A347565 -id:A347565 - cp's OEIS frontend

A241014 Let p be the n-th prime, then a(n) = A/p where A is the smallest number (in absolute value) such that F_{p-(p/5)} == A (mod p^2) with F_n = A000045(n) and (p/5) the Legendre symbol.

1, 1, 1, 3, 5, 3, -1, 3, -8, -3, -6, 13, -2, -4, 16, -25, 10, -13, 7, -16, -15, -30, 21, 5, 37, -4, 22, 24, 26, -53, 13, 64, 58, -22, -29, 60, 44, -3, 44, -43, -5, -50, 94, 31, -56, 5, -99, 3, -73, 18, 29, 5, -59, -1, 2

Offset: 1

Felix Fröhlich, Aug 13 2014

sign

a(n) is the smallest A such that p is a near-Wall-Sun-Sun prime. A gives the value of F_p-(p/5) modulo p^2 and a value of 0 would indicate a Wall-Sun-Sun prime. A244801 is similar but always gives the positive A, while this sequence gives A with the smallest absolute value.

a(1), with p=2, is technically ambiguous between 1 and -1, so a(1)=1 is by convention. Clearly this cannot happen for n>1 (where p^2 is odd). - Jeppe Stig Nielsen, Sep 09 2021

Michael De Vlieger, Table of n, a(n) for n = 1..10000
R. J. McIntosh and E. L. Roettger, A search for Fibonacci-Wieferich and Wolstenholme primes, Math. Comp., 76 (2007), 2087-2094.

Cf. A113650, A195988, A244801, A339855, A347565.

Array[(#3 - #2 Boole[#3 > #2/2])/#1 & @@ {#, #^2, Mod[Fibonacci[# - KroneckerSymbol[#, 5]], #^2]} &@ Prime[#] &, 55] (* Michael De Vlieger, Sep 08 2021 *)

forprime(p=2, 1e2, a=fibonacci(p-kronecker(p, 5))%p^2; if(a>p^2/2, a-=p^2); a=a/p; print1(a, ", "))

a(n)=my(p=prime(n)); centerlift(((Mod([1, 1; 1, 0], p^2))^(p-kronecker(p,5))))[1, 2]/p \\ Charles R Greathouse IV, Aug 21 2014

A339855 Primes p such that the absolute value of the fraction A241014(A000720(p)) / p is a record low.

Original entry on oeis.org

2, 3, 5, 17, 41, 101, 163, 223, 251, 733, 1063, 27191, 77969, 84299, 86813, 123863, 508771, 1677209, 11634179, 91978037, 443127523, 467335159, 1041968177, 2025051311, 13941800291, 24178397183, 762383958397, 766193665711, 1551559563569, 8030311150847

Offset: 1

Jeppe Stig Nielsen, Dec 19 2020

nonn

So-called near-Wall-Sun-Sun primes. Each term is "nearer" to being Wall-Sun-Sun than all smaller primes.
If any Wall-Sun-Sun primes exist, this sequence terminates at the smallest Wall-Sun-Sun prime.
If you start from p=7 (not p=2), then the sequence will start 7, 13, 17, 41, ... instead.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..49
Ulrich Fries and PrimeGrid, PRPNet findlist for project WSS
Reginald McLean and PrimeGrid, WW Statistics
D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly, 67 (1960), 525-532.
Wikipedia, Wall-Sun-Sun prime

Cf. A241014, A113650, A296240, A347565.

rec=+oo;forprime(p=2,,r=abs(centerlift(((Mod([1, 1; 1, 0], p^2))^(p-kronecker(p, 5)-1))[1, 1]))/p^2;if(r

cp's OEIS Frontend

A241014 Let p be the n-th prime, then a(n) = A/p where A is the smallest number (in absolute value) such that F_{p-(p/5)} == A (mod p^2) with F_n = A000045(n) and (p/5) the Legendre symbol.

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A339855 Primes p such that the absolute value of the fraction A241014(A000720(p)) / p is a record low.

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PARI