A007663 -id:A007663 - cp's OEIS frontend

A239640 a(n) is the smallest number such that for n-bonacci constant c_n satisfies round(c_n^prime(m)) == 1 (mod 2*p_m) for every m>=a(n).

3, 3, 4, 5, 7, 7, 10, 13, 14, 14, 19, 23, 23, 31, 34, 34, 46, 50, 60, 65, 73, 79, 88, 92, 107, 113, 126, 139, 149, 168, 182, 198, 210, 227, 244, 265, 276, 292, 317, 340, 369, 384, 408, 436, 444, 480, 516, 540, 565, 606, 628, 669, 704, 735, 759, 810, 829, 895, 925

Offset: 2

Vladimir Shevelev and Peter J. C. Moses, Mar 23 2014

nonn

The n-bonacci constant is a unique root x_1>1 of the equation x^n-x^(n-1)-...-x-1=0. So, for n=2 we have Fibonacci constant phi or golden ratio (A001622); for n=3 we have tribonacci constant (A058265); for n=4 we have tetranacci constant (A086088), for n=5 (A103814), for n=6 (A118427), etc.

			Let n=2, then c_2 = phi (Fibonacci constant). We have round(c_2^2)=3 is not == 1 (mod 4), round(c_2^3)=4 is not == 1 (mod 6), while round(c_2^5)=11 == 1 (mod 10) and one can prove that for p>=5, we have round(c_2^p) == 1 (mod 2*p). Since 5=prime(3), then a(2)=3.

S. Litsyn and V. Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, vol.1, no.4 (2005), 499-512.
V. Shevelev, A property of n-bonacci constant, Seqfan (Mar 23 2014)

Cf. A001622, A007619, A007663, A058265, A086088, A103814, A118427, A118428, A238693, A238697, A238698, A238700, A239502, A239544, A239564, A239565, A239566.

A277167 Prime numbers p such that (-1)^h + (h!)^2 == 0 (mod p^2) where h = (p-1)/2.

Original entry on oeis.org

3, 11, 31, 47, 53

Offset: 1

René Gy, Oct 01 2016

The above congruence is true modulo p for all odd primes. See A089043. But like for Wilson congruence, it is true modulo p^2, for a restricted number of primes. After 53, the next one (if any) seems very far away (>500000).

The fact that the congruence is true modulo p for all odd primes was proved by Lagrange in 1771. Using a theorem of Mathews (1892) and Eisenstein's logarithmetic rule for the Fermat quotient, the condition stated in the definition can be restated as W_p == -2q_p(2) (mod p), where W_p is the Wilson quotient of p (A007619) and q_p(2) is the Fermat quotient of p, base 2 (A007663). - John Blythe Dobson, Jul 31 2017

			(-1)^((11-1)/2)+(((11-1)/2)!)^2 = 14399 = 7*11^2*17.

Lagrange, "Démonstration d’un théoreme nouveau concernant les nombres premiers," Nouveaux Mémoires de l’Académie Royale des Sciences et Belles-Lettres [de Berlin], année 1771 (published 1783), 125-137.
G. B. Mathews, Theory of Numbers, part 1 [all published] (Cambridge, 1892), 318.

René Gy, Find the next "Wilson-like" prime.

Cf. A007540, A089043.

lista(nn) = forprime(p=3, nn, if ((((-1)^((p-1)/2)+(((p-1)/2)!)^2) % p^2) == 0, print1(p, ", "))); \\ Michel Marcus, Oct 02 2016

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]

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A239640 a(n) is the smallest number such that for n-bonacci constant c_n satisfies round(c_n^prime(m)) == 1 (mod 2*p_m) for every m>=a(n).

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A277167 Prime numbers p such that (-1)^h + (h!)^2 == 0 (mod p^2) where h = (p-1)/2.

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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.

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