A394353 Number of distinct primes of the form x*y + x*z + y*z with integers 0 <= x <= y <= z <= n.

0, 1, 3, 5, 6, 11, 13, 18, 22, 27, 30, 41, 42, 57, 62, 70, 74, 89, 93, 115, 122, 134, 143, 169, 174, 192, 200, 215, 224, 252, 254, 284, 292, 308, 319, 345, 356, 386, 401, 422, 433, 478, 489, 532, 545, 562, 577, 620, 627, 661, 675, 707, 730, 779, 787, 825, 840, 872, 890, 936, 944, 1004

Offset

0,3

Comments

a(n) > pi(n), since every prime p <= n could be represented as p = 0*1 + 0*p + 1*p, while an additional prime q = 2*m + 1 > n (that exists between n and 2*n by Bertrand's postulate) could be represented as q = 1*1 + 1*m + 1*m with m <= n.

It appears that n*log(n) < a(n) < n^2/2, n > 3.

Links

Pontus von Brömssen, Table of n, a(n) for n = 0..10000

Formula

a(n) = Sum_{k=1..n} A396668(n). - Pontus von Brömssen, Jun 05 2026

Example

a(2) = 3, as 2, 3, 5 are represented by triples (0,1,2), (1,1,1), (1,1,2), respectively.

Crossrefs

Cf. A225268 (not restricted to primes), A396668.

Sequence in context: A145714 A326310 A047443 * A173593 A268495 A127577

Adjacent sequences: A394350 A394351 A394352 * A394354 A394355 A394356

Keyword

nonn,changed

Author

Michael Shmoish, Apr 14 2026

Status

approved