A014664 -id:A014664 - cp's OEIS frontend

A381924 Multiplicative order of n mod prime(n).

1, 2, 4, 3, 5, 12, 16, 6, 11, 28, 30, 9, 40, 21, 46, 13, 29, 60, 33, 7, 24, 13, 41, 88, 48, 100, 34, 106, 54, 7, 63, 26, 136, 23, 74, 75, 39, 9, 166, 86, 178, 5, 95, 192, 196, 99, 105, 222, 113, 228, 29, 34, 120, 250, 256, 262, 67, 270, 46, 8, 47, 292, 153, 155, 312

Offset: 1

Giorgos Kalogeropoulos, Mar 12 2025

nonn

a(n) is the least k such that prime(n) divides n^k-1.

			a(12) = 9 because the multiplicative order of 12 mod prime(12) is 9.

Vincenzo Librandi, Table of n, a(n) for n = 1..3500

Cf. A226295, A014664, A091185 (n^(-1) mod prime(n)).

[Order(n, NthPrime(n)) : n in [1..65]]; // Vincenzo Librandi, Mar 25 2025

Table[MultiplicativeOrder[n,Prime[n]],{n,65}]

a(n) = znorder(Mod(n, prime(n))); \\ Michel Marcus, Mar 12 2025

If n is a primitive root modulo prime(n), a(n) = prime(n) - 1.

A382414 Primes p such that gcd(ord_p(2), ord_p(5)) = 1.

Original entry on oeis.org

31, 601, 2593, 599479, 204700049, 466344409, 668731841, 11638603429

Offset: 1

Li GAN, Apr 26 2025

'ord_p' here means the multiplicative order modulo p, not to be confused with the p-adic order that is also often denoted by ord_p.

1790799748670521, 58523123221688392679 and 14551915228363037109375001 are also terms. - Giorgos Kalogeropoulos, May 03 2025

Mathematics Stack Exchange, For all alpha,beta in N, are there only finitely many primes so that gcd(ordp(alpha),ordp(beta))=1?, 2021

Cf. A014664, A211241, A344202, A383411.

Select[Range[10000], PrimeQ[#] && CoprimeQ[MultiplicativeOrder[2, #], MultiplicativeOrder[5, #]] &]

forprime(p=13, oo, if(1==gcd(znorder(Mod(5, p)), znorder(Mod(2, p))), print1(p, ", "))); \\ Joerg Arndt, Apr 26 2025

A250203 Numbers n such that the Phi_n(2) is the product of exactly two primes and is divisible by 2n+1.

Original entry on oeis.org

11, 20, 23, 35, 39, 48, 83, 96, 131, 231, 303, 375, 384, 519, 771, 848, 1400, 1983, 2280, 2640, 2715, 3359, 6144, 7736, 7911, 11079, 13224, 16664, 24263, 36168, 130439, 406583

Offset: 1

Eric Chen, Mar 13 2015

Here Phi_n is the n-th cyclotomic polynomial.
Is this sequence infinite?
Phi_n(2)/(2n+1) is only a probable prime for n > 16664.
a(33) > 2000000.
Subsequence of A005097 (2 * a(n) + 1 are all primes)
Subsequence of A081858.
2 * a(n) + 1 are in A115591.
Primes in this sequence are listed in A239638.
A085021(a(n)) = 2.
All a(n) are congruent to 0 or 3 (mod 4). (A014601)
All a(n) are congruent to 0 or 2 (mod 3). (A007494)
Except the term 20, all even numbers in this sequence are divisible by 8.

			Phi_11(2) = 23 * 89 and 23 = 2 * 11 + 1, so 11 is in this sequence.
Phi_35(2) = 71 * 122921 and 71 = 2 * 35 + 1, so 35 is in this sequence.
Phi_48(2) = 97 * 673 and 97 = 2 * 48 + 1, so 48 is in this sequence.

Eric Chen, Gord Palameta, Factorization of Phi_n(2) for n up to 1280
Will Edgington, Factorization of completely factored Phi_n(2) [from Internet Archive Wayback Machine]
Henri Lifchitz and Renaud Lifchitz, PRP records. Search for (2^a-1)/b
Samuel Wagstaff, The Cunningham project

Cf. A239638, A085724, A072226, A005384, A005097, A081858, A014601, A014664, A001917, A115591.

Select[Range[10000], PrimeQ[2*# + 1] && PowerMod[2, #, 2*# + 1] == 1 &&
PrimeQ[Cyclotomic[#, 2]/(2*#+1)] &]

isok(n) = if (((x=polcyclo(n, 2)) % (2*n+1) == 0) && (omega(x) == 2), print1(n, ", ")); \\ Michel Marcus, Mar 13 2015

A305184 Multiplicative order of 2 (mod p^2), where p is the n-th Wieferich prime (A001220).

Original entry on oeis.org

364, 1755

Offset: 1

Felix Fröhlich, May 30 2018

Meissner discovered the congruence 2^364 == 1 (mod 1093^2) and thus proved that 1093 is a Wieferich prime, i.e., a term of A001220 (cf. Meissner, 1913).
Later, Beeger discovered the congruence 2^1755 == 1 (mod 3511^2) and proved that 3511 is also a Wieferich prime (cf. Beeger, 1922).
Let b(n) = (A001220(n)-1)/a(n). Then b(1) = 3 and b(2) = 2.
From the fact that a(1) and a(2) are composite it follows that A001220(1) = 1093 and A001220(2) = 3511 do not divide any terms of A001348 (cf. Dobson).
Curiously, both 364 and 1755 are repdigits in some base. 364 = 444 in base 9 and 1755 = 3333 in base 8. Compare this with Dobson's observation that 1092 and 3510 are 444 in base 16 and 6666 in base 8, respectively (cf. Dobson).

N. G. W. H. Beeger, On a new case of the congruence 2^p-1 == 1 (mod p^2), Messenger of Mathematics 51 (1922), 149-150.
J. B. Dobson, A note on the two known Wieferich primes
W. Meissner, Über die Teilbarkeit von 2^p-2 durch das Quadrat der Primzahl p = 1093, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin, 35 (1913), 663-667. [Annotated scanned copy]

Cf. A001220, A001348, A014664, A243905, A282552, A282902.

forprime(p=1, , if(Mod(2, p^2)^(p-1)==1, print1(znorder(Mod(2, p^2)), ", ")))

a(n) = A014664(A000720(A001220(n))) = A243905(A000720(A001220(n))). [Corrected by Jianing Song, Sep 20 2019]

A342974 Primes p such that the order of 2 modulo p is not divisible by the largest odd divisor of p - 1.

Original entry on oeis.org

31, 43, 109, 127, 151, 157, 223, 229, 241, 251, 277, 283, 307, 331, 397, 431, 433, 439, 457, 499, 571, 601, 631, 641, 643, 673, 683, 691, 727, 733, 739, 811, 911, 919, 953, 971, 997, 1013, 1021, 1051, 1069, 1093, 1103, 1163, 1181, 1321, 1327, 1399, 1423, 1429

Offset: 1

Arkadiusz Wesolowski, Apr 01 2021

nonn

Every prime factor of a composite Fermat number belongs to this sequence.
If a prime of the form 3*2^k + 1 belongs to this sequence, then k is in A204620 (see Golomb).
Primes p such that A014664(primepi(p)) is not divisible by A057023(primepi(p)). - Michel Marcus, Apr 26 2021

Solomon W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663.

Cf. A014664, A023394, A057023, A204620, A226366, A280003, A280004.

Select[Prime@Range@300,Mod[MultiplicativeOrder[2,#],Max@Select[Divisors[#-1],OddQ]]!=0&] (* Giorgos Kalogeropoulos, Apr 02 2021 *)

forprime(p=3, 1429, if(Mod(znorder(Mod(2, p)), (p-1)>>valuation(p-1, 2)), print1(p, ", ")));

A363286 Odd primes p such that the congruence 2^x == 1 (mod p) has no solution for 0 < x < (p - 1)/2.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 47, 53, 59, 61, 67, 71, 79, 83, 97, 101, 103, 107, 131, 137, 139, 149, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 227, 239, 263, 269, 271, 293, 311, 313, 317, 347, 349, 359, 367, 373, 379, 383, 389, 401, 409, 419

Offset: 1

Arkadiusz Wesolowski, May 25 2023

nonn

An odd prime p belongs to this sequence if and only if A001917(A000720(p)) is equal to 1 or 2.

Cf. A001917, A014664.

[p: p in [3..419 by 2] | IsPrime(p) and (p-1)/Modorder(2, p) le 2];

isok(p) = p%2 && isprime(p) && (p-1)/znorder(Mod(2, p))<=2;

from itertools import islice
from sympy import nextprime, n_order
def A363286_gen(startvalue=3): # generator of terms >= startvalue
    p = max(startvalue,3)-1
    while (p:=nextprime(p)):
        if n_order(2,p)<<1 >= p-1:
            yield p
A363286_list = list(islice(A363286_gen(),30)) # Chai Wah Wu, Jul 17 2023

a(n) ~ (3/2)*n*log((3/2)*n).

A383467 Primes p such that gcd(ord_p(2), ord_p(6)) = 1.

Original entry on oeis.org

5, 7, 31, 43, 135607, 153649, 270841, 1489441, 1505447, 25781083, 127236649, 558062249, 745988807, 27989941729, 29512739491, 47206579351

Offset: 1

Li GAN, Apr 27 2025

'ord_p' here means the multiplicative order modulo p, not to be confused with the p-adic order that is also often denoted by ord_p.

Mathematics Stack Exchange, For all alpha,beta in N, are there only finitely many primes so that gcd(ordp(alpha),ordp(beta))=1?, 2021.

Cf. A014664, A211242, A344202, A383411, A382414.

Select[Range[10000], PrimeQ[#] && CoprimeQ[MultiplicativeOrder[2, #], MultiplicativeOrder[6, #]] &]

cp's OEIS Frontend

A381924 Multiplicative order of n mod prime(n).

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A382414 Primes p such that gcd(ord_p(2), ord_p(5)) = 1.

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A250203 Numbers n such that the Phi_n(2) is the product of exactly two primes and is divisible by 2n+1.

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A305184 Multiplicative order of 2 (mod p^2), where p is the n-th Wieferich prime (A001220).

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A342974 Primes p such that the order of 2 modulo p is not divisible by the largest odd divisor of p - 1.

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A363286 Odd primes p such that the congruence 2^x == 1 (mod p) has no solution for 0 < x < (p - 1)/2.

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A383467 Primes p such that gcd(ord_p(2), ord_p(6)) = 1.

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