A061395 Let p be the largest prime factor of n; if p is the k-th prime then set a(n) = k; a(1) = 0 by convention.

0, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 2, 6, 4, 3, 1, 7, 2, 8, 3, 4, 5, 9, 2, 3, 6, 2, 4, 10, 3, 11, 1, 5, 7, 4, 2, 12, 8, 6, 3, 13, 4, 14, 5, 3, 9, 15, 2, 4, 3, 7, 6, 16, 2, 5, 4, 8, 10, 17, 3, 18, 11, 4, 1, 6, 5, 19, 7, 9, 4, 20, 2, 21, 12, 3, 8, 5, 6, 22, 3, 2, 13, 23, 4, 7, 14, 10, 5, 24, 3, 6, 9, 11, 15

Offset: 1

Author

Henry Bottomley, Apr 30 2001

Keywords

• easy
• nice
• nonn

Comments

Records occur at the primes. - Robert G. Wilson v, Dec 30 2007

For n > 1: length of n-th row in A067255. - Reinhard Zumkeller, Jun 11 2013

a(n) = the largest part of 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(20) = 3; indeed, the partition having Heinz number 20 = 2*2*5 is [1,1,3]. - Emeric Deutsch, Jun 04 2015

Examples

			a(20) = 3 since the largest prime factor of 20 is 5, which is the 3rd prime.
		

Links

• Álvar Ibeas, Table of n, a(n) for n = 1..100000 (first 1000 terms from Harry J. Smith)
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000720, A006530, A055396, A061394, A133674, A243055.

Cf. A029837, A087207.

Programs

Haskell

a061395 = a049084 . a006530  -- Reinhard Zumkeller, Jun 11 2013

Maple

with(numtheory):
a:= n-> pi(max(1, factorset(n)[])):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 03 2013

Mathematica

Insert[Table[PrimePi[FactorInteger[n][[ -1]][[1]]], {n, 2, 120}], 0, 1] (* Stefan Steinerberger, Apr 11 2006 *)
f[n_] := PrimePi[ FactorInteger@n][[ -1, 1]]; Array[f, 94] (* Robert G. Wilson v, Dec 30 2007 *)

PARI

a(n) = if (n==1, 0, primepi(vecmax(factor(n)[,1]))); \\ Michel Marcus, Nov 14 2022

Python

from sympy import primepi, primefactors
def a(n): return 0 if n==1 else primepi(primefactors(n)[-1])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, May 14 2017

Formula

A000040(a(n)) = A006530(n); a(n) = A049084(A006530(n)). - Reinhard Zumkeller, May 22 2003

A243055(n) = a(n) - A055396(n). - Antti Karttunen, Mar 07 2017

a(n) = A000720(A006530(n)). - Alois P. Heinz, Mar 05 2020

a(n) = A029837(A087207(n)+1). - Flávio V. Fernandes, Apr 24 2025

Extensions

Definition reworded by N. J. A. Sloane, Jul 01 2008