A086145 -id:A086145 - cp's OEIS frontend

A006093 a(n) = prime(n) - 1.

1, 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 256, 262, 268, 270

Offset: 1

N. J. A. Sloane

These are also the numbers that cannot be written as i*j + i + j (i,j >= 1). - Rainer Rosenthal, Jun 24 2001; Henry Bottomley, Jul 06 2002
The values of k for which Sum_{j=0..n} (-1)^j*binomial(k, j)*binomial(k-1-j, n-j)/(j+1) produces an integer for all n such that n < k. Setting k=10 yields [0, 1, 4, 11, 19, 23, 19, 11, 4, 1, 0] for n = [-1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], so 10 is in the sequence. Setting k=3 yields [0, 1, 1/2, 1/2] for n = [-1, 0, 1, 2], so 3 is not in the sequence. - Dug Eichelberger (dug(AT)mit.edu), May 14 2001
n such that x^n + x^(n-1) + x^(n-2) + ... + x + 1 is irreducible. - Robert G. Wilson v, Jun 22 2002
Records for Euler totient function phi.
Together with 0, n such that (n+1) divides (n!+1). - Benoit Cloitre, Aug 20 2002; corrected by Charles R Greathouse IV, Apr 20 2010
n such that phi(n^2) = phi(n^2 + n). - Jon Perry, Feb 19 2004
Numbers having only the trivial perfect partition consisting of a(n) 1's. - Lekraj Beedassy, Jul 23 2006
Numbers n such that the sequence {binomial coefficient C(k,n), k >= n } contains exactly one prime. - Artur Jasinski, Dec 02 2007
Record values of A143201: a(n) = A143201(A001747(n+1)) for n > 1. - Reinhard Zumkeller, Aug 12 2008
From Reinhard Zumkeller, Jul 10 2009: (Start)
The first N terms can be generated by the following sieving process:
  start with {1, 2, 3, 4, ..., N - 1, N};
  for i := 1 until SQRT(N) do
  (if (i is not striked out) then
  (for j := 2 * i + 1 step i + 1 until N do
  (strike j from the list)));
  remaining numbers = {a(n): a(n) <= N}. (End)
a(n) = partial sums of A075526(n-1) = Sum_{1..n} A075526(n-1) = Sum_{1..n} (A008578(n+1) - A008578(n)) = Sum_{1..n} (A158611(n+2) - A158611(n+1)) for n >= 1. - Jaroslav Krizek, Aug 04 2009
A006093 U A072668 = A000027. - Juri-Stepan Gerasimov, Oct 22 2009
A171400(a(n)) = 1 for n <> 2: subsequence of A171401, except for a(2) = 2. - Reinhard Zumkeller, Dec 08 2009
Numerator of (1 - 1/prime(n)). - Juri-Stepan Gerasimov, Jun 05 2010
Numbers n such that A002322(n+1) = n. This statement is stronger than repeating the property of the entries in A002322, because it also says in reciprocity that this sequence here contains no numbers beyond the Carmichael numbers with that property. - Michel Lagneau, Dec 12 2010
a(n) = A192134(A095874(A000040(n))); subsequence of A192133. - Reinhard Zumkeller, Jun 26 2011
prime(a(n)) + prime(k) < prime(a(k) + k) for at least one k <= a(n): A212210(a(n),k) < 0. - Reinhard Zumkeller, May 05 2012
Except for the first term, numbers n such that the sum of first n natural numbers does not divide the product of first n natural numbers; that is, n*(n + 1)/2 does not divide n!. - Jayanta Basu, Apr 24 2013
BigOmega(a(n)) equals BigOmega(a(n)*(a(n) + 1)/2), where BigOmega = A001222. Rationale: BigOmega of the product on the right hand side factorizes as BigOmega(a/2) + Bigomega(a+1) = BigOmega(a/2) + 1 because a/2 and a + 1 are coprime, because BigOmega is additive, and because a + 1 is prime. Furthermore Bigomega(a/2) = Bigomega(a) - 1 because essentially all 'a' are even. - Irina Gerasimova, Jun 06 2013
Record values of A060681. - Omar E. Pol, Oct 26 2013
Deficiency of n-th prime. - Omar E. Pol, Jan 30 2014
Conjecture: All the sums Sum_{k=s..t} 1/a(k) with 1 <= s <= t are pairwise distinct. In general, for any integers d >= -1 and m > 0, if Sum_{k=i..j} 1/(prime(k)+d)^m = Sum_{k=s..t} 1/(prime(k)+d)^m with 0 < i <= j and 0 < s <= t then we must have (i,j) = (s,t), unless d = m = 1 and {(i,j),(s,t)} = {(4,4),(8,10)} or {(4,7),(5,10)}. (Note that 1/(prime(8)+1)+1/(prime(9)+1)+1/(prime(10)+1) = 1/(prime(4)+1) and Sum_{k=5..10} 1/(prime(k)+1) = 1/(prime(4)+1) + Sum_{k=5..7} 1/(prime(k)+1).) - Zhi-Wei Sun, Sep 09 2015
Numbers n such that (prime(i)^n + n) is divisible by (n+1), for all i >= 1, except when prime(i) = n+1. - Richard R. Forberg, Aug 11 2016
a(n) is the period of Fubini numbers (A000670) over the n-th prime. - Federico Provvedi, Nov 28 2020

