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.

A126432 Double Wieferich primes with q = 5.

Original entry on oeis.org

1645333507, 188748146801
Offset: 1

Views

Author

Alexander Adamchuk, Mar 12 2007

Keywords

Comments

Double Wieferich prime pairs are pairs of primes (p, q) such that q^(p-1) == 1 (mod p^2) and p^(q-1) == 1 (mod q^2).

Links

Crossrefs

Subsequence of A123692.
Cf. A124121, A124122, A125775.

A068535 Numbers k such that 2^k mod k = 2^k mod k^2.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 35, 64, 128, 256, 297, 512, 1024, 1093, 2048, 2186, 2590, 3279, 3511, 4096, 4372, 5465, 6558, 7022, 7651, 8192, 8744, 9837, 10533, 10930, 13116, 14044, 14209, 16384, 21066, 23175, 24012, 24577, 26592, 28088, 31599, 32768, 35110, 38621, 42132
Offset: 1

Views

Author

Benoit Cloitre, Mar 22 2002

Keywords

Comments

Sequence contains all numbers of the form 2^k (A000079) but also many others as the subsequence of primes in A001220.

Links

Crossrefs

Cf. A000079, A001220, A125773, A125774, A125775.

Programs

  • Mathematica
    Select[Range[33000],PowerMod[2,#,#]==PowerMod[2,#,#^2]&] (* Harvey P. Dale, Oct 29 2017 *)

Extensions

More terms from Amiram Eldar, Jun 19 2022

A125774 Numbers k such that 3^k mod k = 3^k mod k^2.

Original entry on oeis.org

1, 2, 3, 4, 9, 11, 20, 22, 27, 33, 81, 99, 220, 243, 644, 729, 1220, 2187, 2420, 5060, 6561, 7128, 8368, 13420, 14740, 19683, 23620, 40573, 55660, 59049, 145420, 147620, 162140, 177147, 237820, 259820, 290620, 308660, 339020, 447740, 531441, 548660
Offset: 1

Views

Author

Alexander Adamchuk, Dec 07 2006

Keywords

Comments

This sequence includes all powers of 3. a(2) = 2, a(3) = 3, a(6) = 11 and a(45) = 1006003 are the only known primes in this sequence.

Links

Crossrefs

Cf. A014127 (Primes p such that p^2 divides 3^(p-1) - 1).
Cf. A068535 (Numbers k such that 2^k mod k = 2^k mod k^2).
Cf. A125773 (Numbers k, that are not powers of 2, such that 2^k mod k = 2^k mod k^2).
Cf. A125775 (Numbers k such that 5^k mod k = 5^k mod k^2).

Programs

  • Mathematica
    Do[f=PowerMod[3,n,n];g=PowerMod[3,n,n^2];If[f==g,Print[n]],{n,1,1100000}]
Select[Range[600000],PowerMod[3,#,#]==PowerMod[3,#,#^2]&] (* Harvey P. Dale, Feb 21 2013 *)

A125773 Numbers k that are not powers of 2 such that 2^k mod k = 2^k mod k^2; or A068535 with powers of 2 excluded.

Original entry on oeis.org

35, 297, 1093, 2186, 2590, 3279, 3511, 4372, 5465, 6558, 7022, 7651, 8744, 9837, 10533, 10930, 13116, 14044, 14209, 21066, 23175, 24012, 24577, 26592, 28088, 31599, 35110, 38621, 42132, 49154, 987704, 3020871, 3074592, 18368834, 22655923, 105713883, 111503202, 1084277175
Offset: 1

Views

Author

Alexander Adamchuk, Dec 07 2006

Keywords

Comments

A068535 includes all powers of 2. a(3) = 1093 and a(7) = 3511 are the only known primes in this sequence. They belong to A001220 = Wieferich primes p: p^2 divides 2^(p-1) - 1. Note that most listed terms of this sequence are the multiples of Wieferich primes 1093 and 3511. No more terms in this sequence up to 6*10^6.

Crossrefs

Cf. A068535 (Numbers k such that 2^k mod k = 2^k mod k^2).
Cf. A001220 (Wieferich primes p: p^2 divides 2^(p-1) - 1).
Cf. A125774 (Numbers k such that 3^k mod k = 3^k mod k^2).
Cf. A125775 (Numbers k such that 5^k mod k = 5^k mod k^2).

Programs

  • Mathematica
    Do[f=PowerMod[2,n,n];g=PowerMod[2,n,n^2];If[f==g&&!IntegerQ[Log[2,n]],Print[n]],{n,1,6000000}]

Extensions

More terms from Amiram Eldar, Jun 19 2022
Showing 1-4 of 4 results.

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