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

Author

Zhi-Wei Sun, May 25 2019

Keywords

• nonn

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

Examples

			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).
		

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.

Crossrefs

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

Programs

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]