A359360 Length times minimum part of the integer partition with Heinz number n. Least prime index of n times number of prime indices of n.

0, 1, 2, 2, 3, 2, 4, 3, 4, 2, 5, 3, 6, 2, 4, 4, 7, 3, 8, 3, 4, 2, 9, 4, 6, 2, 6, 3, 10, 3, 11, 5, 4, 2, 6, 4, 12, 2, 4, 4, 13, 3, 14, 3, 6, 2, 15, 5, 8, 3, 4, 3, 16, 4, 6, 4, 4, 2, 17, 4, 18, 2, 6, 6, 6, 3, 19, 3, 4, 3, 20, 5, 21, 2, 6, 3, 8, 3, 22, 5, 8, 2

Offset: 1

Author

Gus Wiseman, Dec 28 2022

Keywords

• nonn

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). A prime index of n is a number m such that prime(m) divides n.

Examples

			The partition with Heinz number 7865 is (6,5,5,3), so a(7865) = 4*3 = 12.
		

Crossrefs

Difference of A056239 and A359358.

The opposite version is A326846.

A055396 gives minimum prime index, maximum A061395.

A112798 list prime indices, length A001222, sum A056239.

A243055 subtracts the least prime index from the greatest.

A358195 gives Heinz numbers of rows of A358172, even bisection A241916.

Cf. A006141, A124010, A246277, A268192, A316413, A325352, A326836, A326837, A326844, A355534, A356958.

Programs

Mathematica

Table[PrimeOmega[n]*PrimePi[FactorInteger[n][[1,1]]],{n,100}]

PARI

a(n) = if (n==1, 0, my(f=factor(n)); bigomega(f)*primepi(f[1, 1])); \\ Michel Marcus, Dec 28 2022

Formula

a(n) = A001222(n) * A055396(n).