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A120450 Number of ways to express a prime p as 2*p1 + 3*p2, where p1, p2 are primes or 1. 2
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
EXTENSIONS
a(59) and beyond from Michael S. Branicky, Dec 21 2022
STATUS
approved

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Last modified March 28 14:02 EDT 2024. Contains 371254 sequences. (Running on oeis4.)