A374913 -id:A374913 - cp's OEIS frontend

A374912 Primes p such that (p - 1)^p == p (mod 2*p - 1).

3, 7, 19, 31, 79, 139, 199, 211, 271, 307, 331, 367, 379, 439, 499, 547, 607, 619, 691, 727, 811, 967, 1171, 1279, 1399, 1459, 1531, 1627, 1759, 1867, 2011, 2131, 2179, 2311, 2467, 2539, 2551, 2707, 2719, 2791, 2851, 3019, 3067, 3187, 3319, 3331, 3391, 3499, 3607, 3739, 3967

Offset: 1

Juri-Stepan Gerasimov, Jul 23 2024

nonn

Paolo Xausa, Table of n, a(n) for n = 1..10000

Aside from the first term, a subsequence of A068229.

Cf. A374913, A374914.

[p: p in PrimesUpTo(10^4) | (p-1)^p mod (2*p-1) eq p];

Select[Prime[Range[1000]], PowerMod[# - 1, #, 2*# - 1] == # &] (* Paolo Xausa, Jul 24 2024 *)

list(lim)=my(v=List([3])); forprimestep(p=7,lim\1,12, if(Mod(p-1,2*p-1)^p==p, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Jul 23 2024

a(n) == 7 (mod 12) for n>1. - Hugo Pfoertner, Jul 24 2024

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

A380831 Numbers k such that k^(k + 1) == k + 1 (mod 2k + 1) while 2k+1 is not prime.

Original entry on oeis.org

1023, 1638, 14670, 21399, 24570, 40290, 44178, 45375, 52326, 98046, 128499, 135975, 157410, 229494, 244998, 257223, 370875, 400302, 419430, 436590, 458163, 502326, 625974, 686826, 754854, 839270, 905786, 993510, 1102983, 1134546, 1142226, 1152083, 1193898, 1373238, 1374011

Offset: 1

Michel Marcus, Feb 05 2025

nonn

Intersection of A374913 and A047845.

isok(k) = (!isprime(2*k+1)) && (Mod(k, 2*k+1)^(k+1) == k+1);

cp's OEIS Frontend

A374912 Primes p such that (p - 1)^p == p (mod 2*p - 1).

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

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A380831 Numbers k such that k^(k + 1) == k + 1 (mod 2k + 1) while 2k+1 is not prime.

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cp's OEIS Frontend

A374912 Primes p such that (p - 1)^p == p (mod 2*p - 1).

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Mathematica

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

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Author

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Mathematica

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A380831 Numbers k such that k^(k + 1) == k + 1 (mod 2*k + 1) while 2*k+1 is not prime.

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A380831 Numbers k such that k^(k + 1) == k + 1 (mod 2k + 1) while 2k+1 is not prime.