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

%I A308411 #19 Jun 05 2019 18:43:34
%S A308411 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,
%T A308411 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,
%U A308411 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
%N A308411 Number of ways to write n as 2^i*3^j + A008347(k), where i, j and k > 0 are nonnegative integers.
%C A308411 Conjecture: a(n) > 0 for all n > 1.
%C A308411 We have verified this for all n = 2..10^6.
%C A308411 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
%C A308411 Conjecture verified for n up to 10^10. - _Giovanni Resta_, May 28 2019
%H A308411 Zhi-Wei Sun, <a href="/A308411/b308411.txt">Table of n, a(n) for n = 1..10000</a>
%H A308411 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2013.02.003">On functions taking only prime values</a>, J. Number Theory 133(2013), no.8, 2794-2812.
%e A308411 a(2) = 1 with 2 = 2^0*3^0 + A008347(2).
%e A308411 a(3) = 2 with 3 = 2^0*3^0 + A008347(1) = 2^1*3^0 + A008347(2).
%t A308411 f[n_]:=f[n]=FactorInteger[n];
%t A308411 FQ[n_]:=FQ[n]=n>0&&Part[f[n],Length[f[n]]][[1]]<4;
%t A308411 s[0]=0; s[n_]:=s[n]=Prime[n]-s[n-1];
%t A308411 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]
%Y A308411 Cf. A000079, A000244, A008347, A308403, A316141.
%K A308411 nonn
%O A308411 1,3
%A A308411 _Zhi-Wei Sun_, May 25 2019