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 11-17 of 17 results.

A083733 Pseudoprimes to bases 2 and 7.

Original entry on oeis.org

561, 1105, 2465, 3277, 8321, 10585, 18721, 29341, 46657, 62745, 75361, 104653, 115921, 162401, 219781, 226801, 252601, 278545, 294409, 314821, 334153, 340561, 399001, 410041, 449065, 488881, 512461, 530881, 534061, 552721, 574561, 587861
Offset: 1

Views

Author

Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), May 05 2003

Keywords

Examples

			a(1)=561 since it is the first positive integer k(>1) which satisfies 2^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).

Links

Crossrefs

Intersection of A005938 and A001567. - R. J. Mathar, Apr 05 2011

Programs

Formula

a(n) = n-th positive integer k(>1) such that 2^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).

A083735 Pseudoprimes to bases 3 and 7.

Original entry on oeis.org

703, 1105, 2465, 10585, 18721, 19345, 29341, 38503, 46657, 50881, 75361, 76627, 88831, 104653, 115921, 146611, 162401, 188191, 213265, 226801, 252601, 278545, 286903, 294409, 314821, 334153, 340561, 359341, 385003, 385201, 399001, 410041
Offset: 1

Views

Author

Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), May 05 2003

Keywords

Examples

			a(1)=703 since it is the first number such that 3^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).

Links

Crossrefs

Intersection of A005935 and A005938. - R. J. Mathar, Apr 05 2011

Programs

  • Mathematica
    Select[Range[420000],!PrimeQ[#]&&PowerMod[3,#-1,#]==1&&PowerMod[7,#-1,#] == 1&] (* Harvey P. Dale, Mar 08 2014 *)
  • PARI
    is(n)=!isprime(n)&&Mod(7,n)^(n-1)==1&&Mod(3,n)^(n-1)==1 \\ Charles R Greathouse IV, Apr 12 2012

Formula

a(n) = n-th positive integer k(>1) such that 3^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).

A114249 Number of Fermat pseudoprimes to base 7 less than 10^n.

Original entry on oeis.org

1, 2, 6, 16, 73, 234, 659, 1797, 4950, 13070, 33989, 87448
Offset: 1

Views

Author

Eric W. Weisstein, Nov 18 2005

Keywords

Links

Crossrefs

Cf. A005938, A055550, A114245, A114247.

Programs

  • Mathematica
    Table[Count[Select[Range[2, 10^6], ! PrimeQ[#] && PowerMod[7, # - 1, #] == 1 &], x_ /; x < 10^n], {n, 6}]  (* Robert Price, Jun 09 2019 *)

Extensions

a(9)-a(12) from Hiroaki Yamanouchi, Sep 25 2015

A083736 Pseudoprimes to bases 2,5 and 7.

Original entry on oeis.org

561, 29341, 46657, 75361, 115921, 162401, 252601, 294409, 314821, 334153, 340561, 399001, 410041, 488881, 512461, 530881, 552721, 656601, 658801, 710533, 721801, 852841, 1024651, 1141141, 1152271, 1168513, 1193221, 1461241, 1569457, 1615681
Offset: 1

Views

Author

Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), May 05 2003

Keywords

Examples

			a(1)=561 since it is the first number such that 2^(k-1) = 1 (mod k), 5^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).

Links

Crossrefs

Intersection of A083732 and A005938. Intersection of A083733 and A005936. - R. J. Mathar, Apr 05 2011

Programs

  • Mathematica
    Select[Range[1, 10^5, 2], CompositeQ[#] &&  PowerMod[2, #-1,#] == PowerMod[5, #-1,#] == PowerMod[7, #-1,#] == 1&] (* Amiram Eldar, Jun 29 2019 *)

Formula

a(n) = n-th positive integer k(>1) such that 2^(k-1) = 1 (mod k), 5^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).

A083740 Pseudoprimes to bases 3,5 and 7.

