A195988 -id:A195988 - cp's OEIS frontend

A258367 a(n) is the smallest A (in absolute value) such that for p = prime(n), 2^{(p-1)/2} == +-1 + A*p (mod p^2), i.e., such that p is a near-Wieferich prime.

1, 1, 1, 3, 5, 2, 8, 3, 14, 3, 18, 9, 9, 22, 18, 4, 18, 5, 1, 28, 30, 24, 3, 20, 46, 22, 47, 21, 15, 9, 57, 42, 15, 48, 28, 41, 48, 60, 85, 25, 74, 25, 52, 11, 32, 51, 17, 13, 34, 113, 13, 71, 2, 16, 64, 130, 81, 35, 37, 29, 39, 147, 68, 60, 71, 96, 92, 99, 12

Offset: 2

Felix Fröhlich, May 28 2015

nonn

p is in A001220 iff a(n) = 0. This is the case iff A014664(n) = A243905(n), which happens for n = 183 and n = 490.

Is a(n) = 0 for any other n, and, if yes, are there infinitely many such n?

Felix Fröhlich, Table of n, a(n) for n = 2..10000
R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Mathematics of Computation, 66 (1997), 433-449.

Cf. A001220, A195988, A241014, A244801, A246568, A250406, A250407, A306885, A338646, A343410.

a(n,p=prime(n))=abs(centerlift(Mod(2,p^2)^((p-1)/2))\/p)
apply(p->a(0,p), primes(100)[2..100]) \\ Charles R Greathouse IV, Jun 15 2015

a(n) = min(b(n) mod p, -b(n) mod p) where p = prime(n) and b(n) = Sum_{i=1..ceiling((p-1)/4)} (2i-1)^(p-2). - Daniel Chen, Sep 01 2022

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.

Original entry on oeis.org

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

A246568 Near-Wieferich primes (primes p satisfying 2^((p-1)/2) == +-1 + A*p (mod p^2)) with |A| < 10.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 31, 41, 43, 59, 67, 71, 89, 127, 251, 379, 569, 571, 1093, 1427, 1451, 1733, 2633, 2659, 2903, 3511, 13463, 15329, 15823, 26107, 60631, 546097, 2549177, 110057537, 165322639, 209227901, 671499313, 867457663, 3520624567

Offset: 1

Felix Fröhlich, Aug 30 2014

The data section gives all terms up to 10^10. There are eight more terms up to 3*10^15 (see b-file).

A is essentially (A007663(n) modulo A000040(n))/2 (see Crandall et al. (1997), p. 437). The choice of the bound for A is rather arbitrary and selecting a larger A will result in more terms in a specific interval. For any p there exist two values of A whose sum is p, except when p is in A001220, in which case A = 0.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..50 (terms n= 1..49 by Felix Fröhlich)
R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Math. Comp. Vol. 66, Num. 217 (1997), 433-449.
J. Knauer and J. Richstein, The continuing search for Wieferich primes, Math. Comp. Vol. 74, Num. 251 (2005), 1559-1563.
PrimeGrid, Wieferich & near Wieferich Primes p < 11e15
PrimeGrid, WW Statistics

Cf. A001220, A195988, A241014, A244801.

a258367(n) = abs(centerlift(Mod(2, n^2)^((n-1)/2))\/n) \\ after Charles R Greathouse IV in A258367
forprime(p=3, , if(a258367(p) < 10, print1(p, ", "))) \\ Felix Fröhlich, Apr 26 2022

A244801 Smallest m such that for the prime p = prime(n) the congruence F_(p-(p/5)) == mp (mod p^2) holds (i.e., smallest m such that prime(n) is a near-Wall-Sun-Sun prime), where F_k is the k-th Fibonacci number and (p/5) is the Legendre symbol.

