A065564 -id:A065564 - cp's OEIS frontend

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

Benoit Cloitre, Nov 29 2001

nonn

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).

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

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

Cf. A065554, A065564.

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

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

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

A081723 Let b(n)=floor((3/2)^n), c(n)=floor((4/3)^n); sequence gives values of n such that b(n+1)/b(n)=3/2 and c(n+1)/c(n)=4/3.

Original entry on oeis.org

13, 24, 49, 67, 79, 88, 102, 126, 132, 162, 172, 194, 199, 204, 217, 234, 253, 255, 261, 271, 297, 320, 325, 328, 338, 351, 365, 377, 403, 411, 414, 462, 477, 482, 525, 533, 537, 541, 567, 569, 579, 601, 613, 638, 706, 740, 749, 761, 762, 773, 804, 809, 817

Offset: 1

Benoit Cloitre, Apr 06 2003

nonn

Cf. A065554, A065564.

It seems that a(n) is asymptotic to C*n where 12<=C<=13

A081724 Let b(n)=floor((3/2)^n), c(n)=floor((4/3)^n), d(n)=floor((5/4)^n); sequence gives values of n such that b(n+1)/b(n)=3/2, c(n+1)/c(n)=4/3 and d(n+1)/d(n)=5/4.

Original entry on oeis.org

162, 172, 204, 328, 403, 414, 809, 835, 840, 854, 1111, 1117, 1160, 1188, 1192, 1270, 1294, 1311, 1351, 1409, 1469, 1478, 1508, 1605, 1614, 1769, 1842, 1961, 2065, 2226, 2425, 2456, 2460, 2486, 2581, 2597, 2635, 2638, 2642, 2650, 2679, 2720, 2880, 2932

Offset: 1

Benoit Cloitre, Apr 06 2003

nonn

Cf. A065554, A065564.

bcdQ[n_]:=Module[{b=Floor[(3/2)^n],b1=Floor[(3/2)^(n+1)],c=Floor[ (4/3)^n], c1=Floor[(4/3)^(n+1)],d=Floor[(5/4)^n],d1=Floor[(5/4)^(n+1)]}, b1/b==3/2&&c1/c==4/3&&d1/d==5/4]; Select[Range[3000],bcdQ] (* Harvey P. Dale, Jun 08 2013 *)

It seems that a(n) is asymptotic to C*n where C is around 60.

cp's OEIS Frontend

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

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A081723 Let b(n)=floor((3/2)^n), c(n)=floor((4/3)^n); sequence gives values of n such that b(n+1)/b(n)=3/2 and c(n+1)/c(n)=4/3.

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A081724 Let b(n)=floor((3/2)^n), c(n)=floor((4/3)^n), d(n)=floor((5/4)^n); sequence gives values of n such that b(n+1)/b(n)=3/2, c(n+1)/c(n)=4/3 and d(n+1)/d(n)=5/4.

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