A365920 - OEIS

%I #12 Dec 13 2024 09:37:49

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

%T 6,0,11,7,7,0,12,0,13,4,6,8,14,0,7,5,8,5,15,0,7,0,9,9,16,0,17,10,7,0,

%U 8,4,18,6,10,6,19,0,20,11,7,7,8,5,21,0,7,12

%N Greatest non-subset-sum of the prime indices of n, or 0 if there is none.

%C This is the greatest element of {0,...,A056239(n)} that is not equal to A056239(d) for any divisor d|n, d>1. This definition is analogous to the Frobenius number of a numerical semigroup (see link), but it looks only at submultisets of a finite multiset, not all multisets of elements of a set.

%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

%H Steven R. Finch, <a href="/A066062/a066062.pdf">Monoids of natural numbers</a>, March 17, 2009.

%e The prime indices of 156 are {1,1,2,6}, with subset-sums 0, 1, 2, 3, 4, 6, 7, 8, 9, 10, so a(156) = 5.

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];

%t Table[Max@@Prepend[nmz[prix[n]],0],{n,100}]

%Y For binary indices instead of sums we have A063250.

%Y Positions of first appearances > 2 are A065091.

%Y Zeros are A325781, nonzeros A325798.

%Y For prime indices instead of sums we have A339662, minimum A257993.

%Y For least instead of greatest non-subset-sum we have A366128.

%Y A055932 lists numbers whose prime indices cover an initial interval.

%Y A056239 adds up prime indices, row sums of A112798.

%Y A073491 lists numbers with gap-free prime indices.

%Y A238709/A238710 count partitions by least/greatest difference.

%Y A342050/A342051 have prime indices with odd/even least gap.

%Y Cf. A001223, A001522, A005117, A079068, A098743, A264401, A286469 or A286470, A339737, A339886.

%K nonn

%O 1,5

%A _Gus Wiseman_, Sep 30 2023