A120450 Number of ways to express a prime p as 2*p1 + 3*p2, where p1, p2 are primes or 1.

0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 4, 3, 3, 3, 5, 4, 4, 4, 4, 4, 4, 6, 3, 5, 4, 4, 4, 6, 5, 5, 5, 5, 7, 5, 6, 6, 6, 6, 7, 5, 6, 7, 6, 7, 9, 8, 8, 6, 7, 7, 8, 7, 9, 6, 10, 8, 6, 9, 7, 9, 8, 10, 9, 10, 9, 10, 11, 8, 7, 10, 8, 7, 10, 7, 11, 9, 8, 10, 10, 10

Offset

1,6

Comments

It seems that every prime p > 3 can be expressed as 2*p1 + 3*p2, where p1, p2 are primes or 1. I have tested it for the first 1500 primes (up to 12553) and it is true.

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Example

a(11)=2 because we can write prime(11)=31 as 2*5 + 3*7 OR 2*11 + 3*3.
a(12)=3 because we can write prime(12)=37 as 2*2 + 3*11 OR 2*11 + 3*5 OR 2*17 + 3*1.

Prog

(Python)
from collections import Counter
from sympy import prime, primerange
def aupton(nn):
    primes, c = list(primerange(2, prime(nn)+1)), Counter()
    p2, p3 = [2] + [2*p for p in primes], [3] + [3*p for p in primes]
    for p in p2:
        if p > primes[-1]: break
        for q in p3:
            if p + q > primes[-1]: break
            c[p+q] += 1
    return [c[p] for p in primes]
print(aupton(83)) # Michael S. Branicky, Dec 21 2022

Crossrefs

Sequence in context: A030603 A092670 A210528 * A127238 A072114 A090621

Adjacent sequences: A120447 A120448 A120449 * A120451 A120452 A120453

Keyword

nonn

Author

Vassilis Papadimitriou, Jul 20 2006

Extensions

a(59) and beyond from Michael S. Branicky, Dec 21 2022

Status

approved