A325223 Sum of the prime indices of n minus the greater of the number of prime factors of n counted with multiplicity and the largest prime index of n.

0, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 1, 0, 1, 2, 0, 0, 2, 0, 2, 2, 1, 0, 1, 3, 1, 3, 2, 0, 3, 0, 0, 2, 1, 3, 2, 0, 1, 2, 2, 0, 3, 0, 2, 4, 1, 0, 1, 4, 4, 2, 2, 0, 3, 3, 3, 2, 1, 0, 3, 0, 1, 4, 0, 3, 3, 0, 2, 2, 4, 0, 2, 0, 1, 5, 2, 4, 3, 0, 2, 4, 1, 0, 4, 3, 1, 2

Offset

1,9

Comments

A prime index of n is a number m such that prime(m) divides n.

Also the number of squares in the Young diagram of the integer partition with Heinz number n after the first row or the first column, whichever is larger, is removed. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Links

Table of n, a(n) for n=1..87.

FindStat, St000784: The maximum of the length and the largest part of the integer partition

Formula

a(n) = A056239(n) - max(A001222(n), A061395(n)) = A056239(n) - A263297(n).

Example

88 has 4 prime indices {1,1,1,5} with sum 8 and maximum 5, so a(88) = 8 - max(4,5) = 3.

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Total[primeMS[n]]-Max[Length[primeMS[n]], Max[primeMS[n]]], {n, 100}]

Crossrefs

Positions of 0's are A174090. Positions of 1's are A325231.

Cf. A001222, A051924, A052126, A056239, A061395, A064989, A065770, A112798, A257990, A263297, A325134, A325169, A325224, A325225, A325227.

Sequence in context: A307776 A341027 A050317 * A141095 A175599 A179759

Adjacent sequences: A325220 A325221 A325222 * A325224 A325225 A325226

Keyword

nonn

Author

Gus Wiseman, Apr 12 2019

Status

approved