cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

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Author

Jon E. Schoenfield, Feb 07 2021

Keywords

Comments

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.

Examples

			(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).

Crossrefs

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

Views

Author

Jon E. Schoenfield, Feb 07 2021

Keywords

Comments

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.

Examples

			(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).

Crossrefs

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).

Extensions

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