A339855 -id:A339855 - 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

A347565 Primes p such that A241014(A000720(p)) is +1 or -1.

Original entry on oeis.org

2, 3, 5, 17, 251, 733, 1063, 123863, 1677209, 6336823451747417, 104868559750360787, 7665762181374748069

Offset: 1

Jeppe Stig Nielsen, Sep 06 2021

Very near misses for Wall-Sun-Sun primes.

Cf. A241014, A113650, A296240, A339855.

A338646 Primes p such that 47^(p-1) == 1 + A*p (mod p^2) and |A/p| is a new record low.

Original entry on oeis.org

2, 3, 5, 19, 37, 47, 38693, 44657, 148091, 178621, 692521, 4584379, 262148693, 347850691, 502176491, 1139746919, 1387837067, 5291181761, 92653098679, 202259581243

Offset: 1

Felix Fröhlich, Apr 22 2021

47 is the smallest b such that no base-b Wieferich prime, i.e., prime p such that b^(p-1) == 1 (mod p^2) is known (cf. Fischer).
The known terms of the sequence are base-47 near-Wieferich primes matching a definition of "nearness" introduced by Dorais and Klyve (cf. Dorais, Klyve, 2011).
If a base-47 Wieferich prime exists, then the sequence is finite and terminates at that prime.

			             p | abs(A/p) (frac) | abs(A/p) (dec)
----------------------------------------------------
             2 |  1/2            | 0.5
             3 |  1/3            | 0.333333333333333
             5 |  1/5            | 0.2
            19 |  2/19           | 0.105263157894736
            37 |  2/37           | 0.054054054054054
            47 |  1/2209         | 0.000452693526482
         38693 | 10/38693        | 0.000258444679916
         44657 |  4/44657        | 0.000089571623709
        148091 | 13/148091       | 0.000087783862625
        178621 |  1/178621       | 0.000005598445871
        692521 |  1/692521       | 0.000001443999532
       4584379 |  1/4584379      | 0.000000218132052
     262148693 | 39/262148693    | 0.000000148770530
     347850691 | 47/347850691    | 0.000000135115442
     502176491 | 51/502176491    | 0.000000101557920
    1139746919 | 75/1139746919   | 0.000000065804082
    1387837067 |  8/1387837067   | 0.000000005764365
    5291181761 |  3/5291181761   | 0.000000000566981
   92653098679 |  7/92653098679  | 0.000000000075550
  202259581243 |  5/202259581243 | 0.000000000024720

F. G. Dorais and D. Klyve, A Wieferich Prime Search up to 6.7 × 10^15, Journal of Integer Sequences, Vol. 14 (2011), Article 11.9.2.
Richard Fischer, Thema: Fermatquotient B^(P-1) == 1 (mod P^2)

Cf. A339855.

my(a=0, ab=0, r=0); forprime(p=1, , a = (lift(Mod(47, p^2)^(p-1))-1)/p; ab=abs(a/p); if(r==0, r=ab; print1(p, ", "), if(ab < r, r=ab; print1(p, ", "))))

a(19) from Felix Fröhlich, Jul 01 2021

a(20) from Felix Fröhlich, Jul 02 2021

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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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A347565 Primes p such that A241014(A000720(p)) is +1 or -1.

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A338646 Primes p such that 47^(p-1) == 1 + A*p (mod p^2) and |A/p| is a new record low.

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