A341229 -id:A341229 - cp's OEIS frontend

A341230 Primes p such that (p^128 + 1)/2 is prime.

113, 499, 2081, 2287, 5807, 6151, 7823, 9203, 9629, 11069, 11497, 13463, 16987, 17891, 18049, 19889, 24091, 26981, 27259, 27953, 28319, 28597, 31219, 35899, 39047, 41381, 41603, 43403, 44839, 45343, 49529, 50753, 50857, 55079, 60793, 62219, 66721, 72679, 76771

Offset: 1

Jon E. Schoenfield, Feb 07 2021

nonn

Expressions of the form m^j + 1 can be factored (e.g., m^3 + 1 = (m + 1)*(m^2 - m + 1)) for any positive integer j except when j is a power of 2, so (p^j + 1)/2 for prime p cannot be prime unless j is a power of 2. A005383, A048161, A176116, A340480, A341210, A341224, A341229, and this sequence list primes of the form (p^j + 1)/2 for j=2^0=1, j=2^1=2, ..., j=2^7=128, respectively.

			(3^128 + 1)/2 = 5895092288869291585760436430706259332839105796137920554548481 = 257*275201*138424618868737*3913786281514524929*153849834853910661121, so 3 is not a term.
(113^128 + 1)/2 = 3111793506...0421698561 (a 263-digit number) is prime, so 113 is a term. Since 113 is the smallest prime p such that (p^128 + 1)/2 is prime, it is a(1) and is also A341211(7).

Jon E. Schoenfield, Table of n, a(n) for n = 1..1000

Primes p such that (p^(2^k) + 1)/2 is prime: A005383 (k=0), A048161 (k=1), A176116 (k=2), A340480 (k=3), A341210 (k=4), A341224 (k=5), A341229 (k=6), (this sequence) (k=7).

Cf. A341211 (Smallest prime p such that (p^(2^n) + 1)/2 is prime).

isok(p) = (p>2) && isprime(p) && ispseudoprime((p^128 + 1)/2); \\ Michel Marcus, Feb 07 2021

A341234 Primes p such that (p^256 + 1)/2 is prime.

Original entry on oeis.org

331, 1783, 2591, 2791, 7127, 8443, 9007, 9859, 10133, 10883, 10889, 11621, 12101, 13183, 15391, 17737, 19309, 19571, 21863, 24043, 24203, 31159, 32717, 33377, 34267, 35023, 35531, 38177, 39929, 42397, 43499, 46867, 49499, 49943, 50087, 51137, 53101, 53377

Offset: 1

Jon E. Schoenfield, Feb 07 2021

nonn

Expressions of the form m^j + 1 can be factored (e.g., m^3 + 1 = (m + 1)*(m^2 - m + 1)) for any positive integer j except when j is a power of 2, so (p^j + 1)/2 for prime p cannot be prime unless j is a power of 2. A005383, A048161, A176116, A340480, A341210, A341224, A341229, A341230, and this sequence list primes of the form (p^j + 1)/2 for j=2^0=1, j=2^1=2, ..., j=2^8=256, respectively.

			(3^256 + 1)/2 = 6950422618...4449717761 (a 122-digit number) = 12289 * 8972801 * 891206124520373602817 * (a 90-digit prime), so 3 is not a term.
(331^256 + 1)/2 = 5955749334...7416010241 (a 645-digit number) is prime, so 331 is a term. Since 331 is the smallest prime p such that (p^256 + 1)/2 is prime, it is a(1) and is also A341211(8).

Jon E. Schoenfield, Table of n, a(n) for n = 1..100

Primes p such that (p^(2^k) + 1)/2 is prime: A005383 (k=0), A048161 (k=1), A176116 (k=2), A340480 (k=3), A341210 (k=4), A341224 (k=5), A341229 (k=6), A341230 (k=7), (this sequence) (k=8).

Cf. A341211 (Smallest prime p such that (p^(2^n) + 1)/2 is prime).

Select[Range[20000], PrimeQ[#] && PrimeQ[(#^256 + 1)/2] &] (* Amiram Eldar, Feb 07 2021 *)

A341264 Primes p such that (p^512 + 1)/2 is prime.

