A365920 Greatest non-subset-sum of the prime indices of n, or 0 if there is none.

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

Offset: 1

Author

Gus Wiseman, Sep 30 2023

Keywords

• nonn

Comments

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.

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.

Examples

			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.
		

Links

• Steven R. Finch, Monoids of natural numbers, March 17, 2009.

Crossrefs

For binary indices instead of sums we have A063250.

Positions of first appearances > 2 are A065091.

Zeros are A325781, nonzeros A325798.

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

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

A055932 lists numbers whose prime indices cover an initial interval.

A056239 adds up prime indices, row sums of A112798.

A073491 lists numbers with gap-free prime indices.

A238709/A238710 count partitions by least/greatest difference.

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

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

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[Max@@Prepend[nmz[prix[n]],0],{n,100}]