A001917 -id:A001917 - cp's OEIS frontend

A152598 Records in A001917.

1, 2, 6, 8, 18, 24, 31, 38, 72, 105, 129, 630, 1285, 1542, 2048, 3112, 3512, 7710, 7760, 9728, 27594, 48834, 60787, 104694, 140896, 282224, 384800, 683720, 1205998, 1240672, 1407592, 4012472, 4429250, 4628480, 16657248, 69273666, 107700228, 254290896, 319233186

Offset: 1

Klaus Brockhaus, Dec 09 2008

nonn

			First few terms of A001917 are 1, 1, 2, 1, 1, 2, 1, 2, 1, 6, so a(1) to a(3) are 1, 2, 6.

Amiram Eldar, Table of n, a(n) for n = 1..47

Cf. A001917 ((p-1)/x, where p = prime(n) and x = smallest positive integer such that 2^x == 1 mod p), A152597 (where records occur in A001917).

R:=[]; r:=0; for n in [2..100000] do p:=NthPrime(n); a:=(p-1)/Modorder(2, p); if r lt a then r:=a; Append(~R,a); end if; end for; print R;

ord[n_]:=Module[{x=1},While[PowerMod[2,x,n]!=1,x++];(n-1)/x];DeleteDuplicates[ ord/@Prime[Range[2,10^5]],GreaterEqual] (* The program generates the first 21 terms of the sequence. *) (* Harvey P. Dale, Oct 09 2022 *)

a(n) = A001917(A152597(n)). - Amiram Eldar, Nov 16 2023

More terms from Vassilis Papadimitriou, Mar 06 2010
More terms from Vassilis Papadimitriou, Mar 19 2010
a(37) from Amiram Eldar, Mar 08 2019
a(38)-a(39) from Amiram Eldar, Nov 16 2023

A152597 Where records occur in A001917.

Original entry on oeis.org

2, 4, 11, 21, 31, 110, 124, 185, 279, 399, 716, 1028, 4552, 6207, 6543, 11424, 11557, 12251, 16199, 23043, 43390, 155798, 203095, 457523, 699782, 865318, 1294026, 2918851, 5635889, 6459777, 8999147, 9213126, 22383796, 28194383, 32131750, 105097565, 404165580

Offset: 1

Klaus Brockhaus, Dec 09 2008

nonn

			First few terms of A001917 (has offset 2) are 1, 1, 2, 1, 1, 2, 1, 2, 1, 6, so a(1) to a(3) are 2, 4, 11.

Amiram Eldar, Table of n, a(n) for n = 1..47

Cf. A001917 ((p-1)/x, where p = prime(n) and x = smallest positive integer such that 2^x == 1 mod p), A152598 (records in A001917), A000720, A226216.

W:=[]; r:=0; for n in [2..100000] do p:=NthPrime(n); a:=(p-1)/Modorder(2, p); if r lt a then r:=a; Append(~W,n); end if; end for; print W;

from itertools import islice
from sympy import nextprime, n_order
def agen():
    record, v, p = -1, 1, 3
    while True:
        if v > record: record = v; yield record
        v, p = (p-1)//n_order(2, p), nextprime(p)
print(list(islice(agen(), 20))) # Michael S. Branicky, Oct 09 2022

a(n) = A000720(A226216(n)). - Amiram Eldar, Nov 16 2023

a(27)-a(37) from Amiram Eldar, Mar 08 2019

A001220 Wieferich primes: primes p such that p^2 divides 2^(p-1) - 1.

