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A308950 Number of ways to write n as (p-1)/6 + 2^a*3^b, where p is a prime, and a and b are nonnegative integers.

%I A308950 #17 Jul 03 2019 09:55:03
%S A308950 0,1,2,3,3,3,4,4,5,4,5,4,6,7,6,4,5,6,9,6,6,6,5,6,7,6,7,7,10,7,6,5,8,
%T A308950 10,8,7,8,8,11,5,10,8,8,7,6,6,6,9,10,8,6,5,10,9,8,7,9,7,11,7,8,8,7,13,
%U A308950 10,7,10,5,10,10,10,8,8,13,9,8,8,10,11,9,8,11,8,10,10,8,8,10,9,8,8,8,10,10,8,5,11,8,15,7
%N A308950 Number of ways to write n as (p-1)/6 + 2^a*3^b, where p is a prime, and a and b are nonnegative integers.
%C A308950 Conjecture: Let r be 1 or -1. Then, any integer n > 1 can be written as (p-r)/6 + 2^a*3^b, where p is a prime, and a and b are nonnegative integers; in other words, 6*n+r can be written as p + 2^k*3^m, where p is a prime, and k and m are positive integers.
%C A308950 We have verified this for all n = 2..10^9.
%C A308950 Conjecture verified up to n = 10^11. - _Giovanni Resta_, Jul 03 2019
%H A308950 Zhi-Wei Sun, <a href="/A308950/b308950.txt">Table of n, a(n) for n = 1..10000</a>
%e A308950 a(2) = 1 since 2 = (7-1)/6 + 2^0*3^0 with 7 prime.
%e A308950 a(3) = 2 since 3 = (13-1)/6 + 2^0*3^0 = (7-1)/6 + 2^1*3^0 with 13 and 7 prime.
%t A308950 tab={};Do[r=0;Do[If[PrimeQ[6(n-2^a*3^b)+1],r=r+1],{a,0,Log[2,n]},{b,0,Log[3,n/2^a]}];tab=Append[tab,r],{n,1,100}];Print[tab]
%Y A308950 Cf. A000040, A000079, A000244, A002476, A003586, A007528, A308411.
%K A308950 nonn
%O A308950 1,3
%A A308950 _Zhi-Wei Sun_, Jul 02 2019