A365706 For n >= 1, a(n) is the least prime p such that the arithmetic mean of (n + 1) consecutive primes starting with p is a perfect square, or a(n) = -1 if no such p exists.

3, 2393, 5, 827, 53, 271, 1063, 23993, 197, 29, 193, 2143, 359, 6829, 397, 17, 433, 661, 2837, 25171, 13597, 563, 10301, 1814233, 51427, 6781, 316817, 7477, 71, 238919, 11491, 3109, 42293, 38653, 6263, 13043, 474497, 21433, 13, 21419, 16963, 5119, 705209, 183761

Offset: 1

Author

Ctibor O. Zizka, Oct 15 2023

Keywords

• nonn

Comments

Does a(n) exists for all n >= 1 ?

From David A. Corneth, Oct 18 2023: (Start)

Let s(n) be the sum of n + 1 consecutive primes starting with a(n). Then s(n)/(n+1) = m^2 for some positive integer m.

This means s(n) = (n + 1) * m^2. If n is even then m is odd if a(n) > 2.

As s(n) >= A007504(n) we have m^2 >= s(n)/(n+1) >= A007504(n)/(n+1) i.e. m >= sqrt(A007504(n)/(n+1)). So for some m we can see if m^2 * (n+1) is the sum of n+1 consecutive primes and if so a(n) is the smallest prime of these n+1 primes after testing all candidates up to m. (End)

s(n) = (n + 1)* a(n) + Sum_{i=0..(n-1)} (n-i)*g(i+1), thus we have Sum_{i=0..(n-1)} (n-i)*g(i+1) = (m^2 - a(n)) * (n + 1), g(j) are the n gaps between n + 1 consecutive primes. - Ctibor O. Zizka, Oct 18 2023

Examples

			n = 2: we search for the least prime(i) such that (prime(i) + prime(i + 1) + prime(i + 2))/3 = m^2, m an integer. This is valid for (2393 + 2399 + 2411)/3 = 49^2 thus a(2) = 2393.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040, A007504, A053001, A225195.

Programs

PARI

isok(x) = (denominator(x)==1) && issquare(x);
a(n) = my(k=1); while (!isok((vecsum(primes(k+n))-vecsum(primes(k-1)))/(n+1)), k++); prime(k); \\ Michel Marcus, Oct 16 2023

PARI

a(n) = {my(m = n + 1, ps = vector(m, i, prime(i)), s); forprime(p = ps[m] + 1, , s = vecsum(ps); if(!(s % m) && issquare(s/m), return(ps[1])); ps = concat(vecextract(ps, "^1"), p));} \\ Amiram Eldar, Oct 18 2023

Extensions

More terms from Amiram Eldar, Oct 18 2023