Original entry on oeis.org

1, 1, 1, 3, 5, 3, 16, 3, 15, 26, 25, 13, 39, 39, 16, 28, 10, 48, 7, 55, 58, 49, 21, 5, 37, 97, 22, 24, 26, 60, 13, 64, 58, 117, 120, 60, 44, 160, 44, 130, 174, 131, 94, 31, 141, 5, 112, 3, 154, 18, 29, 5, 182, 250, 2, 105

Offset: 1

Felix Fröhlich, Jul 06 2014

nonn

A value of 0 indicates a Wall-Sun-Sun prime. No such prime is known and if one exists it is > 4*10^16 (cf. PrimeGrid WSS statistics).

Felix Fröhlich, Table of n, a(n) for n = 1..291043
F. G. Dorais and D. Klyve, A Wieferich Prime Search up to 6.7 x 10^15, J. Integer Seq. Volume 14, Issue 9 (2011).
R. J. McIntosh and E. L. Roettger, A search for Fibonacci-Wieferich and Wolstenholme primes, Math. Comp. 76 (2007), 2087-2094.
PrimeGrid, Wall-Sun-Sun Prime Search statistics
PrimeGrid, Welcome to the Wall-Sun-Sun prime search

Cf. A000045, A113650, A195988.

A= 0; p = 0; While[A < 200, p = NextPrime[p];  A= Mod[(Fibonacci[p-JacobiSymbol[p,5]])/p, p]; Print[A]] (* Javier Rivera Romeu, Jan 11 2022 *)

forprime(p=2, 10^2, a=fibonacci(p-kronecker(p, 5))%p^2; a=a/p; print1(a, ", "))

A, p = 0, 0
while A <200:
  p = next_prime(p)
  A = (fibonacci(p-legendre_symbol(p, 5))/p)%p
  print(A, end=", ") #Javier Rivera Romeu, Jan 08 2022

A354675 a(n) is the number of near-Wieferich primes with |A| <= 10 less than 10^n, where A(k) = A258367(k).

Original entry on oeis.org

3, 15, 21, 29, 34, 35, 36, 36, 41

Offset: 1

Felix Fröhlich, Jun 02 2022

A(k) is A258367(k). I believe this was initially defined in Crandall et al. (1997) (in particular pp. 436-437) and it is now common practice for Wieferich searches to report primes with |A| below some predefined limit (for example, the ongoing search at PrimeGrid uses |A| <= 1000).

			n | a(n) | A006880(n) | a(n)/A006880(n)*100
--------------------------------------------
1 |    3 |          4 | 75.000000
2 |   15 |         25 | 60.000000
3 |   21 |        168 | 12.500000
4 |   29 |       1229 |  2.359642
5 |   34 |       9592 |  0.354462
6 |   35 |      78498 |  0.044587
7 |   36 |     664579 |  0.005417
8 |   36 |    5761455 |  0.000625
9 |   41 |   50847534 |  0.000081

Richard Crandall, Karl Dilcher and Carl Pomerance, A search for Wieferich and Wilson primes, Mathematics of Computation, Vol. 66, No. 217 (1997), pp. 433-449; alternative link.
PrimeGrid, WW Statistics

Cf. A001220, A006880, A195988, A246568, A258367, A353141, A354676, A354677, A354678.

a258367(n) = abs(centerlift(Mod(2, n^2)^((n-1)/2))\/n) \\ after Charles R Greathouse IV in A258367
my(i=0, x=10); forprime(p=3, , if(p > x, print1(i, ", "); x=10*x); if(a258367(p) <= 10, i++))

A354676 a(n) = number of near-Wieferich primes with |A| <= 100 less than 10^n.

Original entry on oeis.org

3, 24, 105, 154, 213, 243, 268, 288, 307

Offset: 1

Felix Fröhlich, Jun 02 2022

n | a(n) | A006880(n) | a(n)/A006880(n)*100
--------------------------------------------
|    3 |          4 | 75.000000
|   24 |         25 | 96.000000
|  105 |        168 | 62.500000
|  154 |       1229 | 12.530512
|  213 |       9592 |  2.220600
|  243 |      78498 |  0.309562
|  268 |     664579 |  0.040326
|  288 |    5761455 |  0.004998
|  307 |   50847534 |  0.000603

Cf. A006880, A195988, A246568, A258367, A353141, A354675, A354677, A354678.

a258367(n) = abs(centerlift(Mod(2, n^2)^((n-1)/2))\/n) \\ after Charles R Greathouse IV in A258367
my(i=0, x=10); forprime(p=3, , if(p > x, print1(i, ", "); x=10*x); if(a258367(p) <= 100, i++))

A354677 a(n) = number of near-Wieferich primes with |A| <= 1000 less than 10^n.

Original entry on oeis.org

3, 24, 167, 698, 1155, 1502, 1812, 2064, 2297

Offset: 1

Felix Fröhlich, Jun 02 2022

n | a(n) | A006880(n) | a(n)/A006880(n)*100
--------------------------------------------
|    3 |          4 | 75.000000
|   24 |         25 | 96.000000
|  167 |        168 | 99.404761
|  698 |       1229 | 56.794141
| 1155 |       9592 | 12.041284
| 1502 |      78498 |  1.913424
| 1812 |     664579 |  0.272653
| 2064 |    5761455 |  0.035824
| 2297 |   50847534 |  0.004517

Cf. A006880, A195988, A246568, A258367, A353141, A354675, A354676, A354678.

a258367(n) = abs(centerlift(Mod(2, n^2)^((n-1)/2))\/n) \\ after Charles R Greathouse IV in A258367
my(i=0, x=10); forprime(p=3, , if(p > x, print1(i, ", "); x=10*x); if(a258367(p) <= 1000, i++))

A354678 a(n) = number of near-Wieferich primes with |A| <= 10000 less than 10^n.

Original entry on oeis.org

3, 24, 167, 1228, 5250, 8851, 11922, 14549, 16863

Offset: 1

Felix Fröhlich, Jun 02 2022

n |  a(n) | A006880(n) | a(n)/A006880(n)*100
---------------------------------------------
|     3 |          4 | 75.000000
|    24 |         25 | 96.000000
|   167 |        168 | 99.404761
|  1228 |       1229 | 99.918633
|  5250 |       9592 | 54.733110
|  8851 |      78498 | 11.275446
| 11922 |     664579 |  1.793917
| 14549 |    5761455 |  0.252523
| 16863 |   50847534 |  0.033163

Cf. A006880, A195988, A246568, A258367, A353141, A354675, A354676, A354677.

a258367(n) = abs(centerlift(Mod(2, n^2)^((n-1)/2))\/n) \\ after Charles R Greathouse IV in A258367
my(i=0, x=10); forprime(p=3, , if(p > x, print1(i, ", "); x=10*x); if(a258367(p) <= 10000, i++))

cp's OEIS Frontend

A258367 a(n) is the smallest A (in absolute value) such that for p = prime(n), 2^{(p-1)/2} == +-1 + A*p (mod p^2), i.e., such that p is a near-Wieferich prime.

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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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A246568 Near-Wieferich primes (primes p satisfying 2^((p-1)/2) == +-1 + A*p (mod p^2)) with |A| < 10.

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A244801 Smallest m such that for the prime p = prime(n) the congruence F_(p-(p/5)) == mp (mod p^2) holds (i.e., smallest m such that prime(n) is a near-Wall-Sun-Sun prime), where F_k is the k-th Fibonacci number and (p/5) is the Legendre symbol.

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A354675 a(n) is the number of near-Wieferich primes with |A| <= 10 less than 10^n, where A(k) = A258367(k).

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A354676 a(n) = number of near-Wieferich primes with |A| <= 100 less than 10^n.

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A354677 a(n) = number of near-Wieferich primes with |A| <= 1000 less than 10^n.

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A354678 a(n) = number of near-Wieferich primes with |A| <= 10000 less than 10^n.

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