A103739 -id:A103739 - cp's OEIS frontend

A283792 Primes of the form (p^2 + q^2) / 2 such that (p^2 - q^2) / 24 is prime, where primes p > q > 3.

109, 157, 229

Offset: 1

Thomas Ordowski and Altug Alkan, Mar 16 2017

Union of primes of the form:
t^2 + 6^2 such that t and p = t+6 and q = t-6 are primes,
(2t)^2 + 3^2 such that t and p = 2t+3 and q = 2t-3 are primes,
(3t)^2 + 2^2 such that t and p = 3t+2 and q = 3t-2 are primes,
(6t)^2 + 1^2 such that t and p = 6t+1 and q = 6t-1 are primes.
Note: this last subset is empty.
We have p*q*(p^2-q^2)*(p^2+q^2) = p^5*q - p*q^5 == 0 (mod 5), so at least one of p, q, p^2-q^2, or p^2+q^2 must be divisible by 5. Thus, this sequence is finite and 229 is the last term. - Robert Israel, Mar 16 2017

			Prime 109 = (13^2 + 7^2)/2 is a term since (13^2 - 7^2)/24 = 5 is prime.
Note: 109 = (2*5)^2 + 3^2, 157 = 11^2 + 6^2, and 229 = (3*5)^2 + 2^2.

Cf. A103739, A283562.

list(lim)=my(v=List(), p2, q2, t); lim\=1; lim=min(max(lim,9),229); forprime(p=3, sqrtint(2*lim-9), p2=p^2; forprime(q=3, min(sqrtint(2*lim-p2), p-2), q2=q^2; if((p2-q2)%24==0 && isprime(t=(p2+q2)/2) && isprime((p2-q2)/24), listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Mar 17 2017

A347530 Primes of the form (p^2 + 9)/2 where p is prime.

Original entry on oeis.org

17, 29, 89, 149, 269, 929, 1109, 1409, 3449, 5309, 6389, 8069, 12329, 14969, 33029, 34589, 42929, 47129, 48989, 60209, 67349, 78809, 98129, 109049, 118589, 136769, 158489, 175829, 213209, 264269, 317609, 338669, 363809, 367229, 389849, 438989, 454109, 467549

Offset: 1

Burak Muslu, Sep 05 2021

nonn

Each p is an odd number, so p^2 == 1 (mod 8), thus (p^2 + 9)/2 == 1 (mod 4).

			17 is in the sequence as 17 = (p^2 + 9)/2 where p = 5 is prime.
29 is in the sequence as 29 = (p^2 + 9)/2 where p = 7 is prime.

Cf. A000040, A045637, A062324.

Subsequence of A076727 and of A103739.

Select[(Select[Range[3, 1000], PrimeQ]^2 + 9)/2, PrimeQ] (* Amiram Eldar, Sep 05 2021 *)

A357439 Sums of squares of two odd primes.

Original entry on oeis.org

18, 34, 50, 58, 74, 98, 130, 146, 170, 178, 194, 218, 242, 290, 298, 314, 338, 370, 386, 410, 458, 482, 530, 538, 554, 578, 650, 698, 722, 818, 850, 866, 890, 962, 970, 986, 1010, 1058, 1082, 1130, 1202, 1250, 1322, 1370, 1378, 1394, 1418, 1490, 1538, 1658, 1682

Offset: 1

Giuseppe Melfi, Oct 06 2022

nonn

Although this is twice A143850, it is important enough to warrant an entry of it own. - N. J. A. Sloane, Oct 10 2022

Cf. A045636, A103739, A143850, A227697.

A360490 a(n) = (1/2) * A241102(n).

Original entry on oeis.org

109, 433, 172801, 238573, 363313, 640333, 1145773, 1968301, 2056753, 3121201, 3577393, 6588973, 11197873, 13079233, 13381633, 15431473, 21676033, 26462701, 34476301, 37340353, 43823053, 48481201, 54749953, 56454733, 90816013, 96038893, 102667501, 128786113

Offset: 1

Ya-Ping Lu, Feb 09 2023

nonn

Primes which are half the difference between 2 cubes of primes.
Primes of the form 3*m^2 + 1, where m is the average of a twin prime pair (A014574).
A subsequence of A243761 and a supersequence of A270249.

			172801 is a term because 172801 = (241^3 - 239^3)/2, and 172801, 239 and 241 are all primes.

Cf. A014574, A030078, A103739, A243761, A270249.

from itertools import islice
from sympy import isprime, nextprime
def A360490_gen(): # generator of terms
    p, q = 3**3, 5
    while True:
        if isprime(k:=(m:=q**3)-p>>1):
            yield k
        p, q = m, nextprime(q)
A360490_list = list(islice(A360490_gen(),20)) # Chai Wah Wu, Feb 27 2023

a(n) = (1/2) * A241102(n).

cp's OEIS Frontend

A283792 Primes of the form (p^2 + q^2) / 2 such that (p^2 - q^2) / 24 is prime, where primes p > q > 3.

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A347530 Primes of the form (p^2 + 9)/2 where p is prime.

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Mathematica

A357439 Sums of squares of two odd primes.

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A360490 a(n) = (1/2) * A241102(n).

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