A074741 -id:A074741 - cp's OEIS frontend

A247177 Primes p with property that the sum of the squares of the successive gaps between primes <= p is a prime number.

5, 13, 29, 41, 89, 97, 139, 173, 179, 263, 269, 281, 307, 337, 353, 431, 439, 461, 487, 499, 509, 569, 607, 613, 641, 643, 661, 709, 739, 761, 809, 823, 839, 857, 919, 941, 967, 991, 1031, 1039, 1061, 1117, 1129, 1163, 1171, 1201, 1229, 1277, 1381, 1399

Offset: 1

Abhiram R Devesh, Nov 22 2014

If A074741(n) is prime, then prime(n+1) is in this sequence. - Michel Marcus, Jan 12 2015

			a(1)=5; primes less than or equal to 5: [2, 3, 5]; squares of prime gaps: [1, 4]; sum of squares of prime gaps: 5.
a(2)=13; primes less than or equal to 13: [2, 3, 5, 7, 11, 13]; squares of prime gaps: [1, 4, 4, 16, 4]; sum of squares of prime gaps: 29.

Abhiram R Devesh, Table of n, a(n) for n = 1..1000

Cf. A074741 (sum of squares of gaps between consecutive primes).

listp(nn) = {my(s = 0); my(precp = 2); forprime (p=3, nn, if (isprime(ns = (s + (p - precp)^2)), print1(p, ", ")); s = ns; precp = p;);} \\ Michel Marcus, Jan 12 2015

from sympy import nextprime, isprime
p = 2
s = 0
while s < 8000:
    np = nextprime(p)
    if isprime(s):
        print(p)
    d = np - p
    s += d*d
    p = np

A135406 Sum of squares of gaps between consecutive semiprimes.

Original entry on oeis.org

4, 13, 14, 30, 31, 67, 68, 77, 78, 127, 128, 129, 138, 139, 188, 197, 201, 217, 221, 222, 238, 247, 263, 288, 297, 322, 331, 332, 333, 349, 353, 354, 355, 476, 501, 517, 526, 527, 531, 532, 533, 569, 585, 586, 635, 636, 637, 641, 642, 723, 732, 733, 737, 762

Offset: 1

Jonathan Vos Post, Dec 09 2007

This is to semiprimes A001358 as A074741 is to primes A000040. What is the semiprime analog of D. R. Heath-Brown's conjecture: Sum_{prime(n)<=N} (prime(n)-prime(n-1))^2 ~ 2*N*log(N) and Marek Wolf's conjecture: Sum_{prime(n)A000720(n).

			a(10) = (6-4)^2 + (9-6)^2 + (10-9)^2 + (14-10)^2 + (15-14)^2 + (21-15)^2 + (22-21)^2 + (25-22)^2 + (26-25)^2 + (33-26)^2 = (2^2) + (3^2) + (1^2) + (4^2) + (1^2) + (6^2) + (1^2) + (3^2) + (1^2) + (7^2) = 127.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A001358, A065516, A074741.

A001358 := proc(n) option remember ; local a ; if n = 1 then 4; else for a from A001358(n-1)+1 do if numtheory[bigomega](a) = 2 then RETURN(a) ; fi ; od: fi ; end: A065516 := proc(n) A001358(n+1)-A001358(n) ; end: A135406 := proc(n) add( (A065516(k))^2,k=1..n) ; end: seq(A135406(n),n=1..80) ; # R. J. Mathar, Jan 07 2008

Accumulate[Differences[Select[Range[200],PrimeOmega[#]==2&]]^2] (* Harvey P. Dale, Mar 05 2015 *)

a(n) = SUM[k=1..n] A065516(k)^2 = SUM[k=1..n] (A001358(n+1) - A001358(n))^2.

More terms from R. J. Mathar, Jan 07 2008

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A247177 Primes p with property that the sum of the squares of the successive gaps between primes <= p is a prime number.

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A135406 Sum of squares of gaps between consecutive semiprimes.

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