A374912 -id:A374912 - cp's OEIS frontend

A374913 Numbers k such that k^(k + 1) == k + 1 (mod 2*k + 1).

2, 3, 6, 11, 14, 15, 18, 23, 26, 30, 35, 39, 50, 51, 54, 63, 74, 75, 78, 83, 86, 90, 95, 98, 99, 111, 114, 119, 131, 134, 135, 138, 146, 155, 158, 174, 179, 183, 186, 191, 194, 198, 210, 215, 219, 230, 231, 239, 243, 251, 254, 270, 278, 299, 303, 306, 315, 323, 326, 330, 338, 350

Offset: 1

Juri-Stepan Gerasimov, Jul 23 2024

nonn

Michel Marcus, Table of n, a(n) for n = 1..10000

Supersequence of A002515 and A374914.

Cf. A374912.

[n: n in [0..350] | n^(n+1) mod (2*n+1) eq n+1];

Select[Range[350],Mod[#^(#+1),2#+1]==#+1 &] (* Stefano Spezia, Jul 23 2024 *)

isok(k) = Mod(k, 2*k+1)^(k+1) == k+1; \\ Michel Marcus, Feb 05 2025

Conjecture (Superseeker): a(n) = A263458(n)/2. - R. J. Mathar, Aug 02 2024

The conjectured formula is false. There exist numbers k such that 2*k + 1 is composite and k^(k + 1) == k + 1 (mod 2*k + 1). For example, when k = 1023: 1023^1024 == 1024 (mod 2047) and 2047 = 23*89 is composite. - Jedrzej Miarecki, Jan 16 2025

A374914 Primes p == 2, 3 (mod 4) with 2*p+1 prime.

Original entry on oeis.org

2, 3, 11, 23, 83, 131, 179, 191, 239, 251, 359, 419, 431, 443, 491, 659, 683, 719, 743, 911, 1019, 1031, 1103, 1223, 1439, 1451, 1499, 1511, 1559, 1583, 1811, 1931, 2003, 2039, 2063, 2339, 2351, 2399, 2459, 2543, 2699, 2819, 2903, 2939, 2963, 3023, 3299, 3359, 3491

Offset: 1

Juri-Stepan Gerasimov, Jul 23 2024

2 together with Lucasian primes (A002515).

Primes p such that p^(p + 1) == p + 1 (mod 2*p + 1).

			2 is in this sequence because 2^(2 + 1) = 8 and 8 = 3 (mod 2*2 + 1) where 2 prime.

Supersequence of A002515. Subsequence of A374913.

Cf. A374912.

Select[Prime[Range[490]],Mod[#^(#+1),2#+1]==#+1 &] (* Stefano Spezia, Jul 23 2024 *)

list(lim)=my(v=List([2])); forprimestep(p=3,lim\1,4, if(isprime(2*p+1), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Jul 25 2024

a(n) >> n log^2 n. - Charles R Greathouse IV, Jul 25 2024

A374919 Primes p such that -(p - 1)^p == p (mod 2*p - 1).

Original entry on oeis.org

2, 37, 97, 157, 229, 281, 337, 577, 601, 661, 829, 877, 937, 953, 997, 1009, 1069, 1237, 1297, 1429, 1609, 1657, 2017, 2029, 2089, 2137, 2221, 2281, 2341, 2557, 2617, 2731, 3037, 3061, 3109, 3169, 3181, 3301, 3529, 3697, 3709, 3769, 3877, 4177, 4241, 4261, 4357, 4621, 4801, 4861, 4909

Offset: 1

Juri-Stepan Gerasimov, Jul 23 2024

nonn

Cf. A374912.

[p: p in PrimesUpTo(5000) | -(p-1)^p mod (2*p-1) eq p];

Select[Prime[Range[700]], PowerMod[# - 1, #, 2*# - 1] == # - 1 &] (* Amiram Eldar, Jul 23 2024 *)

cp's OEIS Frontend

A374913 Numbers k such that k^(k + 1) == k + 1 (mod 2*k + 1).

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A374914 Primes p == 2, 3 (mod 4) with 2*p+1 prime.

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A374919 Primes p such that -(p - 1)^p == p (mod 2*p - 1).

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