Original entry on oeis.org

1093, 3511

Offset: 1

N. J. A. Sloane

Sequence is believed to be infinite.
Joseph Silverman showed that the abc-conjecture implies that there are infinitely many primes which are not in the sequence. - Benoit Cloitre, Jan 09 2003
Graves and Murty (2013) improved Silverman's result by showing that for any fixed k > 1, the abc-conjecture implies that there are infinitely many primes == 1 (mod k) which are not in the sequence. - Jonathan Sondow, Jan 21 2013
The squares of these numbers are Fermat pseudoprimes to base 2 (A001567) and Catalan pseudoprimes (A163209). - T. D. Noe, May 22 2003
Primes p that divide the numerator of the harmonic number H((p-1)/2); that is, p divides A001008((p-1)/2). - T. D. Noe, Mar 31 2004
In a 1977 paper, Wells Johnson, citing a suggestion from Lawrence Washington, pointed out the repetitions in the binary representations of the numbers which are one less than the two known Wieferich primes; i.e., 1092 = 10001000100 (base 2); 3510 = 110110110110 (base 2). It is perhaps worth remarking that 1092 = 444 (base 16) and 3510 = 6666 (base 8), so that these numbers are small multiples of repunits in the respective bases. Whether this is mathematically significant does not appear to be known. - John Blythe Dobson, Sep 29 2007
A002326((a(n)^2 - 1)/2) = A002326((a(n)-1)/2). - Vladimir Shevelev, Jul 09 2008, Aug 24 2008
It is believed that p^2 does not divide 3^(p-1) - 1 if p = a(n). This is true for n = 1 and 2. See A178815, A178844, A178900, and Ostafe-Shparlinski (2010) Section 1.1. - Jonathan Sondow, Jun 29 2010
These primes also divide the numerator of the harmonic number H(floor((p-1)/4)). - H. Eskandari (hamid.r.eskandari(AT)gmail.com), Sep 28 2010
1093 and 3511 are prime numbers p satisfying congruence 429327^(p-1) == 1 (mod p^2). Why? - Arkadiusz Wesolowski, Apr 07 2011. Such bases are listed in A247208. - Max Alekseyev, Nov 25 2014. See A269798 for all such bases, prime and composite, that are not powers of 2. - Felix Fröhlich, Apr 07 2018
A196202(A049084(a(1))) = A196202(A049084(a(2))) = 1. - Reinhard Zumkeller, Sep 29 2011
If q is prime and q^2 divides a prime-exponent Mersenne number, then q must be a Wieferich prime. Neither of the two known Wieferich primes divide Mersenne numbers. See Will Edgington's Mersenne page in the links below. - Daran Gill, Apr 04 2013
There are no other terms below 4.97*10^17 as established by PrimeGrid (see link below). - Max Alekseyev, Nov 20 2015. The search was done via PrimeGrid's PRPNet and the results were not double-checked. Because of the unreliability of the testing, the search was suspended in May 2017 (cf. Goetz, 2017). - Felix Fröhlich, Apr 01 2018. On Nov 28 2020, PrimeGrid has resumed the search (cf. Reggie, 2020). - Felix Fröhlich, Nov 29 2020. As of Dec 29 2022, PrimeGrid has completed the search to 2^64 (about 1.8 * 10^19) and has no plans to continue further. - Charles R Greathouse IV, Sep 24 2024
Are there other primes q >= p such that q^2 divides 2^(p-1)-1, where p is a prime? - Thomas Ordowski, Nov 22 2014. Any such q must be a Wieferich prime. - Max Alekseyev, Nov 25 2014
Primes p such that p^2 divides 2^r - 1 for some r, 0 < r < p. - Thomas Ordowski, Nov 28 2014, corrected by Max Alekseyev, Nov 28 2014
For some reason, both p=a(1) and p=a(2) also have more bases b with 1 < b < p that make b^(p-1) == 1 (mod p^2) than any smaller prime p; in other words, a(1) and a(2) belong to A248865. - Jeppe Stig Nielsen, Jul 28 2015
Let r_1, r_2, r_3, ..., r_i be the set of roots of the polynomial X^((p-1)/2) - (p-3)! * X^((p-3)/2) - (p-5)! * X^((p-5)/2) - ... - 1. Then p is a Wieferich prime iff p divides sum{k=1, p}(r_k^((p-1)/2)) (see Example 2 in Jakubec, 1994). - Felix Fröhlich, May 27 2016
Arthur Wieferich showed that if p is not a term of this sequence, then the First Case of Fermat's Last Theorem has no solution in x, y and z for prime exponent p (cf. Wieferich, 1909). - Felix Fröhlich, May 27 2016
Let U_n(P, Q) be a Lucas sequence of the first kind, let e be the Legendre symbol (D/p) and let p be a prime not dividing 2QD, where D = P^2 - 4*Q. Then a prime p such that U_(p-e) == 0 (mod p^2) is called a "Lucas-Wieferich prime associated to the pair (P, Q)". Wieferich primes are those Lucas-Wieferich primes that are associated to the pair (3, 2) (cf. McIntosh, Roettger, 2007, p. 2088). - Felix Fröhlich, May 27 2016
Any repeated prime factor of a term of A000215 is a term of this sequence. Thus, if there exist infinitely many Fermat numbers that are not squarefree, then this sequence is infinite, since no two Fermat numbers share a common factor. - Felix Fröhlich, May 27 2016
If the Diophantine equation p^x - 2^y = d has more than one solution in positive integers (x, y), with (p, d) not being one of the pairs (3, 1), (3, -5), (3, -13) or (5, -3), then p is a term of this sequence (cf. Scott, Styer, 2004, Corollary to Theorem 2). - Felix Fröhlich, Jun 18 2016
Odd primes p such that Chi_(D_0)(p) != 1 and Lambda_p(Q(sqrt(D_0))) != 1, where D_0 < 0 is the fundamental discriminant of the imaginary quadratic field Q(sqrt(1-p^2)) and Chi and Lambda are Iwasawa invariants (cf. Byeon, 2006, Proposition 1 (i)). - Felix Fröhlich, Jun 25 2016
If q is an odd prime, k, p are primes with p = 2*k+1, k == 3 (mod 4), p == -1 (mod q) and p =/= -1 (mod q^3) (Jakubec, 1998, Corollary 2 gives p == -5 (mod q) and p =/= -5 (mod q^3)) with the multiplicative order of q modulo k = (k-1)/2 and q dividing the class number of the real cyclotomic field Q(Zeta_p + (Zeta_p)^(-1)), then q is a term of this sequence (cf. Jakubec, 1995, Theorem 1). - Felix Fröhlich, Jun 25 2016
From Felix Fröhlich, Aug 06 2016: (Start)
Primes p such that p-1 is in A240719.
Prime terms of A077816 (cf. Agoh, Dilcher, Skula, 1997, Corollary 5.9).
p = prime(n) is in the sequence iff T(2, n) > 1, where T = A258045.
p = prime(n) is in the sequence iff an integer k exists such that T(n, k) = 2, where T = A258787. (End)
Conjecture: an integer n > 1 such that n^2 divides 2^(n-1)-1 must be a Wieferich prime. - Thomas Ordowski, Dec 21 2016
The above conjecture is equivalent to the statement that no "Wieferich pseudoprimes" (WPSPs) exist. While base-b WPSPs are known to exist for several bases b > 1 other than 2 (see for example A244752), no base-2 WPSPs are known. Since two necessary conditions for a composite to be a base-2 WPSP are that, both, it is a base-2 Fermat pseudoprime (A001567) and all its prime factors are Wieferich primes (cf. A270833), as shown in the comments in A240719, it seems that the first base-2 WPSP, if it exists, is probably very large. This appears to be supported by the guess that the properties of a composite to be a term of A001567 and of A270833 are "independent" of each other and by the observation that the scatterplot of A256517 seems to become "less dense" at the x-axis parallel line y = 2 for increasing n. It has been suggested in the literature that there could be asymptotically about log(log(x)) Wieferich primes below some number x, which is a function that grows to infinity, but does so very slowly. Considering the above constraints, the number of WPSPs may grow even more slowly, suggesting any such number, should it exist, probably lies far beyond any bound a brute-force search could reach in the forseeable future. Therefore I guess that the conjecture may be false, but a disproof or the discovery of a counterexample are probably extraordinarily difficult problems. - Felix Fröhlich, Jan 18 2019
Named after the German mathematician Arthur Josef Alwin Wieferich (1884-1954). a(1) = 1093 was found by Waldemar Meissner in 1913. a(2) = 3511 was found by N. G. W. H. Beeger in 1922. - Amiram Eldar, Jun 05 2021
From Jianing Song, Jun 21 2025: (Start)
The ring of integers of Q(2^(1/k)) is Z[2^(1/k)] if and only if k does not have a prime factor in this sequence (k is in A342390). See Theorem 5.3 of the paper of Keith Conrad. For example, we have:
  (1 + 2^(364/1093) + 2^(2*364/1093) + ... + 2^(1092*364/1093))/1093 is an algebraic integer, but it is not in Z[2^(1/1093)];
  (1 + 2^(1755/3511) + 2^(2*1755/3511) + ... + 2^(3510*1755/3511))/3511 is an algebraic integer, but it is not in Z[2^(1/3511)]. (End)

