A374912 -id:A374912

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A374913

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

FORMULA

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

Supersequence of A002515 and A374914.

Cf. A374912.

KEYWORD

nonn

AUTHOR

Juri-Stepan Gerasimov, Jul 23 2024

STATUS

approved

A374914

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

2 together with Lucasian primes (A002515).

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

LINKS

Table of n, a(n) for n=1..49.

FORMULA

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

EXAMPLE

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

MATHEMATICA

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

PROG

(PARI) 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

CROSSREFS

Supersequence of A002515. Subsequence of A374913.

Cf. A374912.

KEYWORD

nonn,easy

AUTHOR

Juri-Stepan Gerasimov, Jul 23 2024

STATUS

approved

A374919

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

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..51.

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A374912.

KEYWORD

nonn

AUTHOR

Juri-Stepan Gerasimov, Jul 23 2024

STATUS

approved

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