A001220 -id:A001220 - cp's OEIS frontend

A244752 Square array read by antidiagonals in which rows are indexed by composite numbers w and row w gives n such that n^(w-1) == 1 (mod w^2).

17, 33, 37, 49, 73, 65, 65, 109, 129, 80, 81, 145, 193, 82, 101, 97, 181, 257, 161, 201, 145, 113, 217, 321, 163, 301, 289, 197, 129, 253, 385, 242, 401, 433, 393, 26, 145, 289, 449, 244, 501, 577, 589, 199, 257, 161, 325, 513, 323, 601, 721, 785, 224, 513

Offset: 2

Felix Fröhlich, Jul 05 2014

We can say that "w is a Wieferich pseudoprime to base n".

Any prime factor of w is a Wieferich prime to base n.

			Table starts
w=4: 17, 33, 49, 65, 81, 97, 113, ....
w=6: 37, 73, 109, 145, 181, 217, ....
w=8: 65, 129, 193, 257, 321, 385, ....
w=9: 80, 82, 161, 163, 242, 244, ....
w=10: 101, 201, 301, 401, 501, 601, ....
w=12: 145, 289, 433, 577, 721, 865, ....
w=14: 197, 393, 589, 785, 981, ....
....

Robert Price, Table of n, a(n) for n = 2..1276

Cf. A001220, A240719, A244249.

T = {};
For[w = 4, w <= 100, w++,
  If[PrimeQ[w], Continue[]];
  t = {};
  For [n = 2, n <= 10^5, n++,
   If[Mod[n^(w - 1), w^2] == 1, AppendTo[t, n]]];
  AppendTo[T, t]];
Print[TableForm[T]];
A244752 = {};
For[c = 1, c <= 50, c++,
  For[r = 1, r <= c, r++, AppendTo[A244752, T[[r]][[c - r + 1]]]]];
A244752 (* Robert Price, Sep 07 2019 *)

forcomposite(w=2, 20, print1("w=", w, ": "); for(n=2, 10^3, if(Mod(n, w^2)^(w-1)==1, print1(n, ", "))); print(""))

a(17)-a(55) from Robert Price, Sep 07 2019

A084653 Pseudoprimes whose prime factors do not divide any smaller pseudoprime.

Original entry on oeis.org

341, 1387, 2047, 8321, 13747, 18721, 19951, 31621, 60701, 83333, 88357, 219781, 275887, 422659, 435671, 513629, 514447, 587861, 604117, 653333, 680627, 710533, 722261, 741751, 769757, 916327, 1194649, 1252697, 1293337, 1433407, 1441091

Offset: 1

T. D. Noe, Jun 02 2003

nonn

Here pseudoprime means a Fermat base-2 pseudoprime; sequence A001567, a composite number n such that n divides 2^(n-1) - 1. All numbers in this sequence seem to have only two prime factors - a conjecture that has been tested for all pseudoprimes < 10^15. The two prime factors are given in A084654 and A084655. The two prime factors are the same when the pseudoprime is the square of a Wieferich prime (A001220).

			a(2) = 1387 because 1387 = 19*73 and the smaller pseudoprimes (341, 561, 645, 1105) do not have the factors 19 or 73.

T. D. Noe, Table of n, a(n) for n = 1..10000
R. G. E. Pinch, Pseudoprimes and their factors (FTP)
Eric Weisstein's World of Mathematics, Pseudoprime
Index entries for sequences related to pseudoprimes

Cf. A001220, A001567, A084654, A084655.

A178871 2nd Wieferich prime base prime(n).

Original entry on oeis.org

3511, 1006003, 20771, 491531

Offset: 1

Jonathan Sondow, Jun 20 2010, Jun 24 2010

2nd prime p such that p^2 divides prime(n)^(p-1) - 1.
2nd prime p such that p divides the Fermat quotient q_p(p_n) = ((p_n)^(p-1) - 1)/p, where p_n = prime(n).
a(5) is unknown: 71 is the only known prime p that divides q_p(11).
If a(5) is found, the sequence continues a(6) = 863, a(7) = 3, a(8) = 7, a(9) = 2481757.
See additional comments, references, links, and cross-refs in A039951 and A174422.

			a(1) = 3511 is the 2nd Wieferich prime A001220(2).
a(2) = 1006003 is the 2nd Mirimanoff prime A014127(2).

Wikipedia, Generalized Wieferich primes

Cf. A039951 = smallest prime p such that p^2 divides n^(p-1) - 1, A174422 = first Wieferich prime base prime(n).

{default(primelimit, 10^7); for(n=1, 9, a=prime(n); c=0; forprime(p=2, 10^7, if(Mod(a, p^2)^(p-1)==1, c++; if(c==2, print1(p, ", "); next(2)))); print1(">10^7, "))} \\ Jens Kruse Andersen, Jun 18 2014

A182297 Wieferich numbers (2): positive odd integers q such that q and (2^A002326((q-1)/2)-1)/q are not relatively prime.

Original entry on oeis.org

21, 39, 55, 57, 105, 111, 147, 155, 165, 171, 183, 195, 201, 203, 205, 219, 231, 237, 253, 273, 285, 291, 301, 305, 309, 327, 333, 355, 357, 385, 399, 417, 429, 453, 465, 483, 489, 495, 497, 505, 507, 525, 543, 555, 579, 597, 605, 609, 615, 627, 633, 651, 655

Offset: 1

Felix Fröhlich, Apr 23 2012

nonn

The primes in this sequence are A001220, the Wieferich primes. - Charles R Greathouse IV, Feb 02 2014

Odd prime p is a Wieferich prime if and only if A002326((p^2-1)/2) = A002326((p-1)/2). See the sixth comment to A001220 and my formula below. - Thomas Ordowski, Feb 03 2014

			21 is in the sequence because the multiplicative order of 2 mod 21 is 6, and (2^6-1)/21 = 3, which is not coprime to 21.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Z. Franco and C. Pomerance, On a conjecture of Crandall concerning the qx + 1 problem, Math. Comp. Vol. 64, No. 211 (1995), 1333-1336.

For another definition of Wieferich numbers, see A077816.

Cf. A002326.

with(numtheory):
a:= proc(n) option remember; local q;
      for q from 2 +`if`(n=1, 1, a(n-1)) by 2
        while igcd((2^order(2, q)-1)/q, q)=1 do od; q
    end:
seq (a(n), n=1..60);  # Alois P. Heinz, Apr 23 2012

Select[Range[1, 799, 2], GCD[#, (2^MultiplicativeOrder[2, #] - 1)/#] > 1 &] (* Alonso del Arte, Apr 23 2012 *)

is(n)=n%2 && gcd(lift(Mod(2,n^2)^znorder(Mod(2,n))-1)/n,n)>1 \\ Charles R Greathouse IV, Feb 02 2014

Odd numbers q such that A002326((q^2-1)/2) < q * A002326((q-1)/2). Other positive odd integers satisfy the equality. - Thomas Ordowski, Feb 03 2014

Odd numbers q such that gcd(A165781((q-1)/2), q) > 1. - Thomas Ordowski, Feb 12 2014

A238736 Balancing Wieferich primes: primes p that divide their Pell quotients, where the Pell quotient of p is A000129(p - (2/p))/p and (2/p) is a Jacobi symbol.

Original entry on oeis.org

13, 31, 1546463

Offset: 1

John Blythe Dobson, Mar 04 2014

Williams 1982 (p. 86), notes that p = 13, 31 and 1546463 are the only primes less than 10^8 for which the Pell quotient vanishes mod p. Elsenhans and Jahnel, "The Fibonacci sequence modulo p^2," p. 5, report in effect that there are no more such primes p < 10^9.
Williams 1991 (p. 440), and Sun 1995 pt. 3, Theorem 3.3, together prove a set of formulas connecting the Pell quotient with the Fermat quotient (base 2) (A007663) and harmonic numbers like H(floor(p/8)) (see example in the Formula section below). As is well known, the vanishing of the Fermat quotient (base 2) is a necessary condition for the failure of the first case of Fermat's Last Theorem (see discussion under A001220); and in light of a corresponding result of Dilcher and Skula concerning this type of harmonic number, the vanishing of the Pell quotient mod p is also a necessary condition for the failure of the first case of Fermat's Last Theorem.
There are no more terms up to 10^10.
Using the PARI script by Charles R Greathouse IV, I have extended the search from 10^10 to 10^12 without finding a further solution. - John Blythe Dobson, Mar 30 2015
Also primes p such that p^2 divides A001109((p - (2/p))/2). - Jianing Song, Oct 08 2018
From Felix Fröhlich, May 18 2019: (Start)
The term "balancing Wieferich prime" comes from Rout, 2016.
Primes p that satisfy the congruence B_{p-(8/p)} == 0 (mod p^2), where B_i denotes the i-th balancing number A001109(i) and (a/b) denotes the Jacobi symbol (cf. Rout, 2016, (1.6)).
Primes p such that the period of the balancing sequence (A001109) modulo p is equal to the period of the balancing sequence modulo p^2 (cf. Panda, Rout, 2014, p. 275).
Under the abc conjecture for the number field Q(sqrt(2)) there exist at least (log(x)/log(log(x)))*(log(log(log(x))))^m balancing non-Wieferich primes <= x such that p == 1 (mod k) for any integers k > 2, m > 0 (cf. Dutta, Patel, Ray, 2019). This is an improvement of an earlier result stating there are at least log(x)/log(log(x)) balancing non-Wieferich primes p == 1 (mod k) less than x (cf. Theorem 3.2 in Rout 2016). (End)

			PellQuotient(13) = 6214 = 13*478; PellQuotient(31) = 3470274850 = 31*111944350.

Zakariae Bouazzaoui, On Periods of Fibonacci Sequences and Real Quadratic p-rational Fields, Fibonacci Quart. 58 (2020), no. 5, 103-110. See p. 7.
Karl Dilcher and Ladislav Skula, A new criterion for the first case of Fermat's Last Theorem, Mathematics of Computation, 64 (1995), 363-392.
Utkal Keshari Dutta, Bijan Kumar Patel and Prasanta Kumar Ray, A brief remark on balancing-Wieferich primes, Mathematica, Vol. 60 (83), No. 1 (2018), 48-53 [Subscription required].
Utkal Keshari Dutta, Bijan Kumar Patel and Prasanta Kumar Ray, Balancing non-Wieferich primes in arithmetic progressions, Proceedings - Mathematical Sciences, Vol. 129, No. 2 (2019), Article 21, DOI:10.1007/s12044-018-0459-3.
Andreas-Stephan Elsenhans and Jörg Jahnel, The Fibonacci sequence modulo p^2 -- An investigation by computer for p < 10^14, arXiv 1006.0824 [math.NT], 2010.
Georges Gras, On the structure of the Galois group of the Abelian closure of a number field, arXiv 1212.3588 [math.NT], 2013.
Hao Pan, Lehmer's type congruences for lacunary harmonic sums, arXiv 0905.0941 [math.NT], 2009.
G. K. Panda and S. S. Rout, Periodicity of Balancing Numbers, Acta Mathematica Hungarica 143 (2014), 274-286. Also on ResearchGate.
Sudhansu Sekhar Rout, Balancing non-Wieferich primes in arithmetic progression and abc conjecture, Proc. Japan Acad. Ser. A Math. Sci., Volume 92, Number 9 (2016), 112-116.
Zhi-Hong Sun, Combinatorial sum ... and its applications in Number Theory, III (English version), originally published in Chinese in Journal of Nanjing University Mathematical Biquarterly, 12 (1995), 90-102.
Zhi-Hong Sun, Five congruences for primes, Fibonacci Quarterly, 40 (2002), 345-351.
H. C. Williams, The influence of computers in the development of number theory, Computers & Mathematics with Applications, 8 (1982), 75-93.
H. C. Williams, Some formulas concerning the fundamental unit of a real quadratic field, Discrete Mathematics, 92 (1991), 431--440.

Cf. A000129, A001109.

Select[Prime[Range[1000]], Mod[Fibonacci[# - JacobiSymbol[2, #], 2]/#, #] == 0 &]

is(n)=isprime(n) && (Mod([2,1;1,0],n^2)^(n-kronecker(2,n)))[2,1]==0 \\ Charles R Greathouse IV, Mar 04 2014

The condition for p to be a member of this sequence is A000129(p-e)/p == F(p-e, 2)/p == 0 (mod p), where F(p-e, 2) is the p-e'th Fibonacci polynomial evaluated at the argument 2, and e = (2/p) is a Jacobi Symbol.

