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.

Previous Showing 11-15 of 15 results.

A262053 Euler pseudoprimes to base 6: composite integers such that abs(6^((n - 1)/2)) == 1 mod n.

Original entry on oeis.org

185, 217, 301, 481, 1111, 1261, 1333, 1729, 2465, 2701, 3421, 3565, 3589, 3913, 5713, 6533, 8365, 10585, 11041, 11137, 12209, 14701, 15841, 17329, 18361, 20017, 21049, 22049, 29341, 31021, 31621, 34441, 36301, 38081, 39305, 39493, 41041, 43621, 44801, 46657
Offset: 1

Views

Author

Daniel Lignon, Sep 09 2015

Keywords

Links

Crossrefs

Cf. A006970 (base 2), A262051 (base 3), A262052 (base 5), this sequence (base 6), A262054 (base 7), A262055 (base 8).

Programs

  • Mathematica
    eulerPseudoQ[n_?PrimeQ, b_] = False; eulerPseudoQ[n_, b_] := Block[{p = PowerMod[b, (n - 1)/2, n]}, p == Mod[1, n] || p == Mod[-1, n]]; Select[2 Range[25000] + 1, eulerPseudoQ[#, 6] &] (* Michael De Vlieger, Sep 09 2015, after Jean-François Alcover at A006970 *)
  • PARI
    for(n=1, 1e5, if( Mod(6, (2*n+1))^n == 1 ||  Mod(6, (2*n+1))^n == 2*n && bigomega(2*n+1) != 1 , print1(2*n+1", "))); \\ Altug Alkan, Oct 11 2015

A262054 Euler pseudoprimes to base 7: composite integers such that abs(7^((n - 1)/2)) == 1 mod n.

Original entry on oeis.org

25, 325, 703, 817, 1825, 2101, 2353, 2465, 3277, 4525, 6697, 8321, 10225, 11041, 11521, 12025, 13665, 14089, 19345, 20197, 20417, 20425, 25829, 29857, 29891, 35425, 38081, 39331, 46657, 49241, 49321, 50881, 58825, 64681, 75241, 75361, 76627, 78937, 79381
Offset: 1

Views

Author

Daniel Lignon, Sep 09 2015

Keywords

Links

Crossrefs

Cf. A006970 (base 2), A262051 (base 3), A262052 (base 5), A262053 (base 6), this sequence (base 7), A262055 (base 8).

Programs

  • Mathematica
    eulerPseudoQ[n_?PrimeQ, b_] = False; eulerPseudoQ[n_, b_] := Block[{p = PowerMod[b, (n - 1)/2, n]}, p == Mod[1, n] || p == Mod[-1, n]]; Select[2 Range[25000] + 1, eulerPseudoQ[#, 7] &] (* Michael De Vlieger, Sep 09 2015, after Jean-François Alcover at A006970 *)
  • PARI
    for(n=1, 1e5, if( Mod(7, (2*n+1))^n == 1 ||  Mod(7, (2*n+1))^n == 2*n && bigomega(2*n+1) != 1 , print1(2*n+1", "))); \\ Altug Alkan, Oct 11 2015

A262055 Euler pseudoprimes to base 8: composite integers such that abs(8^((n - 1)/2)) == 1 mod n.

Original entry on oeis.org

9, 21, 65, 105, 133, 273, 341, 481, 511, 561, 585, 1001, 1105, 1281, 1417, 1541, 1661, 1729, 1905, 2047, 2465, 2501, 3201, 3277, 3641, 4033, 4097, 4641, 4681, 4921, 5461, 6305, 6533, 6601, 7161, 8321, 8481, 9265, 9709, 10261, 10585, 10745, 11041, 12545
Offset: 1

Views

Author

Daniel Lignon, Sep 09 2015

Keywords

Links

Crossrefs

Cf. A006970 (base 2), A262051 (base 3), A262052 (base 5), A262053 (base 6), A262054 (base 7), this sequence (base 8).

