A065560 -id:A065560 - cp's OEIS frontend

A081725 A065560(n) + n - prime(n).

1, 2, 4, 3, 5, 5, 7, 7, 6, 8, 15, 7, 10, 10, 9, 8, 11, 10, 11, 14, 13, 14, 13, 11, 12, 15, 16, 20, 21, 13, 14, 14, 17, 13, 16, 16, 16, 17, 17, 17, 20, 16, 20, 22, 26, 19, 13, 15, 19, 21, 21, 25, 21, 21, 21, 21, 25, 25, 27, 31, 27, 20, 22, 26, 28, 20, 21, 17, 21, 23, 24, 22, 22, 22

Offset: 1

Benoit Cloitre, Apr 06 2003

nonn

Conjecture : a(n)>0

A031435 Reversal point for powers of consecutive natural numbers.

Original entry on oeis.org

1, 2, 4, 7, 9, 12, 15, 18, 21, 25, 28, 32, 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

Offset: 1

Donald Mintz (djmintz(AT)home.com)

nonn

a(n+1) is the smallest k such that floor((1+1/n)^k) == 0 (mod n). A065560(n) is not a strictly increasing sequence, but this one is. - Benoit Cloitre, May 23 2002

			a(2) = 2: 3^2 > 2^3 but 3^1 < 2^2.
a(3) = 4: 4^4 > 3^5 but 4^3 < 3^4.
a(4) = 7: 5^7 > 4^8 but 5^6 < 4^7.
a(5) = 9: 6^9 > 5^10 but 6^8 < 5^9.
a(6) = 12: 7^12 > 6^13 but 7^11 < 6^12.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A065560.

nn = 60; Table[SelectFirst[Range[5 nn], Mod[Floor[(1 + 1/n)^#], n] == 0 &], {n, nn}] (* Michael De Vlieger, Mar 30 2016, Version 10 *)

for(n=1,100,print1(ceil((n+1/2)*log(n)),",")) \\ Valid for 1

a(n) = my(k=1); while((1+1/n)^k < n, k++); k; \\ Michel Marcus, Mar 30 2019

def a(n)
  (1..Float::INFINITY).find { |i| (i * Math.log(n, n + 1)).to_i < i - 1 } - 1
end # Peter Kagey, Mar 29 2016

If bases are N, N+1, the reversal point is floor( log(1+N)/log(1+1/N) ).
For n>1, ceiling((n+1/2)*log(n)) is an approximation to a(n) which is valid for all n <= 1000 except n=77 and n=214. - Benoit Cloitre, May 23 2002; corrected by Franklin T. Adams-Watters, Dec 16 2005
a(n) = floor(1/(1-log(n)/log(n+1))), empirical observation verified for n = 1 to 10000. - Fred Patrick Doty, Jul 13 2023

More terms from Benoit Cloitre, May 23 2002

A081726 Let f(x,y)=floor((1+1/y)^x); sequence gives least k such that f(k+2,n)/f(k,n)=(1+1/n)^2.

Original entry on oeis.org

1, 36, 20, 117, 37, 42, 70, 68, 57, 139, 186, 149, 119, 542, 389, 135, 328, 297, 327, 1055, 299, 273, 782, 328, 1117, 445, 1916, 1951, 493, 1148, 1238, 4156, 1616, 2328, 1003, 3106, 312, 536, 3320, 383, 1470, 528, 375, 3870, 555, 990, 1278, 1846, 851

Offset: 1

Benoit Cloitre, Apr 06 2003

nonn

Cf. A065560.

cp's OEIS Frontend

A081725 A065560(n) + n - prime(n).

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A031435 Reversal point for powers of consecutive natural numbers.

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A081726 Let f(x,y)=floor((1+1/y)^x); sequence gives least k such that f(k+2,n)/f(k,n)=(1+1/n)^2.

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