Archimedeans Problems Drive, Eureka, 40 (1979), 28.
Harvey Dubner, Generalized Fermat primes, J. Recreational Math., 18 (1985): 279-280.
M. Gardner, The Colossal Book of Mathematics, pp. 31, W. W. Norton & Co., NY, 2001.
M. Gardner, Mathematical Circus, pp. 251-2, Alfred A. Knopf, NY, 1979.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
Thomas F. Bloom, Unit fractions with shifted prime denominators, arXiv:2305.02689 [math.NT], 2023.
R. P. Boas & N. J. A. Sloane, Correspondence, 1974
Harvey Dubner, Generalized Fermat primes, J. Recreational Math. 18.4 (1985-1986), 279. (Annotated scanned copy)
Armel Mercier, Problem E 3065, American Mathematical Monthly, 1984, p. 649.
Armel Mercier, S. K. Rangarajan, J. C. Binz and Dan Marcus, Problem E 3065, American Mathematical Monthly, No. 4, 1987, pp. 378.
Poo-Sung Park, Additive uniqueness of PRIMES-1 for multiplicative functions, arXiv:1708.03037 [math.NT], 2017.
J. R. Rickard and J. J. Hitchcock, Problem Drive 4, Archimedeans Problems Drive, Eureka, 40 (1979), 28-29, 40. (Annotated scanned copy)
Index entries for sequences generated by sieves

a(n) = K(n, 1) and A034693(K(n, 1)) = 1 for all n. The subscript n refers to this sequence and K(n, 1) is the index in A034693. - Labos Elemer
Cf. A000040, A034694. Different from A075728.
Complement of A072668 (composite numbers minus 1), A072670(a(n))=0.
Essentially the same as A039915.
Cf. A084920, A006093, A050997, A008864, A060800, A131991, A131992, A131993.
Cf. A101301 (partial sums), A005867 (partial products).
Column 1 of the following arrays/triangles: A087738, A249741, A352707, A378979, A379010.
The last diagonal of A162619, and of A174996, the first diagonal in A131424.
Row lengths of irregular triangles A086145, A124223, A212157.

Filtered([1..280],IsPrime)-1; # Muniru A Asiru, Nov 25 2018

a006093 = (subtract 1) . a000040  -- Reinhard Zumkeller, Mar 06 2012

[NthPrime(n)-1: n in [1..100]]; // Vincenzo Librandi, Nov 17 2015

for n from 2 to 271 do if (n! mod n^2 = n*(n-1) and (n<>4) then print(n-1) fi od; # Gary Detlefs, Sep 10 2010
# alternative
A006093 := proc(n)
    ithprime(n)-1 ;
end proc:
seq(A006093(n),n=1..100) ; # R. J. Mathar, Feb 06 2019