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Cf. similar primes related to the first case of Fermat's last theorem: A007540, A088164.
Sequences "primes p such that p^2 divides X^(p-1)-1": A014127 (X=3), A123692 (X=5), A212583 (X=6), A123693 (X=7), A045616 (X=10), A111027 (X=12), A128667 (X=13), A234810 (X=14), A242741 (X=15), A128668 (X=17), A244260 (X=18), A090968 (X=19), A242982 (X=20), A298951 (X=22), A128669 (X=23), A306255 (X=26), A306256 (X=30).
Cf. A001567, A002323, A077816, A001008, A039951, A049094, A126196, A126197, A178815, A178844, A178871, A178900, A246503, A247208, A269798.

Filtered([1..50000],p->IsPrime(p) and (2^(p-1)-1) mod p^2 =0); # Muniru A Asiru, Apr 03 2018

import Data.List (elemIndices)
a001220 n = a001220_list !! (n-1)
a001220_list = map (a000040 . (+ 1)) $ elemIndices 1 a196202_list
-- Reinhard Zumkeller, Sep 29 2011

[p : p in PrimesUpTo(310000) | IsZero((2^(p-1) - 1) mod (p^2))]; // Vincenzo Librandi, Jan 19 2019

wieferich := proc (n) local nsq, remain, bin, char: if (not isprime(n)) then RETURN("not prime") fi: nsq := n^2: remain := 2: bin := convert(convert(n-1, binary),string): remain := (remain * 2) mod nsq: bin := substring(bin,2..length(bin)): while (length(bin) > 1) do: char := substring(bin,1..1): if char = "1" then remain := (remain * 2) mod nsq fi: remain := (remain^2) mod nsq: bin := substring(bin,2..length(bin)): od: if (bin = "1") then remain := (remain * 2) mod nsq fi: if remain = 1 then RETURN ("Wieferich prime") fi: RETURN ("non-Wieferich prime"): end: # Ulrich Schimke (ulrschimke(AT)aol.com), Nov 01 2001

Select[Prime[Range[50000]],Divisible[2^(#-1)-1,#^2]&]  (* Harvey P. Dale, Apr 23 2011 *)
Select[Prime[Range[50000]],PowerMod[2,#-1,#^2]==1&] (* Harvey P. Dale, May 25 2016 *)

N=10^4; default(primelimit,N);
forprime(n=2,N,if(Mod(2,n^2)^(n-1)==1,print1(n,", ")));
\\ Joerg Arndt, May 01 2013

from sympy import prime
from gmpy2 import powmod
A001220_list = [p for p in (prime(n) for n in range(1,10**7)) if powmod(2,p-1,p*p) == 1]
# Chai Wah Wu, Dec 03 2014

(A178815(A000720(p))^(p-1) - 1) mod p^2 = A178900(n), where p = a(n). - Jonathan Sondow, Jun 29 2010

Odd primes p such that A002326((p^2-1)/2) = A002326((p-1)/2). See A182297. - Thomas Ordowski, Feb 04 2014

A001122 Primes with primitive root 2.

Original entry on oeis.org

3, 5, 11, 13, 19, 29, 37, 53, 59, 61, 67, 83, 101, 107, 131, 139, 149, 163, 173, 179, 181, 197, 211, 227, 269, 293, 317, 347, 349, 373, 379, 389, 419, 421, 443, 461, 467, 491, 509, 523, 541, 547, 557, 563, 587, 613, 619, 653, 659, 661, 677, 701, 709, 757, 773, 787, 797

