cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A005938 Pseudoprimes to base 7.

Original entry on oeis.org

6, 25, 325, 561, 703, 817, 1105, 1825, 2101, 2353, 2465, 3277, 4525, 4825, 6697, 8321, 10225, 10585, 10621, 11041, 11521, 12025, 13665, 14089, 16725, 16806, 18721, 19345, 20197, 20417, 20425, 22945, 25829, 26419, 29234, 29341, 29857, 29891, 30025, 30811, 33227
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

According to Karsten Meyer, May 16 2006, 6 should be excluded, following the strict definition in Crandall and Pomerance.
Theorem: If both numbers q & 2q-1 are primes(q is in the sequence A005382) and n=q*(2q-1) then 7^(n-1)==1 (mod 7)(n 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. This sequence is a subsequence of A122784. - Farideh Firoozbakht, Sep 14 2006
Composite numbers n such that 7^(n-1) == 1 (mod n).

References

  • R. Crandall and C. Pomerance, "Prime Numbers - A Computational Perspective", Second Edition, Springer Verlag 2005, ISBN 0-387-25282-7 Page 132 (Theorem 3.4.2. and Algorithm 3.4.3)
  • R. K. Guy, Unsolved Problems in Number Theory, A12.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Pseudoprimes to other bases: A001567 (2), A005935 (3), A005936 (5), A005937 (6), A005939 (10).
Cf. A005382, A122784.

Programs

  • Mathematica
    Select[Range[31000], ! PrimeQ[ # ] && PowerMod[7, (# - 1), # ] == 1 &] (* Farideh Firoozbakht, Sep 14 2006 *)
  • Python
    from sympy import isprime
def ok(n): return pow(7, n-1, n) == 1 and not isprime(n)
print(list(filter(ok, range(1, 34000)))) # Michael S. Branicky, Jun 25 2021

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

Original entry on oeis.org

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

Views

Author

Farideh Firoozbakht, Sep 11 2006

Keywords

Comments

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]

Examples

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

Links

Crossrefs

Cf. A001567, A005935, A122014.
Cf. A122781, A122782, A122783, A122784, A122785, A122786.

Programs

  • Maple
    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
  • Mathematica
    Select[Range[30000], ! PrimeQ[ # ] && Mod[3^#, # ] == Mod[3, # ] &]
Join[{1},Select[Range[20000],!PrimeQ[#]&&PowerMod[3,#,#]==3&]] (* Harvey P. Dale, Apr 30 2023 *)
  • PARI
    is_A122780(n)={n>0 & Mod(3, n)^n==3 & !ispseudoprime(n)} \\ M. F. Hasler, Jul 19 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

Views

Author

Arkadiusz Wesolowski, Aug 05 2017

Keywords

Comments

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

Crossrefs

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

Programs

  • Magma
    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;
  • Mathematica
    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 *)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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