A229835 - OEIS

%I #10 Dec 19 2013 10:26:34

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

%T 8,9,7,7,10,10,9,8,7,10,7,10,3,7,12,8,10,6,8,9,6,10,8,11,7,11,8,7,9,8,

%U 12,10,8,12,7,9,10,10,8,11,10,7,10,9,14,9,9,9,8,10,10,9,7,8,9,9,8,10,9,10,10,9,7,8,7,12,8

%N Number of ways to write n = (p - 1)/6 + q, where p is a prime, and q is a term of the sequence A000009.

%C Conjecture: a(n) > 0 for all n > 1. Also, any integer n > 1 can be written as (p + 1)/6 + q, where p is a prime and q is a term of A000009.

%C We have verified this for n up to 2*10^8. Note that 26128189 cannot be written as (p - 1)/4 + q with p a prime and q a term of A000009. Also, 65152682 cannot be written as (p + 1)/4 + q with p a prime and q a term of A000009.

%H Zhi-Wei Sun, <a href="/A229835/b229835.txt">Table of n, a(n) for n = 1..10000</a>

%e a(2) = 1 since 2 = (7 - 1)/ 6 + 1 with 7 prime, and 1 = A000009(i) for i = 0, 1, 2.

%e a(3) = 2 since 3 = (7 - 1 )/6 + 2 with 7 prime and 2 = A000009(3) = A000009(4), and 3 = (13 - 1 )/6 + 1 with 13 prime and 1 = A000009(i) for i = 0, 1, 2.

%t Do[m=0;Do[If[PartitionsQ[k]>=n,Goto[aa]];If[k>1&&PartitionsQ[k]==PartitionsQ[k-1],Goto[bb]];

%t If[PrimeQ[6(n-PartitionsQ[k])+1],m=m+1];Label[bb];Continue,{k,1,2n}];

%t Label[aa];Print[n," ",m];Continue,{n,1,100}]

%Y Cf. A000009, A000040, A232398, A232463, A233307, A233346, A233390, A233393, A233417, A233359.

%K nonn

%O 1,3

%A _Zhi-Wei Sun_, Dec 19 2013