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 1-4 of 4 results.

A005939 Pseudoprimes to base 10.

Original entry on oeis.org

9, 33, 91, 99, 259, 451, 481, 561, 657, 703, 909, 1233, 1729, 2409, 2821, 2981, 3333, 3367, 4141, 4187, 4521, 5461, 6533, 6541, 6601, 7107, 7471, 7777, 8149, 8401, 8911, 10001, 11111, 11169, 11649, 12403, 12801, 13833, 13981, 14701, 14817, 14911, 15211
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

This sequence is a subsequence of A121014 & A121912. In fact the terms are composite terms n of these sequences such that gcd(n,10)=1. Theorem: If both numbers q & 2q-1 are primes(q is in the sequence A005382) and n=q*(2q-1) then 10^(n-1) == 1 (mod n) (n is in the sequence A005939) iff mod(q, 20) is in the set {1, 7, 19}. 91,703,12403,38503,79003,188191,269011,... are such terms. - Farideh Firoozbakht, Sep 15 2006
Composite numbers n such that 10^(n-1) == 1 (mod n). - Michel Lagneau, Feb 18 2012
Composite numbers n such that the number of digits of the period of 1/n divides n-1. - Davide Rotondo, Dec 16 2020

References

  • 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

Cf. A001567 (pseudoprimes to base 2), A005382, A121014, A121912.

Programs

  • Mathematica
    Select[Range[15300], ! PrimeQ[ # ] && PowerMod[10, (# - 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

Views

Author

Farideh Firoozbakht, Sep 12 2006

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

A121912 Numbers k such that 10^k == 10 (mod k).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 15, 17, 18, 19, 23, 29, 30, 31, 33, 37, 41, 43, 45, 47, 53, 55, 59, 61, 67, 71, 73, 79, 83, 89, 90, 91, 97, 99, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 165, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227
Offset: 1

Views

Author

Zak Seidov, Sep 02 2006

Keywords

Comments

By Fermat, all primes are members.
Numbers k not divisible by 4 or 25 such that the multiplicative order of 10 mod (k/gcd(k,10)) divides k-1. - Robert Israel, Feb 10 2019
10^2^k + 1, 10^5^k + 1 and 10^10^k + 1 are terms for k >= 0. - Jinyuan Wang, Feb 11 2019

Examples

			13 is a term because 10^13 = 13*769230769230 + 10.

Links

Crossrefs

Cf. A056969 (10^n modulo n), A121014 (Nonprime terms in A121912).

Programs

  • Maple
    filter:= n -> (10 &^ n - 10 mod n = 0):
select(filter, [$1..1000]); # Robert Israel, Feb 10 2019
  • Mathematica
    Select[Range[250], PowerMod[10, #, # ] == Mod[10, # ] &] (* Ray Chandler, Sep 02 2006 *)
  • PARI
    is(n) = Mod(10, n)^n == Mod(10, n) \\ Jinyuan Wang, Feb 11 2019

A153519 Nonprime numbers k such that 10^k == 10 (mod k) and are not Carmichael numbers.

Original entry on oeis.org

1, 6, 9, 10, 15, 18, 30, 33, 45, 55, 90, 91, 99, 165, 246, 259, 370, 385, 451, 481, 495, 505, 657, 703, 715, 909, 1035, 1045, 1233, 1626, 2035, 2409, 2981, 3333, 3367, 3585, 4005, 4141, 4187, 4521, 4545, 5005, 5461, 6533, 6541, 6565, 7107, 7471, 7777, 8149
Offset: 1

Views

Author

Artur Jasinski, Dec 28 2008

Keywords

Comments

Old name: Members of A121014 which are not Carmichael numbers A002997.

Links

Crossrefs

Cf. A002997, A122780, A153508, A153513, A153514, A153515.

Programs

  • Mathematica
    Select[Range[8000], !PrimeQ[#] && PowerMod[10, #, #] == Mod[10, #] && !(# > 1 && Divisible[# - 1, CarmichaelLambda[#]]) &] (* Amiram Eldar, Mar 19 2020 *)
  • PARI
    isok(n) = !isprime(n) && !is_A002997(n) && (Mod(10^n, n) == Mod(10, n)); \\ Michel Marcus, Nov 06 2013

Extensions

New name from Michel Marcus, Nov 06 2013
Showing 1-4 of 4 results.

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

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