A241014 -id:A241014 - cp's OEIS frontend

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

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

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.

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.

Original entry on oeis.org

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

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

A250406 Values of B such that p = prime(n) satisfies (p-1)! == -1-B*p (mod p^2), i.e., p is a near-Wilson prime.

Original entry on oeis.org

1, 2, 0, 2, 10, 0, 12, 17, 15, 11, 12, 30, 25, 30, 41, 19, 32, 5, 55, 2, 62, 6, 63, 19, 27, 29, 46, 106, 79, 18, 56, 12, 81, 72, 55, 65, 6, 55, 146, 67, 131, 109, 32, 158, 50, 81, 38, 43, 114, 98, 64, 132, 45, 37, 80, 190, 148, 101, 252, 4, 119, 62, 36, 52, 25

Offset: 1

Felix Fröhlich, Nov 22 2014

nonn

p is in A007540 iff a(n) == 0.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
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, A241014, A244801, A250407.

f[n_] := Block[{k = 0, m, p = Prime[n]}, m = Mod[(p - 1)!, p^2]; While[ Mod[-1 - k*p, p^2] != m, k++]; k]; Array[f, 70] (* Robert G. Wilson v, Dec 03 2014 *)

forprime(p=1, 1e9, b=0; while(Mod((p-1)!, p^2)!=-1-b*p, b++); print1(b, ", "))

A260209 Values A such that p=prime(n) satisfies binomial(2p-1, p-1) == 1 + A*p (mod p^4).

Original entry on oeis.org

1, 3, 25, 245, 121, 169, 867, 3249, 6877, 9251, 961, 15059, 57154, 61017, 68479, 106742, 201898, 208376, 107736, 176435, 330398, 237158, 158447, 213867, 903264, 856884, 21218, 755634, 1259386, 944906, 161290, 531991, 150152, 656914, 1287658, 592826, 640874

Offset: 1

Felix Fröhlich, Jul 19 2015

nonn

p is a Wolstenholme prime (A088164) iff a(n) == 0. This holds for n = 1944 and n = 157504.
When performing a search for Wolstenholme primes, one can choose an integer constant c >= 0 and record all primes p with A <= c in order to get a larger data set.
The values here asymptotically appear to grow more quickly than those in A260210.
It appears that a(n)/A260210(n) = A001248(n) for all n.

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Cf. A088164, A241014, A244801, A250406, A258367, A260210.

f[n_] := Block[{p = Prime@ n}, (Mod[ Binomial[2p - 1, p - 1], p^4] - 1)/p]; Array[f, 37] (* Robert G. Wilson v, Jul 29 2015 *)

a(n) = p=prime(n); (lift(Mod(binomial(2*p-1, p-1), p^4))-1)/p

A260210 A034602(n) modulo prime(n).

Original entry on oeis.org

1, 5, 1, 1, 3, 9, 13, 11, 1, 11, 34, 33, 31, 38, 58, 56, 24, 35, 62, 38, 23, 27, 96, 84, 2, 66, 106, 74, 10, 31, 8, 34, 58, 26, 26, 144, 150, 140, 167, 137, 31, 107, 78, 157, 1, 103, 165, 97, 111, 60, 196, 48, 97, 259, 155, 175, 244, 13, 269, 34, 184, 222, 54

Offset: 3

Felix Fröhlich, Jul 19 2015

nonn

p is a Wolstenholme prime (A088164) iff a(n) = 0. This holds for n = 1944 and n = 157504.
When performing a search for Wolstenholme primes, one can choose an integer constant c >= 0 and record all primes with a(n) <= c in order to get a larger data set.
The values here appear to have a nicer asymptotic growth behavior than those in A260209.
It appears that A260209(n)/a(n) = A001248(n).
The formula only returns integers for primes greater than 3. - Robert G. Wilson v, Jul 29 2015

Felix Fröhlich, Table of n, a(n) for n = 3..10000

Cf. A034602, A088164, A241014, A244801, A250406, A258367, A260209.

f[n_] := Block[{p = Prime@ n}, (Mod[ Binomial[2p - 1, p - 1], p^4] - 1)/p^3]; Array[f, 60, 3] (* Robert G. Wilson v, Jul 29 2015 *)

a(n) = p=prime(n); lift(Mod(binomial(2*p-1, p-1)\p^3, p))

A034602(n)/prime(n) = A260209(n)/prime(n)^2, for n>2. - Robert G. Wilson v, Jul 29 2015

A271782 Smallest n-Wall-Sun-Sun prime.

Original entry on oeis.org

13, 241, 2, 3, 191, 5, 2, 3, 2683

Offset: 2

Felix Fröhlich, Apr 18 2016

A prime p is a k-Wall-Sun-Sun prime iff p^2 divides F_k(pi_k(p)), where F_k(n) is the k-Fibonacci numbers, i.e., a Lucas sequence of first kind with (P,Q) = (k,-1), and pi_k(p) is the Pisano period of k-Fibonacci numbers modulo p (cf. A001175, A175181-A175185).
For prime p > 2 not dividing k^2 + 4, it is a k-Wall-Sun-Sun prime iff p^2 | F_k(p-((k^2+4)/p)), where ((k^2+4)/p) is the Kronecker symbol.
a(1) would be the smallest Wall-Sun-Sun prime whose existence is an open question.
a(12)..a(16) = 2, 3, 3, 29, 2. a(18)..a(33) = 3, 11, 2, 23, 3, 3, 2, 5, 79, 3, 2, 7, 23, 3, 2, 239. a(36)..a(38) = 2, 7, 17. a(40), a(41) = 2, 3. a(43)..a(46) = 5, 2, 3, 41. - R. J. Mathar, Apr 22 2016
a(17) = 1192625911, a(35) = 153794959, a(39) = 30132289567, a(47)..a(50) = 139703, 2, 3, 3. If they exist, a(11), a(34), a(42) are greater than 10^12. - Max Alekseyev, Apr 26 2016
Column k = 1 of table T(n, k) of k-th n-Wall-Sun-Sun prime (that table is not yet in the OEIS as a sequence). - Felix Fröhlich, Apr 25 2016
From Richard N. Smith, Jul 16 2019: (Start)
a(n) = 2 if and only if n is divisible by 4.
a(n) = 3 if and only if n == 5, 9, 13, 14, 18, 22, 23, 27, 31 (mod 36). (End)

