A143850 -id:A143850 - cp's OEIS frontend

A103739 Primes which are half the sum of 2 squares of primes.

17, 29, 37, 73, 89, 97, 109, 149, 157, 193, 229, 241, 269, 277, 349, 409, 433, 541, 601, 661, 709, 769, 829, 853, 929, 937, 1009, 1021, 1069, 1109, 1117, 1129, 1249, 1321, 1409, 1429, 1489, 1549, 1609, 1669, 1753, 1789, 1801, 1873, 2029, 2089, 2161, 2221

Offset: 1

Giovanni Teofilatto, Mar 28 2005

Primes of the form x^2 + y^2, where x > y > 0, such that x-y = p and x+y = q are primes. Proof: (p^2+q^2)/2 = ((x-y)^2+(x+y)^2)/2 = x^2+y^2 so we have x = (p+q)/2 and y = (q-p)/2. - Thomas Ordowski, Sep 24 2012
All terms == 1 or 5 (mod 12). - Thomas Ordowski, Jun 28 2013
Or, primes in A143850. - Zak Seidov, Jun 06 2015

			17 is in the sequence because (3^2 + 5^2) / 2 = 17.

T. D. Noe, Table of n, a(n) for n = 1..1000

Intersection of A143850 and A000040.

Cf. A001248, A002313.

Primes:= select(isprime,[seq(2*i+1,i=1..400)]):
Psq:= map(`^`,Primes,2):
M:= max(Psq):
S:= select(t -> t <= M/2 and isprime(t),{seq(seq((Psq[i]+Psq[j])/2, j=1..i-1),i=1..nops(Psq))}):
sort(convert(S,list)); # Robert Israel, Jun 08 2015

list(lim)=my(v=List(), p2, t); lim\=1; if(lim<9, lim=9); forprime(p=3, sqrtint(2*lim-9), p2=p^2; forprime(q=3, min(sqrtint(2*lim-p2), p), if(isprime(t=(p2+q^2)/2), listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Feb 14 2017

Corrected and extended by Walter Nissen, Jul 19 2005

A075892 Average of squares of successive primes: a(n) = (prime(n+1)^2 + prime(n)^2)/2, with n >= 2.

Original entry on oeis.org

17, 37, 85, 145, 229, 325, 445, 685, 901, 1165, 1525, 1765, 2029, 2509, 3145, 3601, 4105, 4765, 5185, 5785, 6565, 7405, 8665, 9805, 10405, 11029, 11665, 12325, 14449, 16645, 17965, 19045, 20761, 22501, 23725, 25609, 27229, 28909, 30985, 32401

Offset: 2

Zak Seidov, Oct 17 2002

a(n) is prime for n in A240749. - Robert Israel, Jul 06 2017

If p and q are primes such that p > q > 3, then ((p^2 - q^2)/2, p*q, (p^2 + q^2)/2) is a primitive Pythagorean triple. - César Aguilera, Jun 02 2022

			a(2)=17 because (prime(3)^2 + prime(2)^2)/2 = (5^2 + 3^2)/2 = 17.

Zak Seidov, Table of n, a(n) for n = 2..10001

Cf. A006094, A124434, A143850, A240749.

[(NthPrime(n+1)^2+NthPrime(n)^2)/2: n in [2..50]]; // Vincenzo Librandi, Mar 07 2015

seq((ithprime(i)^2 + ithprime(i+1)^2)/2, i=2..100); # Robert Israel, Jul 06 2017

Table[(Prime[n + 1]^2 + Prime[n]^2)/2, {n, 2, 50}] (* Vincenzo Librandi, Mar 07 2015 *)
p=2;q=3;Table[p=q;q=NextPrime[q];(q^2+p^2)/2,{100}] (* Zak Seidov, Jul 06 2017 *)

a(n) = (prime(n+1)^2+prime(n)^2)/2; \\ Michel Marcus, Oct 03 2013

a(n)^2 = A124434(n)^2 + A006094(n)^2. - César Aguilera, Jun 02 2022

A227697 Numbers of the form ((p^2 + q^2)/2 - 1)/4, where p and q are odd primes.

Original entry on oeis.org

2, 4, 6, 7, 9, 12, 16, 18, 21, 22, 24, 27, 30, 36, 37, 39, 42, 46, 48, 51, 57, 60, 66, 67, 69, 72, 81, 87, 90, 102, 106, 108, 111, 120, 121, 123, 126, 132, 135, 141, 150, 156, 165, 171, 172, 174, 177, 186, 192, 207, 210

Offset: 1

Richard R. Forberg, Sep 22 2013

nonn

a(n) mod 3 = 0 if neither prime = 3.
a(n) mod 3 = 1 if one prime = 3.
a(1) mod 3 = 2 (where both primes = 3).

a(n) = (A143850(n)-1)/4.

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.

cp's OEIS Frontend

A103739 Primes which are half the sum of 2 squares of primes.

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A075892 Average of squares of successive primes: a(n) = (prime(n+1)^2 + prime(n)^2)/2, with n >= 2.

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A227697 Numbers of the form ((p^2 + q^2)/2 - 1)/4, where p and q are odd primes.

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A357439 Sums of squares of two odd primes.

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