A308411 Number of ways to write n as 2^i*3^j + A008347(k), where i, j and k > 0 are nonnegative integers.

0, 1, 2, 3, 4, 4, 4, 4, 5, 5, 5, 3, 5, 5, 4, 5, 5, 3, 6, 7, 6, 6, 5, 6, 7, 4, 5, 8, 6, 8, 7, 5, 5, 8, 8, 6, 7, 5, 7, 8, 7, 6, 6, 6, 8, 6, 5, 8, 6, 7, 7, 5, 8, 7, 8, 10, 9, 9, 6, 7, 8, 8, 6, 7, 10, 8, 8, 6, 9, 6, 7, 9, 8, 7, 7, 8, 9, 4, 8, 11, 10, 6, 9, 7, 11, 8, 8, 10, 9, 6, 4, 7, 10, 7, 7, 3, 11, 9, 10, 7

Offset

1,3

Comments

Conjecture: a(n) > 0 for all n > 1.

We have verified this for all n = 2..10^6.

Qing-Hu Hou at Tianjin University extended the verification to 2*10^7. Then I used Hou's program to verify a(n) > 0 for n up to 10^9. - Zhi-Wei Sun, May 28 2019

Conjecture verified for n up to 10^10. - Giovanni Resta, May 28 2019

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 2794-2812.

Example

a(2) = 1 with 2 = 2^0*3^0 + A008347(2).
a(3) = 2 with 3 = 2^0*3^0 + A008347(1) = 2^1*3^0 + A008347(2).

Mathematica

f[n_]:=f[n]=FactorInteger[n];
FQ[n_]:=FQ[n]=n>0&&Part[f[n], Length[f[n]]][[1]]<4;
s[0]=0; s[n_]:=s[n]=Prime[n]-s[n-1];
tab={}; Do[r=0; Do[If[FQ[n-s[k]], r=r+1], {k, 1, 2n-1}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]

Crossrefs

Cf. A000079, A000244, A008347, A308403, A316141.

Sequence in context: A195850 A253593 A174901 * A087848 A087844 A287635

Adjacent sequences: A308408 A308409 A308410 * A308412 A308413 A308414

Keyword

nonn

Author

Zhi-Wei Sun, May 25 2019

Status

approved