A081308 Number of ways to write n as sum of a prime and an 3-smooth number.

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

Offset: 1

Author

Reinhard Zumkeller, Mar 17 2003

Keywords

• nonn

Comments

Asymptotically, for n coprime to 6, a(n) ~ C*n on the average, with C=3/(2*log(2)*log(3))~1.969796..., see the link. - M. F. Hasler, Oct 21 2011

a(A081310(n)) = 0; a(A081311(n)) > 0; a(A081312(n)) = 1; a(A081313(n)) > 1.

Examples

			a(12)=2: 12=11+1=3+3^2; a(13)=3: 13=11+2=7+2*3=5+2^3.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Mark Underwood, another goldbachian theme, Mar. 16, 2009 and follow-up on Oct. 21, 2011.
• Mark Underwood and others, Another goldbachian theme, digest of 64 messages in primenumbers Yahoo group, Mar 16, 2009 - Oct 30, 2011. [Cached copy]

Crossrefs

Cf. A081310, A081311, A081309, A000040, A003586.

Cf. A010051.

Programs

Haskell

a081308 n = sum $ map (a010051' . (n -)) $ takeWhile (< n) a003586_list
-- Reinhard Zumkeller, Jul 04 2012

Mathematica

nmax = 1000;
S = Select[Range[nmax], Max[FactorInteger[#][[All, 1]]] <= 3&];
a[n_] := Count[TakeWhile[S, #Jean-François Alcover, Oct 13 2021 *)

PARI

A081308(n)=my(L2=log(2));sum(e3=0,log(n+.5)\log(3), sum(e2=0,log(n\3^e3)\L2, isprime(n-(3^e3)<M. F. Hasler, Oct 21 2011