A160358 -id:A160358 - cp's OEIS frontend

A160359 Primes that are partial sums of squares of the odd primes.

83, 373, 2393, 4723, 56383, 122701, 1789391, 1987849, 2706679, 7287991, 27833021, 41765519, 43570537, 55600477, 62283451, 67053359, 104720809, 122353547, 204330311, 347756707, 450504547, 540649093, 677774533, 715849843, 839886301

Offset: 1

M. F. Hasler, May 18 2009

nonn

There is necessarily an odd number of terms in the sum a(n) = 3^2+5^2+...+p(k)^2, i.e. the index k=A160358(n)+1 of the last prime in this sum is even.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A098562, A133547.

P:= select(isprime,[seq(i,i=3..10000,2)]):
S:= ListTools:-PartialSums(map(`^`,P,2)):
select(isprime,S); # Robert Israel, May 13 2024

Select[Accumulate[Prime[Range[2,400]]^2],PrimeQ] (* Harvey P. Dale, Jul 17 2021 *)

s=0; forprime( p=3,9999, isprime(s+=p^2) & print1(s","))

Equals A000040 intersect A133547.

A370633 Numbers k such that A372041(k) = 3.

Original entry on oeis.org

1, 2, 4, 5, 11, 14, 31, 32, 35, 47, 70, 79, 80, 86, 89, 91, 104, 109, 127, 149, 161, 170, 182, 185, 194, 196, 226, 242, 260, 283, 289, 302, 311, 316, 317, 319, 338, 350, 355, 361, 362, 373, 392, 413, 433, 472, 490, 496, 499, 500, 506, 517, 518, 520, 524, 530, 568, 574, 593

Offset: 1

Michel Lagneau, May 01 2024

nonn

3 is the smallest positive value occurring in A372041 and occurs when the sum of the squares of the first 2*k+1 primes starting at 3 is a prime.

			k = 2 is a term since A372041(2) = 3, meaning that the sum of the 2*k+1 = 5 squares of primes starting at 3 is a prime: 3^2 + 5^2 + 7^2 + 11^2 + 13^2 = 373.

Cf. A133547, A160358, A372041.

a(n) = (A160358(n) - 1)/2.

cp's OEIS Frontend

A160359 Primes that are partial sums of squares of the odd primes.

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