Table[Prime[n] - 1, {n, 1, 30}] (* Vladimir Joseph Stephan Orlovsky, Apr 27 2008 *)
a[ n_] := If[ n < 1, 0, -1 + Prime @ n] (* Michael Somos, Jul 17 2011 *)
Prime[Range[60]] - 1 (* Alonso del Arte, Oct 26 2013 *)

isA006093(n) = isprime(n+1) \\ Michael B. Porter, Apr 09 2010

A006093(n) = prime(n)-1 \\ Michael B. Porter, Apr 09 2010

\\ Sieve as described in Rainer Rosenthal's comment.
m=270;s=vector(m);for(i=1,m,for(j=i,m,k=i*j+i+j;if(k<=m,s[k]=1)));for(k=1,m,if(s[k]==0,print1(k,", "))); \\ Hugo Pfoertner, May 14 2019

from sympy import prime
for n in range(1,100): print(prime(n)-1, end=', ') # Stefano Spezia, Nov 30 2018

a(n) = (p-1)! mod p where p is the n-th prime, by Wilson's theorem. - Jonathan Sondow, Jul 13 2010
a(n) = A000010(prime(n)) = A000010(A006005(n)). - Antti Karttunen, Dec 16 2012
a(n) = A005867(n+1)/A005867(n). - Eric Desbiaux, May 07 2013
a(n) = A000040(n) - 1. - Omar E. Pol, Oct 26 2013
a(n) = A033879(A000040(n)). - Omar E. Pol, Jan 30 2014

Correction for change of offset in A158611 and A008578 in Aug 2009 Jaroslav Krizek, Jan 27 2010
Obfuscating comments removed by Joerg Arndt, Mar 11 2010
Edited by Charles R Greathouse IV, Apr 20 2010

A057593 Triangle T(n, k) giving period length of the periodic sequence k^i (i >= imin) mod n (n >= 2, 1 <= k <= n-1).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 4, 4, 2, 1, 2, 1, 1, 2, 1, 3, 6, 3, 6, 2, 1, 1, 2, 1, 2, 1, 2, 1, 6, 1, 3, 6, 1, 3, 2, 1, 4, 4, 2, 1, 1, 4, 4, 2, 1, 10, 5, 5, 5, 10, 10, 10, 5, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 12, 3, 6, 4, 12, 12, 4, 3, 6, 12, 2, 1

Offset: 2

Gottfried Helms, Oct 05 2000

From Wolfdieter Lang, Sep 04 2017: (Start)
i) If gcd(n, k) = 1 then imin = imin(n, k) = 0 and the length of the period P = T(n, k) = order(n, k), given in A216327 corresponding to the numbers of A038566. This is due to Euler's theorem. E.g., T(4, 3) = 2 because A216327(4, 2) = 2 corresponding to A038566(4, 2) = 3.
ii) If gcd(n, k) is not 1 then the smallest nonnegative index imin = imin(n, k) is obtained from A290601 with the corresponding length of the period given in A290602. Also in this case the sequence always becomes periodic, because one of the possible values from {0, 1, ..., n-1} has to appear a second time because the sequence has more than n entries. Example: T(4, 2) = 1 because imin is given by A290601(1, 1) = 2 (corresponding to the present n = 4, k = 2 values) with the length of the period P given by A290602(1, 1) = 1. (End)

