A048161 -id:A048161 - cp's OEIS frontend

A048295 Sequence of 3 Pythagorean triangles, each with a leg and hypotenuse prime. The hypotenuse of each triangle is the leg of the next triangle.

271, 349, 3001, 10099, 11719, 12281, 25889, 39901, 46399, 63659, 169219, 250361, 264169, 287629, 289049, 312581, 353081, 440681, 473009, 502501, 502961, 541951, 594751, 620491, 627911, 632699, 704581, 757111, 762899, 922261, 959269

Offset: 1

Harvey Dubner (harvey(AT)dubner.com)

nonn

			p(5)=271, q=36721, r=674215921, s=227283554064939121.

Ray Chandler, Table of n, a(n) for n = 1..10000
H. Dubner and T. Forbes, Prime Pythagorean triangles, J. Integer Seqs., Vol. 4 (2001), #01.2.3.

Cf. A048161, A048270, A308635, A308636. Primes in A188546.

For each p(n), q=(p*p+1)/2, r=(q*q+1)/2, s=(r*r+1)/2 and p, q, r, s are all prime.

More terms from Ray Chandler, Jun 12 2019

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

A263951 Square numbers in A070552.

Original entry on oeis.org

9, 25, 121, 361, 841, 3481, 3721, 5041, 6241, 10201, 17161, 19321, 32761, 39601, 73441, 121801, 143641, 167281, 201601, 212521, 271441, 323761, 326041, 398161, 410881, 436921, 546121, 564001, 674041, 776161, 863041, 982081, 1062961, 1079521, 1104601, 1142761, 1190281, 1274641, 1324801

Offset: 1

Zak Seidov, Oct 30 2015

nonn

All terms are == 1 (mod 8). For n > 2, a(n) == 1 (mod 120).

This sequence is a subsequence of A247687 and it contains the squares of all those primes p for which the areas of the 3 regions in the symmetric representation of p^2 (p once and (p^2 + 1)/2 twice), are primes; i.e., p^2 and p^2 + 1 are semiprimes (see A070552). The sequence of those primes p is A048161. Cf. A237593. - Hartmut F. W. Hoft, Aug 06 2020

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Subsequence of A070552.

Cf. A048161, A067755, A068485, A237593, A247687, A263990.

a263951[n_] := Select[Map[Prime[#]^2&, Range[n]], PrimeQ[(#+1)/2]&]
a263951[190] (* Hartmut F. W. Hoft, Aug 06 2020 *)

forprime(p=3, 2000, if(isprime((p^2+1)/2), print1(p^2, ", "))) \\ Altug Alkan, Oct 30 2015

a(n) = A048161(n)^2.
From Hartmut F. W. Hoft, Aug 06 2020: (Start)
a(n) = 2 * A067755(n) + 1, n >= 1.
a(n+2) = 120 * A068485(n) + 1, n >= 1.  (End)

A308635 Sequence of 4 Pythagorean triangles, each with a leg and hypotenuse prime. The hypotenuse of each triangle is the leg of the next triangle.

Original entry on oeis.org

169219, 1370269, 5965699, 15227879, 17750981, 19342559, 21828601, 24861761, 27379621, 34602049, 39844619, 48719711, 50049281, 51649019, 52187371, 52816609, 58026659, 73659239, 79782821, 86569771, 91316801, 96842831, 104572009

Offset: 1

Ray Chandler, Jun 12 2019

nonn

			p(1)=169219, q=14317534981, r=102495903966079335181, s=5252705164911878795670904374881472151381, ...

Ray Chandler, Table of n, a(n) for n = 1..2625
H. Dubner and T. Forbes, Prime Pythagorean triangles, J. Integer Seqs., Vol. 4 (2001), #01.2.3.

Cf. A048161, A048270, A048295, A308636. Primes in A188547.

For each p(n), q=(p*p+1)/2, r=(q*q+1)/2, s=(r*r+1)/2, t=(s*s+1)/2 and p, q, r, s, t are all prime.

A308636 Sequence of 5 Pythagorean triangles, each with a leg and hypotenuse prime. The hypotenuse of each triangle is the leg of the next triangle.

Original entry on oeis.org

356498179, 432448789, 5380300469, 10667785241, 11238777509, 12129977791, 23439934621, 28055887949, 33990398249, 34250028521, 34418992099, 34773959159, 34821663421, 36624331189, 40410959231, 43538725229, 47426774869

Offset: 1

Ray Chandler, Jun 12 2019

nonn

			p(1)=356498179, q=63545475815158021, r=63545475815158021, s=2038208257886801569993754841378314277932542447949256249537232302421, ...

Ray Chandler, Table of n, a(n) for n = 1..38
H. Dubner and T. Forbes, Prime Pythagorean triangles, J. Integer Seqs., Vol. 4 (2001), #01.2.3.

Cf. A048161, A048270, A048295, A308635. Primes in A187431.

For each p(n), q=(p*p+1)/2, r=(q*q+1)/2, s=(r*r+1)/2, t=(s*s+1)/2, u=(t*t+1)/2 and p, q, r, s, t, u are all prime.

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

A074173 Numbers n such that n and n+2 are of the form p^2*q where p and q are distinct primes.

Original entry on oeis.org

18, 50, 242, 423, 475, 603, 637, 722, 845, 925, 1682, 1773, 2007, 2523, 2525, 2527, 3175, 3177, 4203, 4475, 4525, 4923, 5823, 6725, 6811, 6962, 7299, 7442, 7675, 8425, 8957, 8973, 9457, 9925, 10051, 10082, 10467, 11673, 11709, 12427, 12482, 12591

Offset: 1

Amarnath Murthy, Aug 30 2002

nonn

			18 is a member as 18 = 3^2*2 and 20 = 2^2*5.

Ray Chandler, Table of n, a(n) for n = 1..10000
Index to sequences related to prime signature

Cf. A048161, A054753, A074172, A074174, A308735, A308736.

lst={}; Do[f1=FactorInteger[n]; If[Sort[Transpose[f1][[2]]]=={1, 2}, f2=FactorInteger[n+2]; If[Sort[Transpose[f2][[2]]]=={1, 2}, AppendTo[lst, n]]], {n, 3, 10000}]; lst

Even terms in sequence are 2*A048161(n)^2. - Ray Chandler, Jun 24 2019

More terms from T. D. Noe, Oct 04 2004

A118940 Primes p such that (p^2+7)/8 is prime.

Original entry on oeis.org

3, 7, 17, 23, 41, 47, 71, 89, 103, 113, 127, 137, 151, 191, 193, 199, 223, 263, 271, 281, 337, 359, 401, 439, 457, 503, 521, 569, 577, 599, 641, 719, 727, 751, 839, 857, 863, 881, 887, 929, 991, 1009, 1033, 1097, 1103, 1151, 1193, 1217, 1231, 1279, 1297, 1303

Offset: 1

T. D. Noe, May 06 2006

nonn

For all primes q>2, we have q=4k+-1 for some k, which makes it easy to show that 8 divides q^2+7.

Similar sequences, with p and (p^2+a)/b both prime, are A048161, A062324, A062326, A062718, A109953, A110589, A118915, A118918, A118939, A118941 and A118942.

Select[Prime[Range[200]],PrimeQ[(#^2+7)/8]&]

lista(nn) = {forprime(p=2, nn, iferr(if (isprime(q=(p^2+7)/8), print1(q, ", ")), E, ););} \\ Michel Marcus, Feb 18 2018

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

A105318 Starting prime for the smallest prime Pythagorean sequence for n triangles.

Original entry on oeis.org

5, 3, 271, 169219, 356498179, 2500282512131, 20594058719087111, 2185103796349763249

Offset: 1

Lekraj Beedassy, Apr 26 2005

Smallest prime p(0) such that the n-chain governed by recurrence p(i+1)=(p(i)^2 + 1)/2 are all primes. Equivalently, least prime p(0) that generates a sequence of n 2-prime triangles, where p(k) is the hypotenuse of the k-th triangle and the leg of the (k+1)-th triangle.

For n>2, the last digit of a(n) is 1 or 9. - Ya-Ping Lu, May 17 2025

			5 is a(1) because (5^2+1)/2 = 13 is prime, but (13^2+1)/2 = 85 is not.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 258.

C. K. Caldwell, The Prime Glossary, Pythagorean triples
H. Dubner, Posting to Number Theory List
H. Dubner and T. Forbes, Prime Pythagorean Triangles
H. Dubner and T. Forbes, Journal of Integer Sequences, Vol. 4(2001) #01.2.3, Prime Pythagorean triangles
T. Forbes, Posting to Number Theory List
T. Forbes, Posting to Number Theory List

Cf. A048161, A048270, A048295, A308635, A308636.

from sympy import isprime, nextprime; m = lambda x: (x*x+1)//2; p = 2; D = {}
while p < 2185103796349763249:
    p = nextprime(p); q = m(p); n = 1
    while isprime(q) and isprime(m(q)): n += 1; q = m(q)
    if n not in D: D.update({n: p})
[print(k, end =', ') for key, k in sorted(D.items())] # Ya-Ping Lu, May 17 2025

a(1) added by T. D. Noe, Jan 29 2011

cp's OEIS Frontend

A048295 Sequence of 3 Pythagorean triangles, each with a leg and hypotenuse prime. The hypotenuse of each triangle is the leg of the next triangle.

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

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A263951 Square numbers in A070552.

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A308635 Sequence of 4 Pythagorean triangles, each with a leg and hypotenuse prime. The hypotenuse of each triangle is the leg of the next triangle.

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A308636 Sequence of 5 Pythagorean triangles, each with a leg and hypotenuse prime. The hypotenuse of each triangle is the leg of the next triangle.

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

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A074173 Numbers n such that n and n+2 are of the form p^2*q where p and q are distinct primes.

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A118940 Primes p such that (p^2+7)/8 is prime.

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