Offset: 1

N. J. A. Sloane

Artin conjectured that this sequence is infinite.
Conjecture: sequence contains infinitely many pairs of twin primes. - Benoit Cloitre, May 08 2003
Pieter Moree writes (Oct 20 2004): Assuming the Generalized Riemann Hypothesis, it can be shown that the density of primes p such that a prescribed integer g has order (p-1)/t, with t fixed, exists and, moreover, it can be computed. This density will be a rational number times the so-called Artin constant. For 2 and 10 the density of primitive roots is A, the Artin constant itself.
It seems that this sequence consists of A050229 \ {1,2}.
Primes p such that 1/p, when written in base 2, has period p-1, which is the greatest period possible for any integer.
Positive integer 2*m-1 is in the sequence iff A179382(m)=m-1. - Vladimir Shevelev, Jul 14 2010
These are the odd primes p for which the polynomial 1+x+x^2+...+x^(p-1) is irreducible over GF(2). - V. Raman, Sep 17 2012 [Corrected by N. J. A. Sloane, Oct 17 2012]
Prime(n) is in the sequence if (and conjecturally only if) A133954(n) = prime(n). - Vladimir Shevelev, Aug 30 2013
Pollack shows that, on the GRH, that there is some C such that a(n+1) - a(n) < C infinitely often (in fact, 1 can be replaced by any positive integer). Further, for any m, a(n), a(n+1), ..., a(n+m) are consecutive primes infinitely often. - Charles R Greathouse IV, Jan 05 2015
From Jianing Song, Apr 27 2019: (Start)
All terms are congruent to 3 or 5 modulo 8. If we define
  Pi(N,b) = # {p prime, p <= N, p == b (mod 8)};
     Q(N) = # {p prime, p <= N, p in this sequence},
then by Artin's conjecture, Q(N) ~ C*N/log(N) ~ 2*C*(Pi(N,3) + Pi(N,5)), where C = A005596 is Artin's constant.
Conjecture: if we further define
   Q(N,b) = # {p prime, p <= N, p == b (mod 8), p in this sequence},
then we have:
   Q(N,3) ~ (1/2)*Q(N) ~ C*Pi(N,3);
   Q(N,5) ~ (1/2)*Q(N) ~ C*Pi(N,5). (End)
Conjecture: for a prime p > 5, p has primitive root 2 iff p == +-3 (mod 8) divides 2^k + 3 for some k < p - 1 and divides 2^m + 5 for some m < p - 1. It seems that all primes of the form 2^k + 3 for k <> 2 (A057732) have primitive root 2. - Thomas Ordowski, Nov 27 2023

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D. Shanks, Solved and Unsolved Problems in Number Theory, 2nd. ed., Chelsea, 1978, p. 81.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Antti Karttunen, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Joerg Arndt, Matters Computational (The Fxtbook), pp. 876-878.
Richard Bartels, Generalized Loewy Length of Cohen-Macaulay Local and Graded Rings, arXiv:2308.14932 [math.AC], 2023. See p. 10.
J. Conde, M. Miller, J. M. Miret, and K. Saurav, On the Nonexistence of Almost Moore Digraphs of Degree Four and Five, Mathematics in Computer Science, 9(2) (2015), 145-149.
Jonathan Detchart and Jérôme Lacan, Improving the coding speed of erasure codes with polynomial ring transforms, arXiv:1709.00178 [cs.IT], 2017.
K. Dilcher and L. Ericksen, Reducibility and irreducibility of Stern (0, 1)-polynomials, Communications in Mathematics, 22 (2014), 77-102.
Gabriele Fici and Estéban Gabory, Generalized De Bruijn Words, Invertible Necklaces, and the Burrows-Wheeler Transform, arXiv:2502.12844 [math.CO], 2025. See p. 12.
R. Gupta and M. R. Murty, A remark on Artin's conjecture, Invent. Math. 78 (1984), 127-230.
C. Hooley, On Artin's conjecture, J. Reine Angewandte Math., 225 (1967), 209-220.
Florian Ingels, Anaïs Denis, and Bastien Cazaux, Decomposing Words for Enhanced Compression: Exploring the Number of Runs in the Extended Burrows-Wheeler Transform, arXiv:2506.04926 [cs.DS], 2025. See p. 15.
Robert Jackson, Dmitriy Rumynin and Oleg V. Zaboronski, An approach to RAID-6 based on cyclic groups, Applied Mathematics & Information Sciences, 5(2) (2011), 148-170.
Jonas Kaiser, On the relationship between the Collatz conjecture and Mersenne prime numbers, arXiv:1608.00862 [math.GM], 2016.
Sihem Mesnager and Jean-Pierre Flori, A note on hyper-bent functions via Dillon-like exponents, IACR, Report 2012/033, 2012.
F. Pillichshammer, Bounds for the quality parameter of digital shift nets over Z_2, Finite Fields Applic., 8 (2002), 444-454.
Pieter Moree, Artin's primitive root conjecture-a survey, arXiv:math/0412262 [math.NT], 2004-2012.
Paul Pollack, Bounded gaps between primes with a given primitive root, arXiv:1404.4007 [math.NT], 2014.
Vladimir Shevelev, On the Newman sum over multiples of a prime with a primitive or semiprimitive root 2, arXiv:0710.1354 [math.NT], 2007.
Stephan Tornier, Groups Acting on Trees With Prescribed Local Action, arXiv:2002.09876 [math.GR], 2020.
Qifu Tyler Sun, Hanqi Tang, Zongpeng Li, Xiaolong Yang, and Keping Long, Circular-shift Linear Network Codes with Arbitrary Odd Block Lengths, arXiv:1806.04635 [cs.IT], 2018.
Eric Weisstein's World of Mathematics, Artin's constant.
Wikipedia, Artin's conjecture on primitive roots.
Index entries for sequences related to Artin's conjecture
Index entries for primes by primitive root

Cf. A001123, A001913, A001917, A005596 (Artin's constant), A050229, A071642, A127209, A163782, A292270.
Cf. A002326 for the multiplicative order of 2 mod 2n+1. (Alternatively, the least positive value of m such that 2n+1 divides 2^m-1).
Cf. A216838 (Odd primes for which 2 is not a primitive root).

