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.

Showing 1-6 of 6 results.

A341224 Primes p such that (p^32 + 1)/2 is prime.

Original entry on oeis.org

3, 163, 181, 191, 229, 251, 839, 971, 1181, 1201, 1489, 1801, 1823, 1847, 1861, 1987, 2069, 2087, 3547, 3691, 3697, 6361, 6637, 6899, 6967, 7793, 7963, 8731, 8737, 10253, 10271, 10613, 10639, 10799, 11981, 12689, 12697, 13697, 13841, 14951, 15299, 16547, 16747
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, and this sequence list primes of the form (p^j + 1)/2 for j=2^0=1, j=2^1=2, ..., j=2^5=32, respectively.

Examples

			(3^32 + 1)/2 = 926510094425921 is prime, so 3 is a term.
(5^32 + 1)/2 = 11641532182693481445313 = 641*75068993*241931001601, so 5 is not a term.

Links

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), (this sequence) (k=5).
Cf. A000040.

Programs

  • Maple
    q:= p-> (q-> q(p) and q((p^32+1)/2))(isprime):
select(q, [$3..20000])[];  # Alois P. Heinz, Feb 07 2021
  • Mathematica
    Select[Range[17000], PrimeQ[#] && PrimeQ[(#^32 + 1)/2] &] (* Amiram Eldar, Feb 07 2021 *)
  • PARI
    isok(p) = (p>2) && isprime(p) && isprime((p^32 + 1)/2); \\ Michel Marcus, Feb 07 2021

A341229 Primes p such that (p^64 + 1)/2 is prime.

Original entry on oeis.org

3, 353, 587, 727, 863, 883, 919, 1217, 1237, 1657, 2029, 2203, 2333, 3209, 3529, 3617, 3889, 4889, 5387, 5557, 5689, 5749, 6701, 6961, 7727, 8443, 9377, 9433, 10009, 10243, 10691, 10799, 11027, 12071, 12451, 13681, 13687, 15569, 15601, 15823, 16759, 17939
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, and this sequence list primes of the form (p^j + 1)/2 for j=2^0=1, j=2^1=2, ..., j=2^6=64, respectively.

Examples

			(3^64 + 1)/2 = 1716841910146256242328924544641 is prime, so 3 is a term.
(5^64 + 1)/2 = 271050543121376108501863200217485427856445313 = 769*3666499598977*96132956782643741951225664001, so 5 is not a term.

Links

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), (this sequence) (k=6).

Programs

  • Maple
    q:= p-> (q-> q(p) and q((p^64+1)/2))(isprime):
select(q, [$3..20000])[];  # Alois P. Heinz, Feb 07 2021
  • Mathematica
    Select[Range[18000], PrimeQ[#] && PrimeQ[(#^64 + 1)/2] &] (* Amiram Eldar, Feb 07 2021 *)
  • PARI
    isok(p) = (p>2) && isprime(p) && ispseudoprime((p^64 + 1)/2); \\ Michel Marcus, Feb 07 2021

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

Original entry on oeis.org

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

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, 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.

Examples

			(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).

Links

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), (this sequence) (k=7).
Cf. A341211 (Smallest prime p such that (p^(2^n) + 1)/2 is prime).

Programs

  • PARI
    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

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, 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.

Examples

			(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).

Links

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), (this sequence) (k=8).
Cf. A341211 (Smallest prime p such that (p^(2^n) + 1)/2 is prime).

Programs

  • Mathematica
    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

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, 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
Showing 1-6 of 6 results.

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