login
Length times minimum part of the integer partition with Heinz number n. Least prime index of n times number of prime indices of n.
8

%I #8 Dec 28 2022 15:42:39

%S 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,

%T 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,

%U 6,3,19,3,4,3,20,5,21,2,6,3,8,3,22,5,8,2

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

%C 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.

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

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

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

%o (PARI) a(n) = if (n==1, 0, my(f=factor(n)); bigomega(f)*primepi(f[1, 1])); \\ _Michel Marcus_, Dec 28 2022

%Y Difference of A056239 and A359358.

%Y The opposite version is A326846.

%Y A055396 gives minimum prime index, maximum A061395.

%Y A112798 list prime indices, length A001222, sum A056239.

%Y A243055 subtracts the least prime index from the greatest.

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

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

%K nonn

%O 1,3

%A _Gus Wiseman_, Dec 28 2022