A120451 Number of ways a prime number p can be expressed as 2*(p1-p2) + 3*p3, where p1, p2, p3 are primes or 1, less than or equal to p.

0, 3, 4, 7, 9, 12, 13, 16, 18, 20, 23, 30, 32, 32, 33, 42, 43, 51, 50, 57, 64, 61, 69, 83, 84, 93, 89, 92, 110, 115, 114, 123, 133, 133, 153, 143, 157, 154, 163, 176, 179, 211, 197, 220, 233, 216, 227, 230, 233, 269, 278, 268, 310, 274, 314

Offset

1,2

Comments

At least for the first 200 primes, it is true that every prime p > 2 can be expressed as 2*(p1-p2) + 3*p3, where p1, p2, p3 are primes or 1, less than or equal to p (the proof would be straightforward if both (a) Levy's conjecture and (b) the conjecture saying that every prime p > 3 can be expressed as 2*p1 + 3*p2, where p1, p2 are primes, were true). It would be interesting to study how the sequence changes when we remove the restriction for p1, p2, p3 to be less than or equal to p.

Links

Table of n, a(n) for n=1..55.

Example

a(12)=30 because 37 (the 12th prime) can be expressed as
2*(1 - 2) + 3*13
OR 2*(1 - 11) + 3*19
OR 2*(1 - 17) + 3*23
OR 2*(1 - 29) + 3*31
OR 2*(2 - 3) + 3*13
OR 2*(3 - 1) + 3*11
OR 2*(3 - 13) + 3*19
OR 2*(3 - 19) + 3*23
OR 2*(3 - 31) + 3*31
OR 2*(5 - 3) + 3*11
OR 2*(7 - 5) + 3*11
OR 2*(7 - 17) + 3*19
OR 2*(7 - 23) + 3*23
OR 2*(11 - 3) + 3*7
OR 2*(13 - 2) + 3*5
OR 2*(13 - 5) + 3*7
OR 2*(13 - 11) + 3*11
OR 2*(13 - 23) + 3*19
OR 2*(13 - 29) + 3*23
OR 2*(17 - 3) + 3*3
OR 2*(19 - 2) + 3*1
OR 2*(19 - 5) + 3*3
OR 2*(19 - 11) + 3*7
OR 2*(19 - 17) + 3*11
OR 2*(19 - 29) + 3*19
OR 2*(31 - 17) + 3*3
OR 2*(31 - 23) + 3*7
OR 2*(31 - 29) + 3*11
OR 2*(37 - 23) + 3*3
OR 2*(37 - 29) + 3*7.

Prog

(PARI) a(n) = {my(vp = concat(1, primes(n)), nb=0, p=prime(n), p1, p2, p3); for (i=1, #vp, p1 = vp[i]; for (j=1, #vp, p2 = vp[j]; for (k=1, #vp, p3 = vp[k]; if (2*(p1-p2) + 3*p3 == p, nb++); ); ); ); nb; } \\ Michel Marcus, Jan 26 2021

Crossrefs

Sequence in context: A280493 A032729 A035270 * A327621 A060428 A352274

Adjacent sequences: A120448 A120449 A120450 * A120452 A120453 A120454

Keyword

nonn

Author

Vassilis Papadimitriou, Jul 20 2006

Status

approved