A246568 -id:A246568 - 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

A195988 Near-Wieferich primes above 10^9: primes p > 10^9 such that 2^((p-1)/2) == +-1 + A*p (mod p^2) with |A| <= 100, i.e., p=prime(i) such that A258367(i) <= 100.

Original entry on oeis.org

1222336487, 1259662487, 1274153897, 1494408397, 1584392531, 1586651309, 1662410923, 1817972423, 1890830857, 2062661389, 2244893621, 2332252547, 2416644757, 2461090421, 2566816313, 2570948153, 2589186937, 2709711233, 2760945133

Offset: 1

Felix Fröhlich, Sep 26 2011

nonn

There are many near-Wieferich primes below 10^9 (including Wieferich primes 1093 and 3511). However, Crandall, Dilcher and Pomerance searched and reported such primes in the interval [10^9, 4*10^12].

The choice of upper bound for |A| is rather arbitrary.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..149 (terms n = 1..139 from 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.
R. McIntosh, E-Mail to Paul Zimmermann
PrimeGrid, Wieferich & near Wieferich Primes p < 11e15
PrimeGrid, WW Statistics

Cf. A001220, A246568, A258367, A258368.

Edited by Max Alekseyev, Dec 21 2011
New b-file from Felix Fröhlich, Aug 26 2015
Definition amended by Felix Fröhlich, Aug 29 2015

A353141 Near-Wieferich primes with abs(A) < 2.

Original entry on oeis.org

3, 5, 7, 71, 379, 1093, 2659, 3511, 110057537, 47004625957, 76843523891

Offset: 1

Felix Fröhlich, Apr 26 2022

Primes p such that A258367(i) < 2, where i is the index of p in A000040.
Subsequence of A246568.
Wieferich primes together with the "closest" near-misses possible that are not actually Wieferich.
Countless such sequences with larger bounds on the value of abs(A) are possible. This is one of the few that I believe should be in the OEIS.
The corresponding sequence of A-values is 1, 1, 1, 1, -1, 0, -1, 0, -1, 1, 1.
I checked the range 3 <= p <= 47004625957 with PARI. 76843523891 is from Crandall, Dilcher, Pomerance, 1997.
There are no near-Wieferich primes with abs(A) < 2 in the range 4*10^12 to 1.25*10^15 (cf. Knauer, Richstein, 2005).
There are no near-Wieferich primes with abs(A) < 2 in the range 1*10^15 to 3*10^15 per information I received from Mark Rodenkirch in 2010.
There are no near-Wieferich primes with abs(A) < 2 in the range 3*10^15 to ~6*10^17 (cf. Goetz, cf. Reggie, cf. Fries).
As of Apr 26 2022, a(12) > ~1.1*10^19 if it exists (cf. WW Statistics).
Heuristically, one would expect about 11 to 12 (3*log(log(10^19))) near-Wieferich primes with |A| <= 1 up to 10^19, a very close match to the actual number of 11 (cf. Crandall, Dilcher, Pomerance, 1997, p. 446).

			The prime p = 110057537 satisfies 2^((p-1)/2) == +1 -p (mod p^2) and is therefore in the sequence.

R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Mathematics of Computation 66 (1997), 433-449.
U. Fries, PRPNet findlist for project WFS
M. Goetz, WW by the numbers, PrimeGrid forum.
J. Knauer and J. Richstein, The continuing search for Wieferich primes, Mathematics of Computation 74 (2005), 1559-1563.
J. S. Nielsen, Check a WW find with PARI/GP, PrimeGrid forum.
Reggie, Which near-finds are shown?, PrimeGrid forum (see Message 146701).
PrimeGrid, WW Statistics

Cf. A000040, A001220, A246568, A258367.

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) < 2, print1(p, ", ")))

\\ see Nielsen link for code to check the A-value and +-1-type of a prime

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++))

A250407 Near-Wilson primes (p = prime(n) satisfying (p-1)! == -1-A250406(n)*p (mod p^2)) with A250406(n) < 10.

