A180511 -id:A180511 - cp's OEIS frontend

A213327 Analog of Fermat quotients: a(n) = (round((phi_2)^p)-2)/p, where phi_2 is silver ratio 1+sqrt(2) and p = prime(n).

2, 4, 16, 68, 1476, 7280, 189120, 986244, 27676612, 4346071600, 23696518916, 3930960079760, 120508933265760, 669708812842692, 20814182249890948, 3654563002853231440, 650000099752136709444, 3664265812073801505200, 660535426260570501876228

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Mar 03 2013

nonn

For similar sequence for base 2, see A007663, for a similar sequence for golden ratio, see A064723.

Cf. A007663, A096060, A146211, A180511, A164740, A222207, A064723, A213328.

A213328 Analog of Fermat quotients: a(n) = (round((phi_3)^p)-3)/p, where phi_3 = (3+sqrt(13))/2 and p = prime(n).

Original entry on oeis.org

4, 11, 78, 612, 46374, 428040, 38948910, 380144556, 37367223558, 38467601033550, 392545092308724, 426897839167539480, 45841425452161683630, 476794964068892779068, 51906117696097060014342, 59746844088106673671809870, 69664778857791165966384195366

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Mar 03 2013

nonn

For similar sequence for base 2, see A007663. For similar sequences for golden ratio and silver ratio, see A064723 and A213327. Note that phi_3 is called "bronze ratio".

Cf. A007663, A096060, A146211, A180511, A164740, A222207, A064723, A213327.

A364883 Consider the Fermat quotient for base n: Fq(n,k) = (n^(p - 1) - 1)/p, where p = prime(k), for k >= 1. a(n) is the least k >= 1 such that Fq(n,j) is divisible by n^2 - 1 for all j >= k.

Original entry on oeis.org

3, 3, 4, 4, 5, 5, 5, 4, 6, 6, 7, 7, 7, 5, 8, 8, 9, 9, 9, 6, 10, 10, 10, 7, 7, 7, 11, 11, 12, 12, 12, 8, 8, 8, 13, 13, 13, 9, 14, 14, 15, 15, 15, 10, 16, 16, 16, 5, 8, 8, 17, 17, 17, 6, 9, 11, 18, 18, 19, 19, 19, 12, 7, 7, 20, 20, 20, 10, 21, 21, 22, 22, 22, 13, 9, 9, 23, 23, 23

Offset: 2

Robert G. Wilson v, Aug 17 2023

nonn

Conjecture: numbers appear in the sequence only a finite number of times. Terms appear in runs of length 1, 2, or 3, never more. The first time a term k appears is when the index is even. The terms appear for the first time in their natural order.

			For a(2), examine A007663 and notice that beginning with the second term, offset is 2, all terms are divisible by 3;
For a(3), examine A146211 and notice that beginning with the first term, offset is 3, all terms are divisible by 8;
For a(4), examine A180511 and notice that beginning with the third term, offset is 2, all terms are divisible by 15; etc.

Jean Bourgain, Kevin Ford, Sergei V. Konyagin, and Igor E. Shparlinski, On the Divisibility of Fermat Quotients, Michigan Mathematical Journal, Vol. 59 (Aug 2010), pp. 313-328.
Chris Caldwell, PrimePages, Fermat quotient.
nLab, Fermat quotient.
H. S. Vandiver, Fermat's Quotient and related arithmetic functions, Proceedings of the National Academy of Sciences of the United States of America, Vol. 31, 1945.
Eric Weisstein's World of Mathematics, Fermat Quotient.
Wikipedia, Fermat quotient.

Cf. A007663, A096060, A146211, A180511.

a[n_] := Block[{k = Floor[(1/2.3) n^(87/100) + 100]}, While[p = Prime@ k; PowerMod[n, p - 1, (n^2 - 1)*p] == 1, k--]; ++k]; Array[a, 79, 2]

cp's OEIS Frontend

A213327 Analog of Fermat quotients: a(n) = (round((phi_2)^p)-2)/p, where phi_2 is silver ratio 1+sqrt(2) and p = prime(n).

Views

Author

Keywords

Comments

Crossrefs

A213328 Analog of Fermat quotients: a(n) = (round((phi_3)^p)-3)/p, where phi_3 = (3+sqrt(13))/2 and p = prime(n).

Views

Author

Keywords

Comments

Crossrefs

A364883 Consider the Fermat quotient for base n: Fq(n,k) = (n^(p - 1) - 1)/p, where p = prime(k), for k >= 1. a(n) is the least k >= 1 such that Fq(n,j) is divisible by n^2 - 1 for all j >= k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica