A122785 -id:A122785 - cp's OEIS frontend

A122780 Nonprimes k such that 3^k == 3 (mod k).

1, 6, 66, 91, 121, 286, 561, 671, 703, 726, 949, 1105, 1541, 1729, 1891, 2465, 2665, 2701, 2821, 3281, 3367, 3751, 4961, 5551, 6601, 7107, 7381, 8205, 8401, 8646, 8911, 10585, 11011, 12403, 14383, 15203, 15457, 15841, 16471, 16531, 18721, 19345

Offset: 1

Farideh Firoozbakht, Sep 11 2006

Theorem: If q!=3 and both numbers q and (2q-1) are primes then k=q*(2q-1) is in the sequence. 6, 91, 703, 1891, 2701, 12403, 18721, 38503, 49141, ... is the related subsequence.

The terms > 1 and coprime to 3 of this sequence are the base-3 pseudoprimes, A005935. - M. F. Hasler, Jul 19 2012 [Corrected by Jianing Song, Feb 06 2019]

			66 is composite and 3^66 = 66*468229611858069884271524875811 + 3 so 66 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001567, A005935, A122014.

Cf. A122781, A122782, A122783, A122784, A122785, A122786.

isA122780 := proc(n)
    if isprime(n) then
        false;
    else
        modp( 3 &^ n,n) = modp(3,n) ;
    end if;
end proc:
for n from 1 do
    if isA122780(n) then
        print(n) ;
    end if;
end do: # R. J. Mathar, Jul 15 2012

Select[Range[30000], ! PrimeQ[ # ] && Mod[3^#, # ] == Mod[3, # ] &]
Join[{1},Select[Range[20000],!PrimeQ[#]&&PowerMod[3,#,#]==3&]] (* Harvey P. Dale, Apr 30 2023 *)

is_A122780(n)={n>0 & Mod(3, n)^n==3 & !ispseudoprime(n)} \\ M. F. Hasler, Jul 19 2012

A020137 Pseudoprimes to base 8.

Original entry on oeis.org

9, 21, 45, 63, 65, 105, 117, 133, 153, 231, 273, 341, 481, 511, 561, 585, 645, 651, 861, 949, 1001, 1105, 1281, 1365, 1387, 1417, 1541, 1649, 1661, 1729, 1785, 1905, 2047, 2169, 2465, 2501, 2701, 2821, 3145, 3171, 3201, 3277, 3605, 3641, 4005, 4033, 4097

Offset: 1

David W. Wilson

nonn

This sequence is a subsequence of the sequence A122785. In fact the terms are odd composite terms of A122785. Theorem: If both numbers q and 2q-1 are primes (q is in the sequence A005382) and n=q*(2q-1) then 8^(n-1)==1 (mod n) (n is in the sequence) iff q is of the form 12k+1. 2701,18721,49141,104653,226801,665281,721801,... is the related subsequence. This subsequence is also a subsequence of the sequence A122785. - Farideh Firoozbakht, Sep 15 2006
Composite numbers k such that 8^(k-1) == 1 (mod k). - Michel Lagneau, Feb 18 2012
If q and 3q-2 are odd primes, then q*(3q-2) is in the sequence. - Davide Rotondo, May 25 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..613 from R. J. Mathar, terms 614..1000 from T. D. Noe)
Index entries for sequences related to pseudoprimes

Cf. A001567 (pseudoprimes to base 2), A005382, A122783, A122785.

Select[Range[4100], ! PrimeQ[ # ] && PowerMod[8, (# - 1), # ] == 1 &] (* Farideh Firoozbakht, Sep 15 2006 *)

A122784 Nonprimes k such that 7^k == 7 (mod k).

Original entry on oeis.org

1, 6, 14, 21, 25, 42, 105, 133, 231, 301, 325, 525, 561, 703, 817, 1105, 1729, 1825, 2101, 2353, 2465, 2821, 3277, 3325, 3486, 3913, 4011, 4525, 4825, 5565, 5719, 5901, 6601, 6697, 7525, 8321, 8911, 9331, 10225, 10325, 10585, 10621, 11041, 11521

Offset: 1

Farideh Firoozbakht, Sep 12 2006

nonn

Theorem: If both numbers q and 2q-1 are primes then q*(2q-1) is in the sequence iff q=2 or mod(q,14) is in the set {1, 5, 13}. 6, 703, 18721, 38503, 88831, 104653, 146611, 188191,... are such terms.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005938, A121014, A122780, A122781, A122782, A122783, A122785, A122786.

Select[Range[20000], ! PrimeQ[#] && PowerMod[7, #, #] == Mod[7, #] &]
With[{nn=12000},Select[Complement[Range[nn],Prime[Range[PrimePi[ nn]]]], PowerMod[7,#,#]==Mod[7,#]&]] (* Harvey P. Dale, Jul 12 2012 *)

A290543 Composite numbers n such that A290542(n) >= 2.

Original entry on oeis.org

28, 65, 66, 85, 91, 105, 117, 121, 124, 133, 145, 153, 154, 165, 185, 186, 190, 205, 217, 221, 231, 244, 246, 247, 259, 273, 276, 280, 286, 292, 301, 305, 310, 325, 341, 343, 344, 357, 364, 366, 369, 370, 377, 385, 396, 418, 425, 427, 429, 430, 435, 451

Offset: 1

Arkadiusz Wesolowski, Aug 05 2017

nonn

Is a(n) ~ n * log n as n -> infinity?

Cf. A001567, A006935, A122780, A122781, A122782, A122783, A122784, A122785, A122786, A290542.

lst:=[]; for n in [4..451] do if not IsPrime(n) then r:=Floor(Sqrt(n)); for k in [2..r] do if Modexp(k, n, n) eq k then Append(~lst, n); break; end if; end for; end if; end for; lst;

Select[Flatten@ Position[#, k_ /; k >= 2], CompositeQ] &@ Table[SelectFirst[Range[2, Sqrt@ n], PowerMod[#, n , n] == Mod[#, n] &], {n, 451}] (* Michael De Vlieger, Aug 09 2017 *)

cp's OEIS Frontend

A122780 Nonprimes k such that 3^k == 3 (mod k).

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Maple

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A020137 Pseudoprimes to base 8.

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A122784 Nonprimes k such that 7^k == 7 (mod k).

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A290543 Composite numbers n such that A290542(n) >= 2.

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Magma

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