A351908 Number of prime quadruples p < q < r < s in arithmetic progression with all members less than or equal to prime(n).

0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 3, 4, 4, 5, 7, 7, 7, 7, 7, 8, 9, 9, 11, 12, 13, 14, 14, 15, 16, 19, 19, 20, 21, 22, 24, 25, 27, 28, 28, 29, 31, 32, 34, 35, 35, 38, 41, 42, 44, 45, 46, 47, 50, 51, 52, 53, 53, 55, 59, 62, 63, 64, 65, 67, 69, 69, 71

Offset

1,10

Links

Table of n, a(n) for n=1..69.

Benjamin Green and Terence Tao, Linear equations in primes, arXiv:math/0606088 [math.NT], 2006-2008; Annals of Mathematics Second Series, Vol. 171, No. 3 (May, 2010), pp. 1753-1850.

Ben J. Green and Terence C. Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Math. 167 (2008), pp. 481-547.

Formula

a(n) ~ Cp^2/log^4 p ~ Cn^2/log^2 n where p is the n-th prime and C = A351909.

Prog

(PARI) a(n, qlim=prime(n))=my(s); forprime(q=23, qlim, forprimestep(p=q%6, q-18, 6, isprime((2*p+q)/3) && isprime((2*q+p)/3) && s++)); s
(PARI) b(q)=my(s); forprimestep(p=q%6, q-18, 6, isprime((2*p+q)/3) && isprime((2*q+p)/3) && s++); s
first(n)=my(v=vector(n), s, i); forprime(p=2, prime(n), v[i++]=s+=b(p)); v

Crossrefs

Partial sums of A351907.

Cf. A351909.

Sequence in context: A018048 A077564 A088044 * A029051 A338826 A274201

Adjacent sequences: A351905 A351906 A351907 * A351909 A351910 A351911

Keyword

nonn

Author

Charles R Greathouse IV, Feb 25 2022

Status

approved