Select[ Prime@Range@200, PrimitiveRoot@# == 2 &] (* Robert G. Wilson v, May 11 2001 *)
pr = 2; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == # - 1 &] (* N. J. A. Sloane, Jun 01 2010 *)

forprime(p=3, 1000, if(znorder(Mod(2, p))==(p-1), print1(p,", "))); \\ [corrected by Michel Marcus, Oct 08 2014]

from itertools import islice
from sympy import nextprime, is_primitive_root
def A001122_gen(): # generator of terms
    p = 2
    while (p:=nextprime(p)):
        if is_primitive_root(2,p):
            yield p
A001122_list = list(islice(A001122_gen(),30)) # Chai Wah Wu, Feb 13 2023

Delta(a(n),2^a(n)*x) = a(n)*Delta(a(n),2*x), where Delta(k,x) is the difference between numbers of evil(A001969) and odious(A000069) integers divisible by k in interval [0,x). - Vladimir Shevelev, Aug 30 2013

For n >= 2, a(n) = 1 + 2*A163782(n-1). - Antti Karttunen, Oct 07 2017

A014664 Order of 2 modulo the n-th prime.

Original entry on oeis.org

2, 4, 3, 10, 12, 8, 18, 11, 28, 5, 36, 20, 14, 23, 52, 58, 60, 66, 35, 9, 39, 82, 11, 48, 100, 51, 106, 36, 28, 7, 130, 68, 138, 148, 15, 52, 162, 83, 172, 178, 180, 95, 96, 196, 99, 210, 37, 226, 76, 29, 119, 24, 50, 16, 131, 268, 135, 92, 70, 94, 292, 102, 155, 156, 316

Offset: 2

N. J. A. Sloane

In other words, a(n), n >= 2, is the least k such that prime(n) divides 2^k-1.
Concerning the complexity of computing this sequence, see for example Bach and Shallit, p. 115, exercise 8.
Also A002326((p_n-1)/2). Conjecture: If p_n is not a Wieferich prime (1093, 3511, ...) then A002326(((p_n)^k-1)/2) = a(n)*(p_n)^(k-1). - Vladimir Shevelev, May 26 2008
If for distinct i,j,...,k we have a(i)=a(j)=...=a(k) then the number N = p_i*p_j*...*p_k is in A001262 and moreover A137576((N-1)/2) = N. For example, a(16)=a(37)=a(255)=52. Therefore we could take N = p_16*p_37*p_255 = 53*157*1613 = 13421773. - Vladimir Shevelev, Jun 14 2008
Also degree of the irreducible polynomial factors for the polynomial (x^p+1)/(x+1) over GF(2), where p is the n-th prime. - V. Raman, Oct 04 2012
Is this the same as the smallest k > 1 not already in the sequence such that p = prime(n) is a factor of 2^k-1 (A270600)? If the answer is yes, is the sequence a permutation of the positive integers > 1? - Felix Fröhlich, Feb 21 2016. Answer: No, it is easy to prove that 6 is missing and obviously 11 appears twice. - N. J. A. Sloane, Feb 21 2016
pi(A112927(m)) is the index at which a given number m first appears in this sequence. - M. F. Hasler, Feb 21 2016

			2^2 == 1 (mod 3) and so a(2) = 2;
2^4 == 1 (mod 5) and so a(3) = 4;
2^3 == 1 (mod 7) and so a(4) = 3;
2^10 == 1 (mod 11) and so a(5) = 10; etc.
[Conway & Guy, p. 166]: Referring to the work of Euler, 1/13 in base 2 = 0.000100111011...; (cycle length of 12). - _Gary W. Adamson_, Aug 22 2009

E. Bach and Jeffrey Shallit, Algorithmic Number Theory, I.
Albert H. Beiler, "Recreations in the Theory of Numbers", Dover, 1966; Table 48, page 98, "Exponents to Which a Belongs, MOD p and MOD p^n.
John H. Conway and Richard Guy, "The Book of Numbers", Springer-Verlag, 1996; p. 166: "How does the Cycle Length Change with the Base?". [From Gary W. Adamson, Aug 22 2009]
S. K. Sehgal, Group rings, pp. 455-541 in Handbook of Algebra, Vol. 3, Elsevier, 2003; see p. 493.

Michael De Vlieger, Table of n, a(n) for n = 2..10001 (terms 2..1001 from Nick Hobson, terms 1002..5001 from Zak Seidov)
Gary W. Adamson, Further comments.
O. N. Karpenkov, On examples of difference operators for {0,1}-valued functions over finite sets, Funct. Anal. Other Math. 1 (2006), 175-180.
O. N. Karpenkov, On examples of difference operators for {0,1}-valued functions over finite sets, arXiv:math/0611940 [math.CO], 2006.

Cf. A002326 (order of 2 mod 2n+1), A001122 (full reptend primes in base 2), A065941, A112927.

In other bases: A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.