Original entry on oeis.org

3631, 5113, 10651, 12391, 13999, 22093, 34687, 38713, 38959, 39199, 39679, 44879, 51229, 57389, 58757, 59651, 60331, 61543, 63389, 64483, 72931, 77023, 80369, 91639, 100787, 115679, 119551, 120713, 121727, 122299, 132109, 135599, 140221, 143387, 143873, 145753

Offset: 1

Jon E. Schoenfield, Feb 07 2021

nonn

Expressions of the form m^j + 1 can be factored (e.g., m^3 + 1 = (m + 1)*(m^2 - m + 1)) for any positive integer j except when j is a power of 2, so (p^j + 1)/2 for prime p cannot be prime unless j is a power of 2. A005383, A048161, A176116, A340480, A341210, A341224, A341229, A341230, A341234, and this sequence list primes of the form (p^j + 1)/2 for j=2^0=1, j=2^1=2, ..., j=2^9=512, respectively.

			(3^512 + 1)/2 = 9661674916...6218270721 (a 244-digit number) = 134382593 * 22320686081 * 12079910333441 * 100512627347897906177 * 2652879528...2021744641 (a 193-digit composite number), so 3 is not a term.
(3631^512 + 1)/2 = 2706508826...0763924481 (an 1823-digit number) is prime, so 3631 is a term. Since 3631 is the smallest prime p such that (p^512 + 1)/2 is prime, it is a(1) and is also A341211(9).

Primes p such that (p^(2^k) + 1)/2 is prime: A005383 (k=0), A048161 (k=1), A176116 (k=2), A340480 (k=3), A341210 (k=4), A341224 (k=5), A341229 (k=6), A341230 (k=7), A341234 (k=8), (this sequence) (k=9).

Cf. A341211 (Smallest prime p such that (p^(2^n) + 1)/2 is prime).

A341272 Primes p such that (p^1024 + 1)/2 is prime.

Original entry on oeis.org

827, 10861, 19501, 22751, 23339, 23663, 26347, 29581, 50077, 62131, 63331, 70657, 72221, 73523, 78301, 85447, 109013, 122363, 127363, 149213, 155461, 170551, 173549, 183877, 188579, 206627, 218149, 220147, 222029, 226099, 227231, 232051, 247601, 248317, 248543

Offset: 1

Jon E. Schoenfield, Feb 07 2021

nonn

Expressions of the form m^j + 1 can be factored (e.g., m^3 + 1 = (m + 1)*(m^2 - m + 1)) for any positive integer j except when j is a power of 2, so (p^j + 1)/2 for prime p cannot be prime unless j is a power of 2. A005383, A048161, A176116, A340480, A341210, A341224, A341229, A341230, A341234, A341264, and this sequence list primes of the form (p^j + 1)/2 for j=2^0=1, j=2^1=2, ..., j=2^10=1024, respectively.

			(3^1024 + 1)/2 = 1866959243...6855178241 (a 489-digit number) = 59393 * 448524289 * 847036417 * 8273923970...2296603649 (a 466-digit composite number), so 3 is not a term.
(827^1024 + 1)/2 = 1677304013...0116613121 (a 2988-digit number) is prime, so 827 is a term. Since 827 is the smallest prime p such that (p^1024 + 1)/2 is prime, it is a(1) and is also A341211(10).

Primes p such that (p^(2^k) + 1)/2 is prime: A005383 (k=0), A048161 (k=1), A176116 (k=2), A340480 (k=3), A341210 (k=4), A341224 (k=5), A341229 (k=6), A341230 (k=7), A341234 (k=8), A341264 (k=9), (this sequence) (k=10).

Cf. A341211 (Smallest prime p such that (p^(2^n) + 1)/2 is prime).

a(17)-a(35) from Jinyuan Wang, Feb 09 2021

cp's OEIS Frontend

A341230 Primes p such that (p^128 + 1)/2 is prime.

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A341234 Primes p such that (p^256 + 1)/2 is prime.

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A341264 Primes p such that (p^512 + 1)/2 is prime.

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A341272 Primes p such that (p^1024 + 1)/2 is prime.

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