A071178 Exponent of the largest prime factor of n.

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Author

Benoit Cloitre, Jun 10 2002

Keywords

• easy
• nonn

Comments

a(n) = the multiplicity of the largest part in the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(18) = 2; indeed, the partition having Heinz number 18 = 2*3*3 is [1,2,2]. - Emeric Deutsch, Jun 04 2015

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001221, A001222, A006530, A053585, A067029, A067255, A070003, A124010, A215366.

Programs

Haskell

a071178 = last . a124010_row -- Reinhard Zumkeller, Aug 27 2011

Maple

with(numtheory): with(padic):
a:= n-> `if`(n=1, 0, ordp(n, max(factorset(n)[]))):
seq(a(n), n=1..120);  # Alois P. Heinz, Jun 04 2015

Mathematica

a[n_] := FactorInteger[n] // Last // Last; Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Jun 12 2015 *)
Join[{0},Table[FactorInteger[n][[-1,2]],{n,2,120}]] (* Harvey P. Dale, Aug 02 2025 *)

PARI

a(n) = if(n == 1, 0, my(e = factor(n)[, 2]); e[#e]); \\ Amiram Eldar, Oct 02 2024

Python

from sympy import factorint
def A071178(n): return max(factorint(n).items())[1] if n>1 else 0 # Chai Wah Wu, Oct 10 2023

Formula

a(n) = A124010(n, A001221(n)); A053585(n) = A006530(n)^a(n). - Reinhard Zumkeller, Aug 27 2011

a(n) = A067255(n, A001222(n)). - Reinhard Zumkeller, Jun 11 2013

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 (since the asymptotic density of A070003 is 0). - Amiram Eldar, Oct 02 2024