P:=Filtered([1..350],IsPrime);; a:=List([2..Length(P)],n->OrderMod(2,P[n]));; Print(a); # Muniru A Asiru, Jan 29 2019

with(numtheory): [ seq(order(2,ithprime(n)), n=2..60) ];

Reap[Do[p=Prime[i];Do[If[PowerMod[2,k,p]==1,Print[{i,k}];Sow[{i,k}];Goto[ni]],{k,1,10^6}];Label[ni],{i,2,5001}]][[2,1]] (* Zak Seidov, Jan 26 2009 *)
Table[MultiplicativeOrder[2, Prime[n]], {n, 2, 70}] (* Jean-François Alcover, Dec 10 2015 *)

a(n)=if(n<0,0,k=1;while((2^k-1)%prime(n)>0,k++);k)

A014664(n)=znorder(Mod(2, prime(n))) \\ Nick Hobson, Jan 08 2007, edited by M. F. Hasler, Feb 21 2016

forprime(p=3, 800, print(factormod((x^p+1)/(x+1), 2, 1)[1, 1])) \\ V. Raman, Oct 04 2012

from sympy import n_order, prime
def A014664(n): return n_order(2,prime(n)) # Chai Wah Wu, Nov 09 2023

a(n) = (A000040(n)-1)/A001917(n); a(A072190(n)) = A001122(n) - 1. - Benoit Cloitre, Jun 06 2004

More terms from Benoit Cloitre, Apr 11 2003

A006694 Number of cyclotomic cosets of 2 mod 2n+1.

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 1, 4, 2, 1, 5, 2, 2, 3, 1, 6, 4, 5, 1, 4, 2, 3, 7, 2, 4, 7, 1, 4, 4, 1, 1, 12, 6, 1, 5, 2, 8, 7, 5, 2, 4, 1, 11, 4, 8, 9, 13, 4, 2, 7, 1, 2, 14, 1, 3, 4, 4, 5, 11, 8, 2, 7, 3, 18, 10, 1, 9, 10, 2, 1, 5, 4, 6, 9, 1, 10, 12, 13, 3, 4, 8, 1, 13, 2, 2, 11, 1, 8, 4, 1, 1, 4, 6, 7, 19, 2, 2, 19, 1, 2

Offset: 0

N. J. A. Sloane, Sep 25 2001

a(0) = 0 by convention.
The number of cycles in permutations constructed from siteswap juggling patterns 1, 123, 12345, 1234567, etc., i.e., the number of ball orbits in such patterns minus one.
Also the number of irreducible polynomial factors of the polynomial (x^(2n+1) - 1) / (x - 1) over GF(2). - V. Raman, Oct 04 2012
Also, a(n) is the number of cycles of the Josephus permutation for n elements and a count of 2. For n >= 1, the Josephus permutation is given by the n-th row of A321298. See Knuth 1997 (exercise 1.3.3-29). - Pontus von Brömssen, Sep 18 2022

			Mod 15 there are 4 cosets: {1, 2, 4, 8}, {3, 6, 12, 9}, {5, 10}, {7, 14, 13, 11}, so a(7) = 4. Mod 13 there is only one coset: {1, 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7}, so a(6) = 1.

Donald E. Knuth, The Art of Computer Programming, Vol. 1, 3rd edition, Addison-Wesley, 1997.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1977, pp. 104-105.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ray Chandler, Table of n, a(n) for n = 0..10000
Christopher Adler and Jean-Paul Allouche, Finite self-similar sequences, permutation cycles, and music composition, Journal of Mathematics and the Arts, 16:3, 244-261, (2022).
J.-P. Allouche, Suites infinies à répétitions bornées, Séminaire de Théorie des Nombres de Bordeaux, 20 (13 April, 1984), 1-11.
J.-P. Allouche, Suites infinies à répétitions bornées, Séminaire de Théorie des Nombres de Bordeaux, 20 (13 April, 1984), 1-11.
Jean-Paul Allouche, Manon Stipulanti, and Jia-Yan Yao, Doubling modulo odd integers, generalizations, and unexpected occurrences, arXiv:2504.17564 [math.NT], 2025.
Michael Assis, Folding Pi, arXiv:2403.09277 [math.NT], 2024.

Cf. A000010, A000374 (number of factors of x^n - 1 over GF(2)), A002326 (order of 2 mod 2n+1), A037226, A064286, A064287, A081844, A139767, A321298.
A001917 gives cycle counts of such permutations constructed only for odd primes.
Second column of A357217.

with(group); with(numtheory); gen_rss_perm := proc(n) local a, i; a := []; for i from 1 to n do a := [op(a), ((2*i) mod (n+1))]; od; RETURN(a); end; count_of_disjcyc_seq := [seq(nops(convert(gen_rss_perm(2*j),'disjcyc')),j=0..)];

Needs["Combinatorica`"]; f[n_] := Length[ToCycles[Mod[2Range[2n], 2n + 1]]]; Table[f[n], {n, 0, 100}] (* Ray Chandler, Apr 25 2008 *)
f[n_] := Length[FactorList[x^(2n + 1) - 1, Modulus -> 2]] - 2; Table[f[n], {n, 0, 100}] (* Ray Chandler, Apr 25 2008 *)
a[n_] := Sum[ EulerPhi[d] / MultiplicativeOrder[2, d], {d, Divisors[2n + 1]}] - 1; Table[a[n], {n, 0, 99}] (* Jean-François Alcover, Dec 14 2011, after Joerg Arndt *)

a(n)=sumdiv(2*n+1, d, eulerphi(d)/znorder(Mod(2, d))) - 1; /* cf. A081844 */
vector(122, n, a(n-1)) \\ Joerg Arndt, Jan 18 2011

from sympy import totient, n_order, divisors
def A006694(n): return sum(totient(d)//n_order(2,d) for d in divisors((n+1<<1)-1,generator=True) if d>1) # Chai Wah Wu, Apr 09 2024

