A125773 -id:A125773 - cp's OEIS frontend

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

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

Alexander Adamchuk, Dec 07 2006

nonn

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.

Amiram Eldar, Table of n, a(n) for n = 1..1000

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

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 *)

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

Benoit Cloitre, Mar 22 2002

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

Amiram Eldar, Table of n, a(n) for n = 1..72

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

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

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

Alexander Adamchuk, Dec 07 2006

nonn

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.

Amiram Eldar, Table of n, a(n) for n = 1..400 (terms 1..100 from Harvey P. Dale)

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).

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 *)

cp's OEIS Frontend

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

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

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

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