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%I #9 Dec 07 2016 11:05:40
%S 0,0,1,2,1,3,1,3,2,2,1,5,1,3,2,3,2,4,1,5,2,3,1,5,2,3,3,4,1,4,1,7,3,2,
%T 1,5,2,4,3,4,1,6,2,6,2,3,2,5,1,5,3,5,2,5,2,4,3,3,1,9,1,6,3,3,2,3,3,7,
%U 3,4,1,7,1,6,2,5,3,5,1,7,4,3,1,6,1,6,6,4,1,5,1,7,3,4,3,5,2,7,2,6,1
%N G.f.: Sum_{k>0} x^prime(k)/(1-x^k).
%C New maxima occur at 2,3,5,11,31,59,211,331,619,1759,2341,3049,4343,12373,15431,18691,31667,66643,67651,...
%C 4343 and 15431 are the only composites in the terms displayed above.
%C If we define a new maximum as greater than or equal to the previous maximum we get
%C 1,2,3,5,7,11,19,23,31,59,131,163,167,197,211,331,467,521,547,...
%C This is very dense with primes and contains the previous list as a subset.
%F G.f.: Sum_{k>0} x^prime(k)/(1-x^k).
%t NN=200;MM=PrimePi[NN]+1; Table[Boole[n>2]+Sum[Boole[(n>Prime[k])&&(Mod[n-Prime[k]+k-1,k] == 0)], {k, 2, MM}], {n, 1, NN}]
%K nonn
%O 0,4
%A _Benedict W. J. Irwin_, Nov 28 2016
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