Conjecture: a((3^n-1)/2) = n. - Vladimir Shevelev, May 26 2008 [This is correct. 2*((3^n-1)/2) + 1 = 3^n and the polynomial (x^(3^n) - 1) / (x - 1) factors over GF(2) into Product_{k=0..n-1} x^(2*3^k) + x^(3^k) + 1. - Joerg Arndt, Apr 01 2019]
a(n) = A081844(n) - 1.
a(n) = A064286(n) + 2*A064287(n).
From Vladimir Shevelev, Jan 19 2011: (Start)
1) a(n)=A037226(n) iff 2n+1 is prime;
2) The only case when a(n) < A037226(n) is n=0;
3) If {C_i}, i=1..a(n), is the set of all cyclotomic cosets of 2 mod (2n+1), then lcm(|C_1|, ..., |C_{a(n)}|) = A002326(n). (End)
a(n) = A000374(2*n + 1) - 1. - Joerg Arndt, Apr 01 2019
a(n) = (Sum_{d|(2n+1)} phi(d)/ord(2,d)) - 1, where phi = A000010 and ord(2,d) is the multiplicative order of 2 modulo d. - Jianing Song, Nov 13 2021

Additional comments from Antti Karttunen, Jan 05 2000
Extended by Ray Chandler, Apr 25 2008
Edited by N. J. A. Sloane, Apr 27 2008 at the suggestion of Ray Chandler

A007663 Fermat quotients: (2^(p-1)-1)/p, where p=prime(n).

Original entry on oeis.org

1, 3, 9, 93, 315, 3855, 13797, 182361, 9256395, 34636833, 1857283155, 26817356775, 102280151421, 1497207322929, 84973577874915, 4885260612740877, 18900352534538475, 1101298153654301589, 16628050996019877513, 64689951820132126215, 3825714619033636628817

Offset: 2

N. J. A. Sloane, Sep 19 1994

The only terms that are squares are a(2) = 1 and a(4) = 9. - Nick Hobson, May 20 2007
From Jonathan Sondow, Jul 19 2010: (Start)
a(n) == 0 (mod 3) if n > 2, since p = prime(n) > 3
and 0 = (-1)^(p-1)-1 == 2^(p-1)-1 (mod 3). (End)
p is in A001220 if and only if p | (2^(p-1)-1)/p, i.e., a(n) is divisible by prime(n). - Felix Fröhlich, Jun 20 2014
In general, every prime p that is 1 mod q-1 will create a numerator that is 0 mod q via Fermat's Little Theorem, meaning every p with this property (except q) will have a Fermat quotient divisible by q. - Roderick MacPhee, May 12 2017

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See pp. 47, 308.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 105.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986, p. 70.

T. D. Noe, Table of n, a(n) for n=2..100
Nick Hobson, Solution to puzzle 158: Fermat squares.
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, in Proceedings of CANT 2011, arXiv:1110.3113 [math.NT], 2011-2012.
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.
Andrei K. Svinin, On some class of sums, arXiv:1610.05387 [math.CO], 2016-2017. See p. 11.
H. S. Vandiver, Fermat's Quotients And Related Arithmetic Functions, PNAS 1945 31 (1) pp. 55-60.
H. S. Vandiver, New Types Of Congruences Involving Bernoulli Numbers and Fermat's Quotient, PNAS 1948 34 (3) pp. 103-110.
H. S. Vandiver, On Congruences Which Relate The Fermat And Wilson Quotients To The Bernoulli Numbers, PNAS 1949 35 (6) pp. 332-337.

Cf. A002322, A001917, A096060, A001045.

A007663:= n-> map (p-> (2^(p-1)-1)/p, ithprime(n)):
seq (A007663(n), n=2..20); # Jani Melik, Jan 24 2011

A007663[n_Integer?Positive]:=(-1+2^(Prime[n]-1))/Prime[n]/;(n>1) (* Enrique Pérez Herrero, Sep 08 2010 *)
Table[(2^(n-1)-1)/n,{n,Prime[Range[2,20]]}] (* Harvey P. Dale, Nov 07 2016 *)

forprime(p=3, 100, print1((2^(p-1)-1)/p ", ")) \\ Satish Bysany, Mar 11 2017

From Alexander Adamchuk, Oct 01 2006: (Start)
a(n) = 3*A096060(n) for n > 2.
a(n) = 3*A001045(prime(n)-1)/prime(n) for n > 1. (End)
a(n) = Sum_{i=0..(p-3)/2} 2^i*(p-i-2)!/((i+1)!*(p-2*(i+1))!) where p = prime(n), for n >= 2. - Vladimir Pletser, Jan 26 2023

A196202 a(n) = 2^(prime(n)-1) mod prime(n)^2.

Original entry on oeis.org

2, 4, 16, 15, 56, 40, 222, 58, 392, 30, 187, 38, 944, 1076, 2069, 1909, 473, 2197, 671, 143, 4089, 1502, 3985, 535, 5530, 9293, 6078, 1392, 7304, 9380, 2287, 2228, 7262, 4171, 14305, 8457, 12875, 10922, 7850, 520, 8951, 26789, 9551, 20073, 34476, 26866

Offset: 1

Reinhard Zumkeller, Sep 29 2011

a(A049084(A001220(1))) = a(A049084(A001220(2))) = 1.

			A001220(1)=1093=A000040(183): a(183)=1, or a(A049084(A001220(1)))=1;
A001220(2)=3511=A000040(490): a(490)=1, or a(A049084(A001220(2)))=1.

N. G. W. H. Beeger, On a new case of the congruence 2^(p-1) ≡ 1 (p^2), Messenger of Mathematics 51, (1922), p. 149-150
Paulo Ribenboim, 1093 (Chap 8), in 'My Numbers, My Friends', Springer-Verlag 2000 NY, page 213ff.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
W. Meissner, Über die Teilbarkeit von 2^p-2 durch das Quadrat der Primzahl p = 1093, Sitzungsberichte Königlich Preussischen Akadamie Wissenschaften Berlin, 35 (1913), 663-667. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Wieferich Prime.
Wikipedia, Wieferich prime.

