A176116 -id:A176116 - cp's OEIS frontend

A340480 Primes p such that (p^8 + 1)/2 is prime.

13, 43, 47, 53, 239, 373, 409, 433, 491, 557, 577, 859, 1021, 1103, 1307, 1531, 1699, 1753, 1777, 1871, 2053, 2083, 2297, 2467, 2503, 2593, 2797, 2957, 3251, 3307, 3323, 3511, 3613, 4099, 4523, 4637, 4951, 4999, 5591, 5657, 5693, 5801, 5827, 5849, 6043, 6163

Offset: 1

Jon E. Schoenfield, Jan 31 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, 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, respectively.

(p^8 + 1)/2 is divisible by 17 when m mod 34 is 3, 5, 7, 11, 23, 27, 29, or 31.

			(3^8 + 1)/2 = 3281 = 17*193, so 3 is not a term.
(13^8 + 1)/2 = 407865361 is prime, so 13 is a term.
(17^8 + 1)/2 = 3487878721 = 18913 * 184417, so 17 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), (this sequence) (k=3).

Prime[Position[Table[(Prime[p]^8 + 1)/2, {p, 1, 803}], Integer?PrimeQ]] // Flatten (* _Robert P. P. McKone, Jan 31 2021 *)

isok(p) = isprime(p) && (p>2) && isprime((p^8 + 1)/2); \\ Michel Marcus, Feb 01 2021

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

Original entry on oeis.org

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

A277201 Primes of the form (p^4 + 1)/2, where p is prime.

Original entry on oeis.org

41, 313, 1201, 7321, 14281, 41761, 139921, 353641, 6922921, 12705841, 14199121, 56275441, 81523681, 784119601, 1984563001, 4798962481, 5049019561, 6448958881, 7763701441, 15410832361, 17253574561, 20321481601, 22977034081, 26321586241

Offset: 1

Lechoslaw Ratajczak, Oct 04 2016

nonn

The sequence is a subsequence of A096170.
Conjecture: the sequence consists of the numbers k such that tau(2k) = 4 and tau(2k-1) = 5. tau(82) = 4 and tau(81) = 5, 82/2 = 41 = a(1). tau(626) = 4 and tau(625) = 5, 626/2 = 313 = a(2). tau(2402) = 4 and tau(2401) = 5, 2402/2 = 1201 = a(3). The conjecture was checked for 10^9 consecutive integers.
The above conjecture is true: since tau(2k-1) = 5, 2k-1 must be the 4th power of some prime p, i.e., k = (p^4 + 1)/2 (so p is odd, so p^4 == 1 (mod 16), so k is odd), and since tau(2k) = 4, 2k must be the product of two distinct primes, so k is an odd prime. Thus, the set of numbers k such that tau(2k) = 4 and tau(2k-1) = 5 is the set of primes of the form (p^4 + 1)/2, where p is prime. - Jon E. Schoenfield, Mar 17 2019
Primes of the form a^2 + b^2 such that a^2 - b^2 = p^2, where p is prime. - Thomas Ordowski, Feb 14 2017

			a(1) = 41 because 3 is prime and (3^4 + 1)/2 = 41 is prime.
a(2) = 313 because 5 is prime and (5^4 + 1)/2 = 313 is prime.
a(3) = 1201 because 7 is prime and (7^4 + 1)/2 = 1201 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A096169, A096170, A176116.

[a: p in PrimesUpTo(1000) | IsPrime(a) where a is (p^4+1) div 2 ]; // Vincenzo Librandi, Nov 07 2016

Select[Map[(#^4 + 1)/2 &, Prime@ Range@ 100], PrimeQ] (* Michael De Vlieger, Oct 04 2016 *)
Select[Table[(p^4+1)/2,{p,Prime[Range[100]]}],PrimeQ] (* Harvey P. Dale, Dec 21 2018 *)

makelist(if primep(k)=true then ((k^4)+1)/2 else 0,k,3,500,1)$ sublist(%,primep);

lista(nn) = {forprime(p=3, nn, if (isprime(q=(p^4+1)/2), print1(q, ", ")););} \\ Michel Marcus, Oct 04 2016

a(n) = (A176116(n)^4 + 1)/2.

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

A340480 Primes p such that (p^8 + 1)/2 is prime.

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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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Maple

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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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A277201 Primes of the form (p^4 + 1)/2, where p is prime.

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Magma