A340480 -id:A340480 - cp's OEIS frontend

A341210 Primes p such that (p^16 + 1)/2 is prime.

3, 29, 41, 73, 113, 157, 167, 173, 199, 599, 607, 617, 1213, 1747, 1979, 2027, 2237, 2377, 2441, 2593, 2659, 2689, 2693, 3061, 3137, 3413, 3457, 3539, 3673, 3733, 3769, 4091, 4157, 4273, 4289, 4547, 4603, 4759, 4877, 4909, 4957, 5039, 5231, 5233, 5303, 5419

Offset: 1

Jon E. Schoenfield, Feb 06 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, and this sequence list primes of the form (p^j + 1)/2 for j=2^0=1, j=2^1=2, j=2^2=4, and j=2^3=8, and j=2^4=16, respectively.

			(3^16 + 1)/2 = 21523361 is prime, so 3 is a term.
(5^16 + 1)/2 = 76293945313 = 2593*29423041, so 5 is not a term.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

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

Select[Prime[Range[750]],PrimeQ[(#^16+1)/2]&] (* Harvey P. Dale, Oct 06 2023 *)

isok(p) = isprime(p) && (p>2) && isprime((p^16 + 1)/2); \\ Michel Marcus, Feb 07 2021

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

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, 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.

			(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.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

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.

q:= p-> (q-> q(p) and q((p^32+1)/2))(isprime):
select(q, [$3..20000])[];  # Alois P. Heinz, Feb 07 2021

Select[Range[17000], PrimeQ[#] && PrimeQ[(#^32 + 1)/2] &] (* Amiram Eldar, Feb 07 2021 *)

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

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, 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.

			(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.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

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).

q:= p-> (q-> q(p) and q((p^64+1)/2))(isprime):
select(q, [$3..20000])[];  # Alois P. Heinz, Feb 07 2021

Select[Range[18000], PrimeQ[#] && PrimeQ[(#^64 + 1)/2] &] (* Amiram Eldar, Feb 07 2021 *)

isok(p) = (p>2) && isprime(p) && ispseudoprime((p^64 + 1)/2); \\ Michel Marcus, Feb 07 2021

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

Original entry on oeis.org

3, 3, 3, 13, 3, 3, 3, 113, 331, 3631, 827, 3109, 4253, 7487, 71

Offset: 0

Jon E. Schoenfield, Feb 06 2021

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.
a(12) <= 4253, a(13) <= 7487, a(14) <= 71. - Daniel Suteu, Feb 07 2021
a(13) > 2500 and a(14) = 71. - Jinyuan Wang, Feb 07 2021

			No term is smaller than 3 (since 2 is the only smaller prime, and (2^(2^n) + 1)/2 is not an integer).
(3^(2^0) + 1)/2 = (3^1 + 1)/2 = (3 + 1)/2 = 4/2 = 2 is prime, so a(0)=3.
(3^(2^1) + 1)/2 = (3^2 + 1)/2 = 5 is prime, so a(1)=3.
(3^(2^2) + 1)/2 = (3^4 + 1)/2 = 41 is prime, so a(2)=3.
(3^(2^3) + 1)/2 = (3^8 + 1)/2 = 3281 = 17*193 is not prime, nor is (p^8 + 1)/2 for any other prime < 13, but (13^8 + 1)/2 = 407865361 is prime, so a(3)=13.

Dario Alpern, Integer factorization calculator

Cf. A005383, A048161, A176116, A340480.

Cf. A093625 and A171381 (both for when p=3).

x=3;x=N(x);NOT IsPrime((x^8192+1)/2);N(x)
# Martin Ehrenstein, Feb 08 2021

a(n) = my(p=3); while (!isprime((p^(2^n) + 1)/2), p=nextprime(p+1)); p; \\ Michel Marcus, Feb 07 2021

from sympy import isprime, nextprime
def a(n):
  p, pow2 = 3, 2**n
  while True:
    if isprime((p**pow2 + 1)//2): return p
    p = nextprime(p)
print([a(n) for n in range(9)]) # Michael S. Branicky, Mar 03 2021

a(11) from Daniel Suteu, Feb 07 2021
a(12) from Jinyuan Wang, Feb 07 2021
a(13)-a(14), using Dario Alpern's integer factorization calculator and prior bounds, from Martin Ehrenstein, Feb 08 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

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

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A341210 Primes p such that (p^16 + 1)/2 is prime.

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

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

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

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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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