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

1,4

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.

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]

Example

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

Mathematica

nmax = 1000;
S = Select[Range[nmax], Max[FactorInteger[#][[All, 1]]] <= 3&];
a[n_] := Count[TakeWhile[S, #<n&], s_ /; PrimeQ[n-s]];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Oct 13 2021 *)

Prog

(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)<<e2)))  \\ M. F. Hasler, Oct 21 2011
(Haskell)
a081308 n = sum $ map (a010051' . (n -)) $ takeWhile (< n) a003586_list
-- Reinhard Zumkeller, Jul 04 2012

Crossrefs

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

Cf. A010051.

Sequence in context: A238459 A109301 A107573 * A070210 A100198 A164996

Adjacent sequences: A081305 A081306 A081307 * A081309 A081310 A081311

Keyword

nonn

Author

Reinhard Zumkeller, Mar 17 2003

Status

approved