Programs

  • Mathematica
    eulerPseudoQ[n_?PrimeQ, b_] = False; eulerPseudoQ[n_, b_] := Block[{p = PowerMod[b, (n - 1)/2, n]}, p == Mod[1, n] || p == Mod[-1, n]]; Select[2 Range[11000] + 1, eulerPseudoQ[#, 8] &] (* Michael De Vlieger, Sep 09 2015, after Jean-François Alcover at A006970 *)
  • PARI
    for(n=1, 1e5, if( Mod(8, (2*n+1))^n == 1 ||  Mod(8, (2*n+1))^n == 2*n && bigomega(2*n+1) != 1 , print1(2*n+1", "))); \\ Altug Alkan, Oct 11 2015

A263239 Euler pseudoprimes to base 9: composite integers such that abs(9^((n - 1)/2)) == 1 mod n.

Original entry on oeis.org

4, 28, 91, 121, 286, 532, 671, 703, 949, 1036, 1105, 1541, 1729, 1891, 2465, 2665, 2701, 2821, 3281, 3367, 3751, 4636, 4961, 5551, 6364, 6601, 7381, 8401, 8911, 10585, 11011, 11476, 12403, 14383, 15203, 15457, 15841, 16471, 16531, 18721, 19345, 19684, 23521, 24046, 24661, 24727
Offset: 1

Views

Author

Daniel Lignon, Oct 12 2015

Keywords

Comments

Even numbers are permitted since 9 is an integer square. - Charles R Greathouse IV, Oct 12 2015

Links

Crossrefs

Cf. A020138 (pseudoprimes to base 9).
Cf. A006970 (base 2), A262051 (base 3), A262052 (base 5), A262053 (base 6), A262054 (base 7), A262055 (base 8).

Programs

  • Mathematica
    eulerPseudo9Q[n_]:=(Mod[9^((n-1)/2)+1,n]==0 ||Mod[9^((n-1)/2)-1,n]==0) && Not[PrimeQ[n]];
Select[Range[2,200000],eulerPseudo9Q]
  • PARI
    is(n) = abs(centerlift(Mod(3, n)^(n-1)))==1 && !isprime(n) && n>1 \\ Charles R Greathouse IV, Oct 12 2015

A345753 Euler-Fibonacci pseudoprimes: odd composites k such that F(k) == 5^((k-1)/2) == +-1 (mod k), where F(k) = A000045(k), the Fibonacci numbers.

Original entry on oeis.org

146611, 252601, 399001, 512461, 556421, 852841, 1024651, 1193221, 1314631, 1857241, 1909001, 2100901, 2165801, 2603381, 2704801, 3470921, 3828001, 3942271, 4504501, 5049001, 5148001, 5481451, 6189121, 6840001, 7267051, 7519441, 7879681, 8086231, 8341201, 8719921, 9439201, 9863461
Offset: 1

Views

Author

Amiram Eldar and Thomas Ordowski, Jun 26 2021

Keywords

Comments

If p is an odd prime except 5, then F(p) == 5^((p-1)/2) == +-1 (mod p).
All terms found satisfy the congruence F(k) == 5^((k-1)/2) == 1 (mod k). They are a proper subset of A094394.
Are there odd composites m such that F(m) == 5^((m-1)/2) == -1 (mod m)? They are a proper subset (maybe empty) of A094395 (they are not in the database, below 4*10^9).

Crossrefs

Cf. A000045, A094394, A094395, A262052.

Programs

  • Mathematica
    Select[Range[1, 10^6, 2], CompositeQ[#] && MemberQ[{1, # - 1}, PowerMod[5, (# - 1)/2, #]] && Divisible[5^((# - 1)/2) - Fibonacci[#], #] &]
Previous Showing 11-15 of 15 results.

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

Content is available under The OEIS End-User License Agreement.