A325224 Sum of prime indices of n minus the lesser of the number of prime factors of n counted with multiplicity and the maximum prime index of n.

0, 0, 1, 1, 2, 1, 3, 2, 2, 2, 4, 2, 5, 3, 3, 3, 6, 3, 7, 2, 4, 4, 8, 3, 4, 5, 4, 3, 9, 3, 10, 4, 5, 6, 5, 4, 11, 7, 6, 3, 12, 4, 13, 4, 4, 8, 14, 4, 6, 4, 7, 5, 15, 5, 6, 3, 8, 9, 16, 4, 17, 10, 5, 5, 7, 5, 18, 6, 9, 5, 19, 5, 20, 11, 5, 7, 7, 6, 21, 4, 6, 12

Offset

1,5

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 smaller, 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..82.

FindStat, St000533: The maximal number of non-attacking rooks on a Ferrers shape

Formula

a(n) = A056239(n) - min(A001222(n), A061395(n)) = A056239(n) - A325225(n).

Example

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

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[If[n==1, 0, Total[primeMS[n]]-Min[Length[primeMS[n]], Max[primeMS[n]]]], {n, 100}]

Crossrefs

The number of times k appears in the sequence is A325232(k).

Cf. A001222, A052126, A056239, A061395, A064989, A065770, A112798, A174090, A257990, A263297, A325134, A325169, A325223, A325225, A325227.

Sequence in context: A035221 A035191 A297167 * A303389 A276166 A177062

Adjacent sequences: A325221 A325222 A325223 * A325225 A325226 A325227

Keyword

nonn

Author

Gus Wiseman, Apr 12 2019

Status

approved