cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, Mar 17 2003

Keywords

Comments

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

Links

Crossrefs

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

Programs

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

Views

Author

Reinhard Zumkeller, Mar 17 2003

Keywords

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

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

A081313 Numbers having more than one representation as sum of a prime and a 3-smooth number.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 53, 55, 56, 57, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84, 85, 86
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 17 2003

Keywords

Links

Crossrefs

Cf. A000040, A003586, A081308, A081312.
Subsequence of A081311.

Programs

  • Haskell
    a081313 n = a081313_list !! (n-1)
a081313_list = filter ((> 1) . a081308) [1..]
-- Reinhard Zumkeller, Jul 04 2012
  • Mathematica
    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[#] > 1 &] (* Jean-François Alcover, Oct 13 2021 *)

Formula

A081308(a(n)) > 1.
Showing 1-3 of 3 results.

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