			If n=7, k=2, (imin = 0) the sequence is 1,2,4,1,2,4,1,2,4,... of period 3, so T(7,2) = 3. The triangle T(n, k) begins:
n \ k 1   2   3  4   5   6   7   8  9  10  11  12  13  14 15 16 17 ...
2:    1
3:    1   2
4:    1   1   2
5:    1   4   4  2
6:    1   2   1  1   2
7:    1   3   6  3   6   2
8:    1   1   2  1   2   1   2
9:    1   6   1  3   6   1   3   2
10:   1   4   4  2   1   1   4   4  2
11:   1  10   5  5   5  10  10  10  5   2
12:   1   2   2  1   2   1   2   2  1   1   2
13:   1  12   3  6   4  12  12   4  3   6  12   2
14:   1   3   6  3   6   2   1   1  3   6   3   6   2
15:   1   4   4  2   2   1   4   4  2   1   2   4   4  2
16:   1   1   4  1   4   1   2   1  2   1   4   1   4  1  2
17:   1   8  16  4  16  16  16   8  8  16  16  16   4  16  8  2
18:   1   6   1  3   6   1   3   2  1   1   6   1   3   6  1  1  2
... Reformatted and extended. - _Wolfdieter Lang_, Sep 04 2017
From _Wolfdieter Lang_, Sep 04 2017: (Start)
The  table imin(n, k) begins:
n \ k 1   2   3   4   5   6   7   8  9  10  11  12  13  14  15  16 17 ...
2:    0
3:    0   0
4:    0   2   0
5:    0   0   0   0
6:    0   1   1   1   0
7:    0   0   0   0   0   0
8:    0   3   0   2   0   3   0
9:    0   0   2   0   0   2   0   0
10:   0   1   0   1   1   1   0   1  0
11:   0   0   0   0   0   0   0   0  0   0
12:   0   2   1   1   0   2   0   1  1   2   0
13:   0   0   0   0   0   0   0   0  0   0   0   0
14:   0   1   0   1   0   1   1   1  0   1   0   1   0
15:   0   0   1   0   1   1   0   0  1   1   0   1   0   0
16:   0   4   0   2   0   4   0   2  0   4   0   2   0   4   0
17:   0   0   0   0   0   0   0   0  0   0   0   0   0   0   0   0
18:   0   1   2   1   0   2   0   1  1   1   0   2   0   1   2   1  0
... (End)

Michael De Vlieger, Table of n, a(n) for n = 2..19901 (rows 2 <= n <= 200).

Cf. A057594, A057595.
Cf. A086145 (prime rows), A216327 (entries with gcd(n,k) = 1), A139366.
Cf. A038566, A216327, A290601, A290602.

period[lst_] := Module[{n, i, j}, n=Length[lst]; For[j=2, j <= n, j++, For[i=1, iJean-François Alcover, Feb 04 2015 *)

Constraint on k changed from 2 <= k <= n to 1 <= k < n, based on comment from Franklin T. Adams-Watters, Jan 19 2006, by David Applegate, Mar 11 2014

Name changed and table extended by Wolfdieter Lang, Sep 04 2017

A216327 Irregular triangle of multiplicative orders mod n for the elements of the smallest positive reduced residue system mod n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 4, 2, 1, 2, 1, 3, 6, 3, 6, 2, 1, 2, 2, 2, 1, 6, 3, 6, 3, 2, 1, 4, 4, 2, 1, 10, 5, 5, 5, 10, 10, 10, 5, 2, 1, 2, 2, 2, 1, 12, 3, 6, 4, 12, 12, 4, 3, 6, 12, 2, 1, 6, 6, 3, 3, 2, 1, 4, 2, 4, 4, 2, 4, 2, 1, 4, 4, 2, 2, 4, 4, 2

Offset: 1

Wolfdieter Lang, Sep 28 2012

The sequence of the row lengths is phi(n) = A000010 (Euler's totient).
For the notion 'reduced residue system mod n' which has, as a set, order phi(n) = A000010(n), see e.g., the Apostol reference p. 113. Here such a system with the smallest positive numbers is used. (In the Apostol reference 'order of a modulo n' is called 'exponent of a modulo n'. See the definition on p. 204.)
See A038566 where the reduced residue system mod n appears in row n.
In the chosen smallest reduced residue system mod n one can replace each element by any congruent mod n one, and the given order modulo n list will, of course, be the same. E.g., n=5, {6, -3, 13, -16} also has the orders modulo 5:  1  4  4  2, respectively.
Each order modulo n divides phi(n). See the Niven et al. reference, Corollary 2.32, p. 98.
The maximal order modulo n is given in A002322(n).
For the analog table of orders Modd n see A216320.

			This irregular triangle begins:
n\k 1  2  3  4  5  6  7  8  9  10 11 12  13 14  15 16 17 18
1:  1
2:  1
3:  1  2
4:  1  2
5:  1  4  4  2
6:  1  2
7:  1  3  6  3  6  2
8:  1  2  2  2
9:  1  6  3  6  3  2
10: 1  4  4  2
11: 1 10  5  5  5 10 10 10  5   2
12: 1  2  2  2
13: 1 12  3  6  4 12 12  4  3   6 12  2
14: 1  6  6  3  3  2
15: 1  4  2  4  4  2  4  2
16: 1  4  4  2  2  4  4  2
17: 1  8 16  4 16 16 16  8  8  16 16 16   4 16   8  2
18: 1  6  3  6  3  2
19: 1 18 18  9  9  9  3  6  9  18  3  6  18 18  18  9  9  2
20: 1  4  4  2  2  4  4  2
...
a(3,2) = 2 because A038566(3,2) = 2 and 2^1 == 2 (mod 3), 2^2 = 4 == 1 (mod 3).
a(7,3) = 6 because A038566(7,3) = 3 and 3^1 == 3 (mod 7), 3^2 = 9 == 2 (mod 7), 3^3 = 2*3 == 6 (mod 7),  3^4 == 6*3 == 4 (mod 7), 3^5 == 4*3 == 5 (mod 7) and  3^6 == 5*3 == 1 (mod 7). The notation == means 'congruent'.
The maximal order modulo 7 is 6 = A002322(7) = phi(7), and it appears twice: A111725(7) = 2.
The maximal order modulo 14 is 6 = A002322(14) = 1*6.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
I. Niven, H. S. Zuckerman, and H. L. Montgomery, An Introduction to the Theory of Numbers, Fifth edition, Wiley, 1991.

Michael De Vlieger, Table of n, a(n) for n = 1..12232 (rows n = 1..200, flattened.)

Cf. A038566, A002322 (maximal order), A111725 (multiplicity of max order), A216320 (Modd n analog).

Cf. A086145

Table[Table[MultiplicativeOrder[k,n],{k,Select[Range[n],GCD[#,n]==1&]}],{n,1,13}]//Grid  (* Geoffrey Critzer, Jan 26 2013 *)

rowa(n) = select(x->gcd(n, x)==1, [1..n]); \\ A038566
row(n) = apply(znorder, apply(x->Mod(x, n), rowa(n))); \\ Michel Marcus, Sep 12 2023

a(n,k) = order A038566(n,k) modulo n, n >= 1, k=1, 2, ..., phi(n) = A000010(n). This is the order modulo n of the k-th element of the smallest reduced residue system mod n (when their elements are listed increasingly).

A128250 LCG periods: periods of the output sequences produced by multiplicative linear congruential generators (LCGs) with prime moduli, for all valid combinations of multiplier and modulus.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 2, 1, 3, 6, 3, 6, 2, 1, 10, 5, 5, 5, 10, 10, 10, 5, 2, 1, 12, 3, 6, 4, 12, 12, 4, 3, 6, 12, 2, 1, 8, 16, 4, 16, 16, 16, 8, 8, 16, 16, 16, 4, 16, 8, 2, 1, 18, 18, 9, 9, 9, 3, 6, 9, 18, 3, 6, 18, 18, 18, 9, 9, 2

Offset: 2

Ross Drewe, May 09 2007, May 11 2007, May 25 2007

The periods of these output sequences need to be known when designing LCG-based pseudorandom number generators. The period of an LCG output is always 1 when a = 1, always 2 when a = m-1 and maximal (i.e. m-1) only if a is a totative of m. There is no fast method for finding totative values when m is large. The sample shows the terms generated by the first 8 moduli (i.e. primes from 2 to 19), as generated by: A = LCG_periods(19) (see program).

Apparently this is A086145 with a top row added. - R. J. Mathar, Jun 14 2008

			Q =
p(2,1) ..................................... [1]
p(3,1) p(3,2) .............................. [1 2]
p(5,1) p(5,2) p(5,3) p(5,4) ................ [1 4 4 2]
p(7,1) p(7,2) p(7,3) p(7,4) p(7,5) p(7,6) .. [1 3 6 3 6 2]
Therefore A = [1] [1 2] [1 4 4 2] [1 3 6 3 6 2] .....

