A192109 -id:A192109 - cp's OEIS frontend

A216822 Numbers n such that 2^n == 2 (mod n*(n+1)).

1, 5, 13, 29, 37, 61, 73, 157, 181, 193, 277, 313, 397, 421, 457, 541, 561, 613, 661, 673, 733, 757, 877, 997, 1093, 1153, 1201, 1213, 1237, 1289, 1321, 1381, 1453, 1621, 1657, 1753, 1873, 1905, 1933, 1993, 2017, 2137, 2341, 2473, 2557, 2593, 2797, 2857, 2917

Offset: 1

V. Raman, Sep 17 2012

a(17) = 561 is the first composite number in the sequence. - Charles R Greathouse IV, Sep 19 2012

Intersection of { A015919(n) } and { A192109(n)-1 }. - Max Alekseyev, Apr 22 2013

V. Raman and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 2000 terms from V. Raman)
Mersenne Forum, Prime Conjecture

Cf. A069051 (prime n such that 2^n == 2 (mod n*(n-1))).
Cf. A217466 (prime terms of the sequence).
Cf. A217465 (composite terms of the sequence)

Select[Range[1, 10000], Mod[2^# - 2, # (# + 1)] == 0 &] (* T. D. Noe, Sep 19 2012 *)
Join[{1},Select[Range[3000],PowerMod[2,#,#(#+1)]==2&]] (* Harvey P. Dale, Oct 05 2022 *)

is(n)=Mod(2,n*(n+1))^n==2; \\ Charles R Greathouse IV, Sep 19 2012

A216822_list = [n for n in range(1,10**6) if n == 1 or pow(2,n,n*(n+1)) == 2] # Chai Wah Wu, Mar 25 2021

a(1)=1 prepended by Max Alekseyev, Dec 29 2017

A216090 Numbers n such that k^(n-1) == k (mod n) for every k = 1, 2, ..., n-1.

Original entry on oeis.org

1, 2, 6, 10, 14, 22, 26, 30, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 182, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482

Offset: 1

Michel Lagneau, Sep 01 2012

nonn

Subsequence of, but different from A197930, for example A197930(11) = 42 with 42 distinct residues, but the set R of the residues k^41 mod 42 is R = {1, 32, 33, 16, 17, 6, …, 9, 10, 41} for k = 1, 2, …, 41 instead R = {1, 2, 3, …, 40, 41}. Terms of A197930 that are not in this sequence: 42, 78, 110, 114, 138, 170, …

Squarefree numbers n such that A002322(n) divides n-2. Contains all doubled odd primes and all doubled Carmichael numbers. - Thomas Ordowski, Apr 23 2017

			a(4) = 10 because x^9  == 1, 2, ..., 9  (mod 10) with 9 distinct residues such that:
1^9 = 1 == 1 (mod 10);
2^9 = 512 == 2 (mod 10);
3^9 = 19683 == 3 (mod 10);
4^9 = 262144 == 4 (mod 10);
5^9 = 1953125 == 5 (mod 10);
6^9 = 10077696 == 6 (mod 10);
7^9 = 40353607 == 7 (mod 10);
8^9 = 134217728 == 8 (mod 10);
9^9 = 387420489 == 9 (mod 10).

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Subsequence of A192109.
Terms > 2 form a subsequence of A050990.
Cf. A197929, A197930, A197943.

with(numtheory):for n from 1  to 500 do:j:=0:for i from 1 to n do: if irem(i^(n-1),n)=i then j:=j+1:else fi:od:if j=n-1 then printf(`%d, `, n):else fi:od:

f[n_] := And @@ Table[PowerMod[k, n - 1, n] == k, {k, n - 1}]; Select[Range[500], f] (* T. D. Noe, Sep 03 2012 *)

isok(n) = {for (k=1, n-1, if (Mod(k, n)^(n-1) != Mod(k, n), return (0));); return (1);} \\ Michel Marcus, Apr 23 2017

from sympy.ntheory.factor_ import core
from sympy import primefactors
def ok(n):
    if n<3: return True
    if core(n) == n:
        for p in primefactors(n):
            if (n - 2)%(p - 1): return False
        return True
    return False
print([n for n in range(1, 501) if ok(n)]) # Indranil Ghosh, Apr 23 2017

cp's OEIS Frontend

A216822 Numbers n such that 2^n == 2 (mod n*(n+1)).

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