A082519 -id:A082519 - cp's OEIS frontend

A036468 Number of ways to represent 2n+1 as a+b with 0 < a < b and a^2 + b^2 prime.

1, 2, 2, 2, 3, 3, 4, 4, 4, 3, 4, 8, 4, 6, 5, 4, 9, 8, 6, 9, 7, 7, 7, 5, 7, 9, 14, 8, 9, 11, 7, 17, 11, 10, 9, 11, 9, 8, 13, 9, 15, 20, 11, 14, 13, 8, 18, 14, 10, 18, 16, 10, 17, 16, 13, 20, 20, 13, 14, 17, 12, 23, 18, 14, 22, 15, 17, 18, 21, 12, 19, 29, 16, 23, 21, 14, 27, 24

Offset: 1

R. K. Guy

nonn

Zhang Ming-Zhi (zamiz(AT)mail.sc.cninfo.net) asks if a(m) is always > 0.
I have confirmed that a(n) > 0 for 0 < n < 10^7. - T. D. Noe, Oct 17 2004
This open problem is mentioned by Guy at the end of section C1. - T. D. Noe, Apr 22 2009
a(n) <= phi(2n+1)/2, where phi(m) = A000010(m), while a(n) = phi(2n+1)/2 only for n = 1, 2, and 7. - Thomas Ordowski, Jan 25 2014
Records in a(n) are for 2n+1 = 3, 5, 11, 15, 25, 35, 55, 65, 85, 125, 145, 185, 205, 215, 235, 265, 295, 325, 365, 415, ... cf. A001750. - Thomas Ordowski, Mar 02 2017
a(n) tends to be larger for n == 2 (mod 5): see plot in Links. - Robert Israel, Mar 02 2017
Number of primes p = ((2n+1)^2 + x^2)/2 for positive integers x < 2n+1. - Thomas Ordowski, Mar 06 2017

R. K. Guy, Unsolved Problems in Theory of Numbers, Section C1.

T. D. Noe, Table of n, a(n) for n = 1..10000
Gordon Hamilton, Unsolved K-12: Grade 7, 2014. (video)
Robert Israel, a(5k+j) for j=0,1,2,3,4

Cf. A082519, A099468, A281543.

a:= n-> add(`if`(isprime(i^2+(2*n+1-i)^2), 1, 0), i=1..n):
seq(a(n), n=1..80);  # Alois P. Heinz, Jul 09 2016

Table[cnt=0; m=2n+1; Do[If[PrimeQ[k^2+(m-k)^2], cnt++ ], {k, n}]; cnt, {n, 100}]

a(n)=sum(k=1,n,isprime(k^2+(2*n-k+1)^2)) \\ Charles R Greathouse IV, Jan 09 2014

a(n) = O(n/log(n)). - Thomas Ordowski, Feb 11 2013

More terms from David W. Wilson and Michael Kleber

A099332 Primes p such that p = a^2 + b^2 for a,b>0 and a+b is prime.

Original entry on oeis.org

2, 5, 13, 17, 29, 37, 61, 73, 89, 97, 101, 109, 149, 157, 181, 193, 229, 241, 257, 269, 277, 293, 349, 409, 421, 433, 461, 521, 541, 593, 601, 641, 661, 701, 709, 733, 769, 797, 829, 853, 881, 929, 937, 953, 997, 1009, 1021, 1049, 1061, 1069, 1109, 1117

Offset: 1

T. D. Noe, Oct 15 2004

nonn

Let q=a+b. For a specific prime q, the number of distinct primes p that are the sum of two squares is A082519(q)/2.

Primes p of the form (q-b)^2 + b^2, where q is prime and 0

			29 is in this sequence because 29=2^2+5^2 and 2+5 is prime.

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

Cf. A082519 (number of times that k^2 + (n-k)^2 is prime for 1 <= k <= n-1).

Needs["NumberTheory`NumberTheoryFunctions`"]; lst={2}; Do[n=4k+1; If[PrimeQ[n], If[PrimeQ[Plus@@QuadraticRepresentation[1, n]], AppendTo[lst, n]]], {k, 500}]; lst

Primes p such that p = (q^2 + x^2)/2, where q is prime and |x| < q. - Thomas Ordowski, Feb 15 2013

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A036468 Number of ways to represent 2n+1 as a+b with 0 < a < b and a^2 + b^2 prime.

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