102466 A102467 this_sequence A102469 A102470 A102471

nonn

Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jan 09 2005

More terms from Ryan Propper (rpropper(AT)stanford.edu), Jul 31 2005