Let PellQuotient(p) = A000129(p-e)/p, q_2 = (2^(p-1) - 1)/p = A007663(p) be the corresponding Fermat quotient of base 2, H(floor(p/8)) be a harmonic number, and e = (2/p) be a Jacobi Symbol. Then a result of Williams (1991), as refined by Sun (1995), shows that 2*PellQuotient(p) == -4*q_2 - H(floor(p/8)) (mod p).

Name amended by Felix Fröhlich, May 26 2019

A270096 Smallest m such that 2^m == 2^n (mod n).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 3, 2, 1, 2, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 4, 5, 2, 9, 4, 1, 2, 1, 5, 3, 2, 11, 6, 1, 2, 3, 4, 1, 6, 1, 4, 9, 2, 1, 4, 7, 10, 3, 4, 1, 18, 15, 5, 3, 2, 1, 4, 1, 2, 3, 6, 5, 6, 1, 4, 3, 10, 1, 6, 1, 2, 15, 4, 17, 6, 1, 4

Offset: 1

Thomas Ordowski, Mar 11 2016

nonn

a(n) = 1 iff n is a prime or a pseudoprime (odd or even) to base 2.
We have a(n) <= n - phi(n) and a(n) <= phi(n), so a(n) <= n/2.
From Robert Israel, Mar 11 2016: (Start)
If n is in A167791, then a(n) = A068494(n).
If n is odd, a(n) = n mod A002326((n-1)/2).
a(n) >= A007814(n).
a(p^k) = p^(k-1) for all k >= 1 and all odd primes p not in A001220.
Conjecture: a(n) <= n/3 for all n > 8. (End)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A001220, A002326, A007814, A051953, A068494, A167791.

Cf. A276976 (a generalization on all integer bases).

f:= proc(n) local d,b,t, m,c;
  d:= padic:-ordp(n,2);
  b:= n/2^d;
  t:= 2 &^ n mod n;
  m:= numtheory:-mlog(t,2,b,c);
  if m < d then m:= m + c*ceil((d-m)/c) fi;
  m
end proc:
f(1):= 0:
map(f, [$1..1000]; # Robert Israel, Mar 11 2016

Table[k = 0; While[PowerMod[2, n, n] != PowerMod[2, k, n], k++]; k, {n, 120}] (* Michael De Vlieger, Mar 15 2016 *)

a(n) = {my(m = 0); while (Mod(2, n)^m != 2^n, m++); m; } \\ Altug Alkan, Sep 23 2016

a(n) < n/2 for n > 4.
a(2^k) = k for all k >= 0.
a(2*p) = 2 for all primes p.

More terms from Michel Marcus, Mar 11 2016

A163209 Catalan pseudoprimes: odd composite integers n=2m+1 satisfying A000108(m) == (-1)^m 2 (mod n).

Original entry on oeis.org

5907, 1194649, 12327121

Offset: 1

Peter Luschny, Jul 24 2009

Also, Wilson spoilers: composite n which divide A056040(n-1) - (-1)^floor(n/2). For the factorial function, a Wilson spoiler is a composite n that divides (n-1)! + (-1). Lagrange proved that no such n exists. For the swinging factorial (A056040), the situation is different.
Also, composite odd integers n=2*m+1 such that A000984(m) == (-1)^m (mod n).
Contains squares of A001220. In particular, a(2) = A001220(1)^2 = 1093^2 = 1194649 = A001567(274) and a(3) = A001220(2)^2 = 3511^2 = 12327121 = A001567(824).
See the Vardi reference for a binomial setup.
Aebi and Cairns 2008, page 9: a(4) either has more than 2 factors or is > 10^10. - Dana Jacobsen, May 27 2015
a(4) > 10^10. - Dana Jacobsen, Mar 03 2018

I. Vardi, Computational Recreations in Mathematica, 1991, p. 66.

C. Aebi, G. Cairns (2008). "Catalan numbers, primes and twin primes". Elemente der Mathematik 63 (4): 153-164. doi:10.4171/EM/103
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.
Index entries for sequences related to pseudoprimes

swing := proc(n) option remember; if n = 0 then 1 elif irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
WS := proc(f,r,n) select(p->(f(p-1)+r(p)) mod p = 0,[$2..n]);
select(q -> not isprime(q),%) end:
A163209 := n -> WS(swing,p->(-1)^iquo(p+2,2),n);

v(n,p)=my(s); n*=2; while(n\=p, s+=n%2); s
is(n)=if(n%2==0,return(0)); my(m=Mod(1,n),a=n\2); fordiv(n,d,if(isprime(d) && v(a,d), return(0))); forprime(p=2,a, m*=p^v(a,p)); forprime(p=a+1,n,m*=p); m==(-1)^a
forcomposite(n=4,2e7, if(is(n), print1(n", "))) \\ Charles R Greathouse IV, Mar 06 2015

# Reasonable for isolated values, slow for the sequence:
use ntheory ":all";
sub is { my $m = ($[0]-1)>>1; (binomial($m<<1,$m) % $[0]) == (($m&1) ? $_[0]-1 : 1); }
foroddcomposites { say if is($) } 2e7;  # _Dana Jacobsen, May 03 2015

# Much faster for sequential testing:
use Math::GMPz; use ntheory ":all"; { my($c,$l)=(Math::GMPz->new(1),1); sub catalan { while ($[0] > $l) { $l++; $c *= 4*$l-2; Math::GMPz::Rmpz_divexact_ui($c,$c,$l+1); } $c; } } my $m; foroddcomposites { $m = ($-1)>>1; say if (catalan($m) % $) == (($m&1) ? $-2 : 2); } 2e7;  # Dana Jacobsen, May 03 2015

a(1) = 5907 = 3*11*179 was found by S. Skiena
Typo corrected Peter Luschny, Jul 25 2009
Edited by Max Alekseyev, Jun 22 2012

A172290 Prime divisors of 2^1092-1, listed with multiplicities.

Original entry on oeis.org

3, 3, 5, 7, 7, 13, 13, 29, 43, 53, 79, 113, 127, 157, 313, 337, 547, 911, 1093, 1093, 1249, 1429, 1613, 2731, 3121, 4733, 5419, 8191, 14449, 21841, 121369, 224771, 503413, 1210483, 1948129, 22366891, 108749551, 112901153, 23140471537, 25829691707, 105310750819, 467811806281, 4093204977277417, 8861085190774909, 556338525912325157, 86977595801949844993, 275700717951546566946854497, 292653113147157205779127526827, 3194753987813988499397428643895659569

Offset: 1

Artur Jasinski, Jan 30 2010

Up to now only two primes p such that p^2 divide 2^(p-1)-1 are known (these two are Wieferich primes, see A001220).
The sequence is finite with A001222(2^1092-1) = 49 terms; A001221(2^1092-1) = 45. - Reinhard Zumkeller, May 14 2010
Terms appearing more than once (in fact twice) are 3, 7, 13, and 1093.

Reinhard Zumkeller, Table of n, a(n) for n = 1..49 (complete sequence)
Dario A. Alpern, Factorization using the Elliptic Curve Method.

Cf. A001220, A172291, A177855, A046051, A046800, A242715.

Missing terms a(34) and a(35) inserted by Reinhard Zumkeller, May 14 2010

Definition clarified and terms corrected by Joerg Arndt, Apr 25 2011

A243905 Multiplicative order of 2 modulo prime(n)^2 for n >= 2.

Original entry on oeis.org

6, 20, 21, 110, 156, 136, 342, 253, 812, 155, 1332, 820, 602, 1081, 2756, 3422, 3660, 4422, 2485, 657, 3081, 6806, 979, 4656, 10100, 5253, 11342, 3924, 3164, 889, 17030, 9316, 19182, 22052, 2265, 8164, 26406, 13861, 29756, 31862, 32580, 18145, 18528, 38612

Offset: 2

Felix Fröhlich, Jun 14 2014

nonn

p=prime(n) is in A001220 if and only if a(n) is equal to A014664(n). So far this is known to hold only for p=1093 and p=3511.
This happens for n=183 and 490, that is for p=prime(183)=1093 and p=prime(490)=3511, with values 364 and 1755 (see b-files). - Michel Marcus, Jun 29 2014
If 2^q-1 is p=prime(n), i.e., for n in A016027, then a(n)=pq and lpf(2^a(n)-1)=p. - Thomas Ordowski, Feb 04 2019

Charles R Greathouse IV, Table of n, a(n) for n = 2..10000

Cf. A001220, A014664.

seq(numtheory:-order(2, ithprime(i)^2), i=2..1000); # Robert Israel, Jul 08 2014

Table[MultiplicativeOrder[2, Prime[n]^2], {n, 2, 100}] (* Jean-François Alcover, Jul 08 2014 *)

forprime(p=3, 10^2, print1(znorder(Mod(2, p^2)), ", "))

a(n) = prime(n)*A014664(n) for all odd primes that are not Wieferich. - Thomas Ordowski, Feb 04 2019

A244550 a(n) = first odd Wieferich prime to base a(n-1) for n > 1, with a(1) = 2.

Original entry on oeis.org

2, 1093, 5, 20771, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71

Offset: 1

Felix Fröhlich, Jun 29 2014

a(2) = 1093 since 1093 is the smallest odd Wieferich prime to base 2.
a(3) = 5 since 5 is the smallest odd Wieferich prime to base 1093.
Subsequence starting at a(5) is periodic with period 3, repeating the terms {3, 11, 71}.
Do values for a(1) exist such that the resulting sequence does not eventually become periodic?
The following table lists the values for a(1) and the resulting cycles those values produce. An entry of the form x-y in first column means all terms from x up to and including y reach the corresponding cycle. An entry of the form {t_1, t_2, t_3, ..., t_n} in second column means the listed terms form a repeating cycle. Entries in second column without curly braces mean the listed terms are reached in order and the term following the last listed term is unknown. A question mark means no further terms have been found in the resulting trajectory of a(1).
a(1)             | resulting terms
----------------------------------
2-13, 15-20,     | {3, 11, 71}
22-28, 30-40,    |
42-46, 48-59,    |
62-71, 73-82,    |
84-87, 89-118,   |
120-132, 134-136,|
138, 140-155,    |
157-185, 188,    |
190-195, 197-199 |
                 |
14, 41, 60, 137, | 29
196              |
                 |
21, 29, 47, 61,  | ?
72, 139, 186-187 |
                 |
83               | {4871, 83}
                 |
88               | 2535619637, 139
                 |
119              | 1741
                 |
133              | 5277179
                 |
156              | 347
                 |
189              | 1847
                 |
Notes
------
The terms of the cycle reached from 83 correspond to A124121(4) and A124122(4), so those terms form a double Wieferich prime pair.

R. Fischer, Thema: Fermatquotient B^(P-1) == 1 (mod P^2)
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A001220, A124121, A124122, A174422, A244546.

[2, 1093, 5, 20771] cat &cat [[3, 11, 71]^^30]; // Wesley Ivan Hurt, Jun 30 2016

2,1093,5,20771,seq(op([3, 11, 71]), n=5..50); # Wesley Ivan Hurt, Jun 30 2016

Join[{2, 1093, 5, 20771},LinearRecurrence[{0, 0, 1},{3, 11, 71},66]] (* Ray Chandler, Aug 25 2015 *)

i=0; a=2; print1(a, ", "); while(i<100, forprime(p=2, 10^6, if(Mod(a, p^2)^(p-1)==1 && p%2!=0, print1(p, ", "); i++; a=p; break({n=1}))))

From Wesley Ivan Hurt, Jun 30 2016: (Start)
G.f.: x*(2+1093*x+5*x^2+20769*x^3-1090*x^4+6*x^5-20700*x^6) / (1-x^3).
a(n) = a(n-3) for n>7.
a(n) = (85 - 52*cos(2*n*Pi/3) + 68*sqrt(3)*sin(2*n*Pi/3))/3 for n>4. (End)

cp's OEIS Frontend

A244752 Square array read by antidiagonals in which rows are indexed by composite numbers w and row w gives n such that n^(w-1) == 1 (mod w^2).

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A084653 Pseudoprimes whose prime factors do not divide any smaller pseudoprime.

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A178871 2nd Wieferich prime base prime(n).

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A182297 Wieferich numbers (2): positive odd integers q such that q and (2^A002326((q-1)/2)-1)/q are not relatively prime.

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A238736 Balancing Wieferich primes: primes p that divide their Pell quotients, where the Pell quotient of p is A000129(p - (2/p))/p and (2/p) is a Jacobi symbol.

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A270096 Smallest m such that 2^m == 2^n (mod n).

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A163209 Catalan pseudoprimes: odd composite integers n=2*m+1 satisfying A000108(m) == (-1)^m * 2 (mod n).

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A163209 Catalan pseudoprimes: odd composite integers n=2m+1 satisfying A000108(m) == (-1)^m 2 (mod n).