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.

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

A125775 Numbers k such that 5^k mod k = 5^k mod k^2.

Original entry on oeis.org

1, 2, 4, 5, 6, 12, 25, 42, 52, 84, 125, 156, 186, 372, 625, 1092, 1218, 1302, 1806, 2436, 2604, 2756, 3125, 3612, 4836, 5334, 7212, 8268, 10668, 12324, 15625, 15918, 18858, 19140, 20771, 24492, 26080, 31668, 31836, 33852, 37716, 37758, 40487, 41542
Offset: 1

Views

Author

Alexander Adamchuk, Dec 07 2006

Keywords

Comments

Includes all powers of 5 (A000351).
a(2) = 2, a(4) = 5, a(35) = 20771 and a(43) = 40487 are the only listed primes. More known primes are listed in A123692.

Links

Crossrefs

Cf. A000351, A123692, A068535, A125773, A125774.

Programs

  • Mathematica
    Do[f=PowerMod[5,n,n];g=PowerMod[5,n,n^2];If[f==g,Print[n]],{n,1,1000000}]
Select[Range[42000],PowerMod[5,#,#]==PowerMod[5,#,#^2]&] (* Harvey P. Dale, Aug 20 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 *)
Showing 1-3 of 3 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.