Cf. A086145.

function A = LCG_periods(N); mlist = primes(N); nprimes = length(mlist); A = []; for i = 1:nprimes; m = mlist(i); for a = 1:m-1; x = 1; count = 0; while 1; count = count + 1; x = mod(a*x, m); if x == 1; break; end; end; A = [A count]; end; end

Multiplicative LCG for modulus m, multiplier a: x(n+1) == a*x(n) mod m. Additional restriction: a < m (as assumed in many applications). The output sequence for any explicit combination of m,a,x0 is always periodic and the period is independent of x0. Therefore denote the period by p(m,a). Let Q be the lower triangular matrix that is produced by tabulating all p(m,a) values, such that the rows represent m values (successive primes) and the columns represent a values (from 1 to m-1). Then A is the sequence obtained by concatenating the rows of this matrix.

A207330 Array of the orders Modd p, p a prime.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 3, 1, 5, 5, 5, 5, 1, 3, 2, 6, 3, 6, 1, 8, 8, 8, 4, 8, 2, 4, 1, 9, 9, 3, 9, 3, 9, 9, 9, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1, 14, 7, 7, 7, 14, 7, 14, 2, 14, 14, 7, 7, 14, 1, 15, 3, 15, 15, 15, 15, 5, 15, 15, 15, 5, 3, 5, 5, 1, 9, 18, 9

Offset: 1

Wolfdieter Lang, Mar 27 2012

For Modd n (not to be confused with mod n) see a comment on A203571.
The row lengths sequence of this array is 1 for row n=1, and (p(n)-1)/2, with p(n):=A000040(n) (the primes), for row n>1.
A primitive root has order delta(p) = (p-1)/2 (delta is given by A055034).

			n, p(n)/m  1  2  3  4  5  6  7  8  9 10 11 12 13 14 ...
     2m-1: 1  3  5  7  9 11 13 15 17 19 21 23 25 27 ...
1,   2:    1
2,   3:    1
3,   5:    1  2
4,   7:    1  3  3
5,  11:    1  5  5  5  5
6,  13:    1  3  2  6  3  6
7,  17:    1  8  8  8  4  8  2  4
8,  19:    1  9  9  3  9  3  9  9  9
9,  23:    1 11 11 11 11 11 11 11 11 11 11
10, 29:    1 14  7  7  7 14  7 14  2 14 14  7  7 14
...
a(6,4) = 6 because 7^1 = 7, 7^2 = 49, 49 (Modd 13) := -49 (mod 13) = 3, 7^3 == 7*3 = 21,
21 (Modd 13) := -21 (mod 13) = 5, 7^4 == 7*5 = 35, 35 (Modd 13) = 35 (mod 13) = 9,
7^5 == 7*9=63, 63 (Modd 13):= 63 (mod 13) = 11, 7^6 == 7*11 = 77, 77 (Modd 13) := -77 (mod 13) = 1.
Row n=5: all 2*m-1, m>1, are primitive roots. The smallest positive one is 3.
Row n=6: only 7 and 11 are primitive roots. The smallest one is 7.

Cf. A086145 (mod n case).

a(n,m) = (multiplicative) order Modd p(n) of 2*m-1, for m=1,...,(p(n)-1)/2, with p(n):= A000040(n) (the primes), n>1, and for a(1,1) = 1 for the prime 2.

cp's OEIS Frontend

A006093 a(n) = prime(n) - 1.

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A057593 Triangle T(n, k) giving period length of the periodic sequence k^i (i >= imin) mod n (n >= 2, 1 <= k <= n-1).

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A216327 Irregular triangle of multiplicative orders mod n for the elements of the smallest positive reduced residue system mod n.

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A128250 LCG periods: periods of the output sequences produced by multiplicative linear congruential generators (LCGs) with prime moduli, for all valid combinations of multiplier and modulus.

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A207330 Array of the orders Modd p, p a prime.

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A359210 Number of m^k == 1 (mod p) for 0 < m,k < p where p is the n-th prime.

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