A081310 -id:A081310 - cp's OEIS frontend

A081311 Numbers that can be written as sum of a prime and an 3-smooth number.

3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

Reinhard Zumkeller, Mar 17 2003

nonn

A081308(a(n))>0; complement of A081310.

Up to 10^n this sequence has 8, 95, 916, 8871, 86974, 858055, 8494293, 84319349, 838308086, ... terms. The lower density is of this sequence is greater than 0.59368 (see Pintz), but seems to be less than 1; can this be proved? Charles R Greathouse IV, Sep 01 2015

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
J. Pintz, A note on Romanov's constant, Acta Mathematica Hungarica 112:1-2 (2006), pp. 1-14.

A118955 is a subsequence.
Union of A081312 and A081313.
Cf. A000040, A003586.

a081310 n = a081310_list !! (n-1)
a081310_list = filter ((== 0) . a081308) [1..]
-- Reinhard Zumkeller, Jul 04 2012

nmax = 1000;
S = Select[Range[nmax], Max[FactorInteger[#][[All, 1]]] <= 3 &];
A081308[n_] := Count[TakeWhile[S, # < n &], s_ /; PrimeQ[n - s]];
Select[Range[nmax], A081308[#] > 0 &] (* Jean-François Alcover, Oct 13 2021 *)

is(n)=for(i=0, logint(n,3), my(k=3^i); while(kCharles R Greathouse IV, Sep 01 2015

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

Original entry on oeis.org

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

Reinhard Zumkeller, Mar 17 2003

nonn

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.

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

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]

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

Cf. A010051.

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

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

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

A081309 Smallest prime p such that n-p is a 3-smooth number, a(n)=0 if no such prime exists.

Original entry on oeis.org

0, 0, 2, 2, 2, 2, 3, 2, 3, 2, 2, 3, 5, 2, 3, 7, 5, 2, 3, 2, 3, 13, 5, 23, 7, 2, 3, 19, 2, 3, 7, 5, 17, 2, 3, 0, 5, 2, 3, 13, 5, 41, 7, 17, 13, 19, 11, 47, 13, 2, 3, 43, 5, 53, 7, 2, 3, 31, 5, 59, 7, 53, 31, 37, 11, 2, 3, 41, 5, 43, 7, 71, 19, 2, 3, 67, 5, 0, 7, 53, 17, 73, 2, 3, 13, 5, 23, 7, 17, 89

Offset: 1

Reinhard Zumkeller, Mar 17 2003

			a(25)=7: 25=7+2*3^2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A081310, A081311, A081308, A000040, A003586.

Cf. A065333.

a081309 n | null ps   = 0
          | otherwise = head ps
          where ps = [p | p <- takeWhile (< n) a000040_list,
                          a065333 (n - p) == 1]
-- Reinhard Zumkeller, Jul 04 2012

smooth3Q[n_] := n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3] == 1;
a[n_] := Module[{p}, For[p = 2, p < n, p = NextPrime[p], If[smooth3Q[n - p], Return[p]]]; 0];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 14 2021 *)

a(n)=0 iff A081308(n)=0.

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A081311 Numbers that can be written as sum of a prime and an 3-smooth number.

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

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Haskell

Mathematica

PARI

A081309 Smallest prime p such that n-p is a 3-smooth number, a(n)=0 if no such prime exists.

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