Original entry on oeis.org

29341, 46657, 75361, 88831, 115921, 146611, 162401, 252601, 294409, 314821, 334153, 340561, 399001, 410041, 488881, 512461, 530881, 552721, 658801, 721801, 852841, 954271, 1024651, 1152271, 1193221, 1314631, 1461241, 1569457, 1615681
Offset: 1

Views

Author

Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), May 05 2003

Keywords

Examples

			a(1)=29341 since it is the first number such that 3^(k-1) = 1 (mod k), 5^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).

Links

Crossrefs

Cf. A005935, A005936, A005938, A083734, A083735.

Programs

  • Mathematica
    Select[Range[1, 10^5, 2], CompositeQ[#] &&  PowerMod[3, #-1, #] == PowerMod[5, #-1, #] == PowerMod[7, #-1, #] == 1&]

Formula

a(n) = n-th positive integer k(>1) such that 3^(k-1) = 1 (mod k), 5^(k-1) = 1 (mod k) and 7^(k-1) = 1 (mod k).
Intersection of A083734 and A005938. Intersection of A083735 and A005936. - R. J. Mathar, Apr 05 2011

A374976 Odd k with p^k mod k != p for all primes p.

Original entry on oeis.org

1, 9, 27, 63, 75, 81, 115, 119, 125, 189, 207, 209, 215, 235, 243, 279, 299, 319, 323, 387, 407, 413, 423, 515, 517, 531, 535, 551, 567, 575, 583, 611, 621, 623, 667, 675, 707, 713, 729, 731, 747, 767, 779, 783, 799, 815, 835, 851, 869, 893, 899, 917, 923, 927
Offset: 1

Views

Author

Francois R. Grieu, Jul 26 2024

Keywords

Comments

Alternatively: 1, and odd composites not a pseudoprime to any prime base.
The sequence contains no primes, no pseudoprimes to any prime base (A001567, A005935, A005936, A005938, A020139, A020141...), and no Carmichael numbers (A002997).

Examples

			k=3 (resp. 5, 7) is not in the sequence because for prime p=2 it holds p^k mod k = 2 which is p.
k=9 is in the sequence because for prime p=2 (resp. 3, 5, 7) it holds p^k mod k = 8 (resp. 0, 8, 1) which is not p, and for all other primes p it holds p>=k therefore p^k mod k can't be p.

Links

Crossrefs

Cf. A000040, A001567, A005935, A005936, A005938, A020139, A020141, A020145, A002997.

Programs

  • Mathematica
    Cases[Range[1, 930, 2], k_/; (For[p=2, p=k)]

A247906 a(n) = n-th pseudoprime to base n.

Original entry on oeis.org

561, 286, 341, 781, 1105, 1105, 133, 364, 703, 793, 1105, 1099, 1891, 6541, 1271, 3991, 1649, 1849, 3059, 7363, 2047, 1738, 4537, 1128, 3145, 2993, 5365, 4069, 4097, 7421, 2465, 11305, 2937, 16589, 4495, 2044, 6601, 26885, 13073, 6892, 22945, 3885, 8695, 10879
Offset: 2

Views

Author

Felix Fröhlich, Sep 26 2014

Keywords

Examples

			a(2) = A001567(2) = 561.
a(3) = A005935(3) = 286.

Crossrefs

Cf. A007535, A098650.
Cf. Pseudoprimes to base b: A001567 (b=2), A005935 (b=3), A020136 (b=4), A005936 (b=5), A005937 (b=6), A005938 (b=7), A020137 (b=8), A020138 (b=9).

Programs

  • PARI
    for(n=2, 20, i=0; forcomposite(c=2, 1e9, if(Mod(n, c)^(c-1)==1, i++; if(i==n, print1(c, ", "); i=0; break({1}))); if(c==1e9, print1(">1e9, "))))
Previous Showing 11-17 of 17 results.

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