Original entry on oeis.org

2, 3, 5, 7, 13, 61, 71, 79, 157, 281, 563, 1277, 1777, 2339, 6311, 8233, 8543, 11047, 22907, 27689

Offset: 1

Felix Fröhlich, Nov 22 2014

A250406(n) is essentially A007619(n) modulo A000040(n) (see Crandall et al. (1997), p. 442).

E. Costa, R. Gerbicz and D. Harvey, A search for Wilson primes, Math. Comp., 83 (2014), 3071-3091.
R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Math. Comp., 66 (1997), 433-449.

Cf. A007540, A246568, A250406.

forprime(p=1, 1e9, for(b=0, 9, if(Mod((p-1)!, p^2)==-1-b*p, print1(p, ", "); break({1}))))

A319314 Numbers k such that 2^phi(k) == phi(k)^2 (mod k^2).

Original entry on oeis.org

1, 3, 4, 5, 6, 8, 10, 12, 384, 640, 768, 896, 960, 24576, 49152, 950272, 1425408, 1572864, 3145728, 10485760, 19398656, 65011712, 100663296, 110057537, 201326592, 220115074, 671088640, 1879048192

Offset: 1

Altug Alkan, Sep 17 2018

Sequence is infinite, i.e., 3*2^(3*(t-1)-(-1)^t) is a term for all t > 0.

Prime terms (5, 110057537, ...) are in A246568 based on case A = +1.

Cf. A077815, A077816, A246568, A292544.

[1] cat [n: n in [1..10^6] | 2^EulerPhi(n) mod n^2 eq EulerPhi(n)^2]; // Vincenzo Librandi, Sep 20 2018

isok(n) = Mod(2, n^2)^eulerphi(n)==eulerphi(n)^2;

A385856 Near-Wieferich primes (primes p satisfying 2^p == 2 + A*p (mod p^2)) with |A| <= 10.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 29, 37, 41, 43, 47, 71, 157, 173, 211, 251, 263, 379, 383, 1093, 1097, 1699, 1753, 2633, 2659, 3373, 3511, 3593, 5501, 8089, 10691, 15823, 27967, 30577, 45827, 46477, 1437049, 1483597, 1897121, 2152849, 6266543, 52368101, 110057537, 126233057, 1683955849, 2001907169, 13211006161, 47004625957

Offset: 1

A. Lamek, Jul 10 2025

nonn

Near-Wieferich primes: 2^p == 2 + A*p (mod p^2) for |A| <= 10. Extends the Wieferich condition by allowing small symmetric offsets.
See also A246568 for a related formulation involving the same congruence structure.
All values verified for p <= 5*10^10.

			p=11: 2^11 == 2048 == 2+(-1)*11 == -9 == 112 (mod 121), so A=-1.
p=5: 2^5 == 32 == 2+1*5 == 7 (mod 25), so A=1.

A. Lamek, User profile with notes on Wieferich-related structures

Cf. A001220, A246568.

isokQ[p_] := Module[{A}, A = Quotient[PowerMod[2, p, p^2] - 2, p]; A <= 10 || p - A <= 10]

isok(p) = lift(Mod(2, p^2)^p-2+10*p) <= 20*p; \\ Michel Marcus, Jul 12 2025

def is_a385856(p):
    A = ((pow(2, p, p*p)-2) // p) % p
    return (A<=10) or (p-A<=10)

a(44)-a(46) from Michel Marcus, Jul 12 2025

a(48) from Jinyuan Wang, Jul 13 2025

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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A195988 Near-Wieferich primes above 10^9: primes p > 10^9 such that 2^((p-1)/2) == +-1 + A*p (mod p^2) with |A| <= 100, i.e., p=prime(i) such that A258367(i) <= 100.

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A353141 Near-Wieferich primes with abs(A) < 2.

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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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A250407 Near-Wilson primes (p = prime(n) satisfying (p-1)! == -1-A250406(n)*p (mod p^2)) with A250406(n) < 10.

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A319314 Numbers k such that 2^phi(k) == phi(k)^2 (mod k^2).

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