A065560 a(n) is the smallest integer k such that floor((1+1/n)^(k+1))/floor((1+1/n)^k) = 1+1/n.

2, 4, 7, 9, 12, 15, 18, 21, 25, 28, 40, 35, 39, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 83, 87, 91, 95, 100, 104, 109, 113, 118, 122, 127, 131, 136, 141, 145, 150, 155, 159, 164, 169, 174, 179, 183, 188, 193, 198, 203, 208, 213, 218, 223, 228, 233, 238, 243, 248, 253

Offset: 2

Author

Benoit Cloitre, Nov 29 2001

Keywords

• nonn

Comments

a(n) is growing roughly like prime(n). a(n) < a(n+1) except for n = 12. (Is this the only exception?)

a(n) < a(n+1) except for n = 12, 108, 266, ... - Boris Gourevitch (boris(AT)pi314.net), Dec 04 2001

Conjecture: a(n)+n > prime(n).

Examples

			a(5) = 9 because 9 is the first integer satisfying floor((6/5)^(9+1))/floor((6/5)^9) = 6/5.
		

Links

• Harry J. Smith, Table of n, a(n) for n=2..800

Crossrefs

Cf. A065554, A065564.

Programs

PARI

a(n) = { my(k=1, f=(n + 1)/n); while((floor(f^(k + 1))/floor(f^k)) != f, k++); k } \\ Harry J. Smith, Oct 22 2009

Formula

Asymptotic (conjectured) formula: a(n)=n*log(n)+o(log(n)).

Extensions

Terms a(53) - a(61) from Harry J. Smith, Oct 22 2009