A260554 -id:A260554 - cp's OEIS frontend

A260553 Primes p such that p = q^2 + 2*r^2 where q and r are also primes.

17, 43, 59, 67, 107, 139, 251, 307, 347, 379, 547, 587, 859, 1699, 1867, 1931, 3371, 3499, 3739, 4507, 5059, 5347, 6907, 6971, 7451, 10091, 10627, 10667, 11467, 12491, 18787, 20411, 21227, 22907, 29947, 32059, 32779, 37547, 38651, 39619, 49307, 49747, 53147

Offset: 1

Colin Barker, Jul 29 2015

nonn

			43 is in the sequence because 43 = 5^2 + 2*3^2 and 43, 5 and 3 are all primes.

Colin Barker and Chai Wah Wu, Table of n, a(n) for n = 1..1873 n = 1..150 from Colin Barker.

Main entry for this sequence is A201613.

Cf. A260554, A260555, A260556, A260557.

Select[#1^2 + 2 #2^2 & @@ # & /@ Tuples[Prime@ Range@ 60, 2], PrimeQ] // Sort (* Michael De Vlieger, Jul 29 2015 *)

lista(nn)=forprime(p=2, nn, forprime(r=2, sqrtint(p\2), if (issquare(q2 = p-2*r^2) && isprime(sqrtint(q2)), print1(p, ", ")););); \\ Michel Marcus, Jul 29 2015

list(lim)=my(v=List()); lim\=1; forprime(q=2, sqrtint((lim-9)\2), my(t=2*q^2); forprime(p=3, sqrtint(lim-t), my(r=t+p^2); if(isprime(r), listput(v, r)))); Set(v) \\ Charles R Greathouse IV, Oct 09 2024

from sympy import prime, isprime
n = 5000
A260553_list, plimit = [], prime(n)**2+8
for i in range(1, n):
    q = 2*prime(i)**2
    for j in range(1, n):
        p = q + prime(j)**2
        if p < plimit and isprime(p):
            A260553_list.append(p)
A260553_list = sorted(A260553_list) # Chai Wah Wu, Jul 30 2015

A260555 Primes p such that p = q^2 + 6*r^2 where q and r are also primes.

Original entry on oeis.org

73, 79, 103, 193, 199, 223, 271, 313, 439, 463, 751, 823, 991, 1039, 1063, 1087, 1303, 1423, 1543, 1567, 1663, 1759, 1783, 1831, 1873, 1999, 2143, 2287, 2383, 2503, 2833, 3343, 3463, 3583, 3631, 3823, 3847, 3943, 4447, 4513, 4639, 4783, 5023, 5167, 5407

Offset: 1

Colin Barker, Jul 29 2015

nonn

Green & Sawhney prove that this sequence is infinite. - Charles R Greathouse IV, Oct 08 2024

			73 is in the sequence because 73 = 7^2 + 6*2^2 and 73, 7 and 2 are all primes.

Colin Barker, Table of n, a(n) for n = 1..1500
Ben Green and Mehtaab Sawhney, Primes of the form p^2 + nq^2, arXiv preprint (2024). arXiv:2410.04189 [math.NT]

Cf. A260553, A260554, A260556, A260557.

Select[#1^2 + 6 #2^2 & @@ # & /@ Tuples[Prime@ Range@ 60, 2], PrimeQ] // Sort (* Michael De Vlieger, Jul 29 2015 *)

list(lim)=my(v=List()); lim\=1; forprime(q=2, sqrtint((lim-9)\6), my(t=6*q^2); forprime(p=3, sqrtint(lim-t), my(r=t+p^2); if(isprime(r), listput(v, r)))); Set(v) \\ Charles R Greathouse IV, Oct 08 2024

A260556 Primes p such that p = q^2 + 8*r^2 where q and r are also primes.

Original entry on oeis.org

41, 97, 193, 241, 401, 433, 601, 977, 1033, 1361, 1753, 2281, 2897, 3793, 4241, 4561, 5113, 6737, 6961, 7993, 10273, 11953, 12841, 13457, 17681, 22273, 22481, 26641, 27961, 32833, 37321, 42641, 49801, 49937, 54361, 57193, 58153, 63073, 63377, 76801, 94321

Offset: 1

Colin Barker, Jul 29 2015

nonn

			601 is in the sequence because 601 = 23^2 + 8*3^2 and 601, 23 and 3 are all primes.

Colin Barker and Chai Wah Wu, Table of n, a(n) for n = 1..1510 (terms for n = 1..100 from Colin Barker).

Cf. A260553, A260554, A260555, A260557.

Select[#1^2 + 8 #2^2 & @@ # & /@ Tuples[Prime@ Range@ 80, 2], PrimeQ] // Sort (* Michael De Vlieger, Jul 29 2015 *)

lista(nn) = {forprime(p=2, nn, forprime(r=2, sqrtint(p\8), if (issquare(q2 = p-8*r^2) && isprime(sqrtint(q2)), print1(p, ", "));););} \\ Michel Marcus, Aug 01 2015

from sympy import prime, isprime
n = 5000
A260556_list, plimit = [], prime(n)**2+32
for i in range(1,n):
    q = 8*prime(i)**2
    for j in range(1,n):
        p = q + prime(j)**2
        if p < plimit and isprime(p):
            A260556_list.append(p)
A260556_list = sorted(A260556_list) # Chai Wah Wu, Jul 30 2015

A260557 Primes p such that p = q^2 + 10*r^2 where q and r are also primes.

Original entry on oeis.org

89, 139, 211, 379, 401, 419, 499, 569, 619, 659, 881, 1019, 1051, 1091, 1259, 1409, 1451, 1459, 1499, 1571, 1619, 1699, 1721, 1811, 1889, 1931, 1979, 2099, 2339, 2459, 2531, 2579, 2699, 2939, 3011, 3251, 3299, 3371, 3539, 3571, 3659, 3761, 3779, 3851, 4019

Offset: 1

Colin Barker, Jul 29 2015

nonn

Green & Sawhney prove that this sequence is infinite. - Charles R Greathouse IV, Oct 08 2024

			419 is in the sequence because 419 = 13^2 + 10*5^2 and 419, 13 and 5 are all primes.

Colin Barker, Table of n, a(n) for n = 1..1750
Ben Green and Mehtaab Sawhney, Primes of the form p^2 + nq^2, arXiv preprint (2024). arXiv:2410.04189 [math.NT]

Cf. A260553, A260554, A260555, A260556.

Select[#1^2 + 10 #2^2 & @@ # & /@ Tuples[Prime@ Range@ 36, 2], PrimeQ] // Sort (* Michael De Vlieger, Jul 29 2015 *)

list(lim)=my(v=List()); lim\=1; forprime(q=2, sqrtint((lim-9)\10), my(t=10*q^2); forprime(p=3, sqrtint(lim-t), my(r=t+p^2); if(isprime(r), listput(v, r)))); Set(v) \\ Charles R Greathouse IV, Oct 08 2024

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A260553 Primes p such that p = q^2 + 2*r^2 where q and r are also primes.

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A260555 Primes p such that p = q^2 + 6*r^2 where q and r are also primes.

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A260556 Primes p such that p = q^2 + 8*r^2 where q and r are also primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

A260557 Primes p such that p = q^2 + 10*r^2 where q and r are also primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI