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%I #7 Dec 21 2022 20:12:17
%S 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,
%T 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,
%U 9,10,11,8,7,10,8,7,10,7,11,9,8,10,10,10
%N Number of ways to express a prime p as 2*p1 + 3*p2, where p1, p2 are primes or 1.
%C 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.
%H Michael S. Branicky, <a href="/A120450/b120450.txt">Table of n, a(n) for n = 1..10000</a>
%e a(11)=2 because we can write prime(11)=31 as 2*5 + 3*7 OR 2*11 + 3*3.
%e 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.
%o (Python)
%o from collections import Counter
%o from sympy import prime, primerange
%o def aupton(nn):
%o primes, c = list(primerange(2, prime(nn)+1)), Counter()
%o p2, p3 = [2] + [2*p for p in primes], [3] + [3*p for p in primes]
%o for p in p2:
%o if p > primes[-1]: break
%o for q in p3:
%o if p + q > primes[-1]: break
%o c[p+q] += 1
%o return [c[p] for p in primes]
%o print(aupton(83)) # _Michael S. Branicky_, Dec 21 2022
%K nonn
%O 1,6
%A _Vassilis Papadimitriou_, Jul 20 2006
%E a(59) and beyond from _Michael S. Branicky_, Dec 21 2022
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