Cf. A061286.

import Math.NumberTheory.Moduli (powerMod)
a196202 n = powerMod 2 (p - 1) (p ^ 2) where p = a000040 n
-- Reinhard Zumkeller, May 18 2015

seq(2 &^ (ithprime(n)-1) mod ithprime(n)^2, n=1..1000); # Robert Israel, Aug 03 2014

PowerMod[2,#-1,#^2]&/@Prime[Range[50]] (* Harvey P. Dale, Apr 25 2012 *)

forprime(p=2, 1e2, print1(lift(Mod(2, p^2)^(p-1)), ", ")) \\ Felix Fröhlich, Aug 03 2014

A292270 Sum of all partial fractions in the algorithm used for calculation of A002326(n).

Original entry on oeis.org

1, 1, 4, 1, 13, 25, 36, 1, 38, 81, 12, 26, 124, 121, 196, 1, 103, 73, 324, 42, 224, 175, 91, 147, 232, 14, 676, 170, 303, 841, 900, 1, 264, 1089, 385, 364, 93, 301, 585, 563, 1093, 1681, 44, 355, 152, 118, 83, 484, 1254, 763, 2500, 1043, 156, 2809, 996, 564, 952, 931, 71, 387, 3325, 176, 3124, 1, 649, 4225, 554, 1081

Offset: 0

Vladimir Shevelev and Antti Karttunen, Oct 05 2017

nonn

This sequence gives important additional insight into the algorithm for the calculation of A002326 (see A179680 for its description). Let us estimate how many steps are required before (the first) 1 will appear. Note that all partial fractions (which are indeed, integers) are odd residues modulo 2*n+1 from the interval [1,2*n-1]. So, if there is no repetition, then the number of steps does not exceed n. Suppose then that there is a repetition before the appearance of 1. Then for an odd residue k from [1, 2*n-1], 2^m_1 == 2^m_2 == k (mod 2*n+1) such that m_2 > m_1. But then 2^(m_2-m_1) == 1 (mod 2*n+1). So, since m_2 - m_1 < m_2, it means that 1 should appear earlier than the repetition of k, which is a contradiction. So the number of steps <= n. For example, for n=9, 2*n+1 = 19, we have exactly 9 steps with all other odd residues <= 17 modulo 19 appearing before the final 1: 5, 3, 11, 15, 17, 9, 7, 13, 1.

A001122 gives the odd numbers k such that a((k-1)/2) = A000290((k-1)/2).

			Let n = 9. According to the comment, a(9) = 5 + 3 + 11 + 15 + 17 + 9 + 7 + 13 + 1 = 81.

Antti Karttunen, Table of n, a(n) for n = 0..10001

Cf. A000265, A000290, A001122, A001917, A002326, A006519, A163782, A179382, A179680, A292239, A292265, A292947, A293218, A293219.

Cf. A000225 (gives the positions of ones), A292938 (of squares), A292939 (and the corresponding odd numbers), A292940 (odd numbers corresponding to squares larger than one), A292379 (odd numbers corresponding to squares less than n^2).

A000265(n) = (n >> valuation(n, 2));
A006519(n) = 2^valuation(n, 2);
A292270(n) = { my(x = n+n+1, z = ((1+x)/A006519(1+x)), m = A000265(1+x)); while(m!=1, z += ((x+m)/A006519(x+m)); m = A000265(x+m)); z; };

(define (A292270 n) (let ((x (+ n n 1))) (let loop ((z (/ (+ 1 x) (A006519 (+ 1 x)))) (k 1)) (let ((m (A000265 (+ x k)))) (if (= 1 m) z (loop (+ z (/ (+ x m) (A006519 (+ x m)))) m))))))

For all n >= 1, A000196(a((A001122(1+n)-1)/2)) = (A001122(1+n)-1)/2, in other words, a(A163782(n)) = A000290(A163782(n)).

A094593 a(n) = (p-1)/x, where p = prime(n) and x = ord(3,p), the smallest positive integer such that 3^x == 1 mod p.

Original entry on oeis.org

1, 1, 2, 4, 1, 1, 2, 1, 1, 2, 5, 1, 2, 1, 2, 6, 3, 2, 6, 1, 2, 1, 2, 1, 3, 2, 4, 1, 1, 2, 1, 1, 1, 3, 2, 1, 2, 1, 2, 4, 2, 12, 1, 1, 1, 1, 2, 4, 1, 2, 2, 2, 1, 2, 1, 9, 4, 1, 1, 1, 9, 2, 8, 1, 1, 2, 2, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 2, 4, 10, 16, 3, 2, 1, 2

Offset: 3

Benoit Cloitre, Jun 06 2004

nonn

T. D. Noe, Table of n, a(n) for n = 3..1000

Cf. A001917.

a(n)=(prime(n)-1)/if(n<0,0,k=1;while((3^k-1)%prime(n)>0,k++);k)

from sympy import prime, n_order
def A094593(n):
    p = prime(n)
    return 1 if n == 3 else (p-1)//n_order(3,p) # Chai Wah Wu, Jan 15 2020

a(n) = (A000040(n)-1)/A062117(n).

cp's OEIS Frontend

A152598 Records in A001917.

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A152597 Where records occur in A001917.

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A001220 Wieferich primes: primes p such that p^2 divides 2^(p-1) - 1.

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A001122 Primes with primitive root 2.

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A014664 Order of 2 modulo the n-th prime.

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A006694 Number of cyclotomic cosets of 2 mod 2n+1.

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