Sergio Falcon and Ángel Plaza, k-Fibonacci sequences modulo m, Chaos, Solit. Fractals 41:1 (2009), 497-504.
Richard N. Smith, Table of n, a(n) for n = 1..1024 (or 0 if unknown, searched up to 2^22)
Richard N. Smith, List of odd n-Wall-Sun-Sun primes <= 2^20 for 1<=n<=1024
Wikipedia, Wall-Sun-Sun prime

Cf. A001177, A039951, A113649, A113650, A113651, A214028, A237517, A237835, A241014, A244801, A253247, A268478.

A271782(k) = forprime(p=2,10^8, if( (([0,1;1,k]*Mod(1,p^2))^(p-kronecker(k^2+4,p)))[1,2]==0, return(p);); ); \\ Max Alekseyev, Apr 22 2016, corrected by Richard N. Smith, Jul 16 2019 to include p=2 and p divides k^2+4

a(4n) = 2.

Edited by Max Alekseyev, Apr 25 2016

A338558 Absolute value q such that tau(p) == q (mod p), where p = prime(n) and tau(i) = A000594(i).

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 7, 7, 1, 5, 10, 6, 11, 12, 20, 24, 14, 12, 3, 19, 6, 37, 20, 33, 20, 27, 50, 34, 36, 29, 18, 64, 4, 2, 66, 32, 3, 64, 61, 51, 60, 84, 95, 83, 63, 97, 42, 28, 61, 67, 32, 10, 29, 73, 37, 92, 16, 120, 31, 107, 120, 141, 145, 39, 12, 74, 150

Offset: 1

Felix Fröhlich, Dec 21 2020

nonn

These are essentially the values that can be used to define "near-misses" in a search of terms for A007659, similar to how "near-Wieferich primes", "near-Wilson primes" and "near-Wall-Sun-Sun primes" are defined in searches for Wieferich primes (A001220), Wilson primes (A007540) and Wall-Sun-Sun (Fibonacci-Wieferich) primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Nik Lygeros and Olivier Rozier, A new solution for the equation tau(p)=0 (mod p), Journal of Integer Sequences 13 (2010), Article 10.7.4.

Cf. A001220, A007540, A007659, A295645.

A-values: A258367 (near-Wieferich), A250406 (near-Wilson), A244801 and A241014 (near-Wall-Sun-Sun), A260209 and A260210 (near-Wolstenholme).

a[n_] := Module[{p = Prime[n]}, Min[Abs[Mod[RamanujanTau[p], {-p, p}]]]]; Array[a, 100] (* Amiram Eldar, Jan 10 2025 *)

a(n) = my(p=prime(n)); abs(centerlift(Mod(ramanujantau(p), p)))

a(n) = 0 iff prime(n) is a term of A007659.

A352858 a(n) = abs(E_{p-3} (mod p)) for p = prime(n), where E_i is the i-th Euler number (A000364).

Original entry on oeis.org

1, 2, 1, 3, 8, 7, 1, 3, 9, 4, 4, 4, 14, 7, 12, 16, 25, 22, 25, 4, 23, 33, 42, 15, 46, 18, 23, 38, 58, 2, 6, 55, 0, 37, 74, 63, 10, 61, 21, 38, 92, 89, 70, 79, 69, 59, 85, 22, 27, 69, 0, 45, 58, 96, 106, 6, 50, 28, 91, 133, 46, 147, 133, 38, 29, 128, 167, 116

Offset: 3

Felix Fröhlich, Apr 06 2022

nonn

a(n) = 0 iff p is a term of A198245.

These are the absolute values of the "A-values" that can be used to define "near-misses" in a search for terms of A198245 (cf. Mestrovic, 2014).

Romeo Mestrovic, A search for primes p such that Euler number E_p-3 is divisible by p, Mathematics of Computation 83 (2014), 2967-2976.

Cf. A000364, A198245.

A-values: A258367 (near-Wieferich), A250406 (near-Wilson), A244801 and A241014 (near-Wall-Sun-Sun), A260209 and A260210 (near-Wolstenholme), A338558 (near-misses for A007659).

eulmod(n) = abs(centerlift(Mod(eulerfrac(n-3), n)))
a(n) = my(p=prime(n)); eulmod(p)

cp's OEIS Frontend

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

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

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

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A250406 Values of B such that p = prime(n) satisfies (p-1)! == -1-B*p (mod p^2), i.e., p is a near-Wilson prime.

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A260209 Values A such that p=prime(n) satisfies binomial(2p-1, p-1) == 1 + A*p (mod p^4).

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A260210 A034602(n) modulo prime(n).

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A271782 Smallest n-Wall-Sun-Sun prime.

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A338558 Absolute value q such that tau(p) == q (mod p), where p = prime(n) and tau(i) = A000594(i).

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