A120355 a(n) = min{k>0: the n-th convergent to e equals m/k! for some m}.

1, 1, 3, 4, 7, 8, 13, 71, 31, 67, 13, 89, 83, 18089, 5441, 17377, 36269, 26021, 4909, 10391023, 1097, 28879, 1846921, 519691, 1329313, 793279, 7553783, 3308341, 65676881, 662407, 677311, 2425388512913, 4403182913, 10832561, 781379079653017, 9120403, 2718061

Offset

0,3

Comments

Conjecture: for n>5, a(n) = A006530(A007677(n)). - Chai Wah Wu, Feb 25 2026

Links

Chai Wah Wu, Table of n, a(n) for n = 0..160

Jonathan Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.

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

Jonathan Sondow and Kyle Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.

Formula

a(n) = A002034(A007677(n)).

Example

The 6th convergent to e is 87/32 and 32 divides 8! but not 7!, so a(6) = 8.

Crossrefs

Cf. A002034, A007677, A006530.

Sequence in context: A060023 A345531 A357644 * A114210 A073271 A117471

Adjacent sequences: A120352 A120353 A120354 * A120356 A120357 A120358

Keyword

hard,nonn

Author

Jonathan Sondow, Aug 16 2006

Extensions

Extended by Max Alekseyev, Jul 28 2009

Missing a(7)=71 inserted by Georg Fischer, Oct 15 2024

a(34)-a(36) from Chai Wah Wu, Feb 25 2026

Status

approved