A028234 If n = p_1^e_1 * ... * p_k^e_k, p_1 < ... < p_k primes, then a(n) = n/p_1^e_1, with a(1) = 1.

1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 7, 5, 1, 1, 9, 1, 5, 7, 11, 1, 3, 1, 13, 1, 7, 1, 15, 1, 1, 11, 17, 7, 9, 1, 19, 13, 5, 1, 21, 1, 11, 5, 23, 1, 3, 1, 25, 17, 13, 1, 27, 11, 7, 19, 29, 1, 15, 1, 31, 7, 1, 13, 33, 1, 17, 23, 35, 1, 9, 1, 37, 25, 19, 11, 39, 1, 5, 1, 41, 1, 21

Offset: 1

Author

Marc LeBrun

Keywords

• nonn
• nice
• easy

Comments

Together with A067029 is useful for defining sequences that are multiplicative with a(p^e) = f(e), as recurrences of the form: a(1) = 1 and for n > 1, a(n) = f(A067029(n)) * a(A028234(n)). - Antti Karttunen, May 29 2017

Links

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

Crossrefs

Cf. A001221, A001222, A002110, A005867, A020639, A028233, A067029.

Programs

GAP

a := List(List(List(List([1..10^3], Factors), Collected), i -> i[1]), j -> j[1]^j[2]);;
A028234 := List([1..Length(a)],i->i/a[i]); # Muniru A Asiru, Jan 27 2018

Haskell

a028234 n = n `div` a028233 n  -- Reinhard Zumkeller, Mar 27 2013

Mathematica

a[n_] := n / Power @@ First[FactorInteger[n]]; Table[a[n], {n, 1, 84}] (* Jean-François Alcover, Jun 12 2012 *)

PARI

a(n) = {my(f = factor(n)); if (#f~, f[1, 1] = 1); factorback(f);} \\ Michel Marcus, Feb 11 2016

Python

from sympy import factorint
def a(n):
    f = factorint(n)
    return 1 if n==1 else n/(min(f)**f[min(f)]) # Indranil Ghosh, May 12 2017

Scheme

(define (A028234 n) (/ n (A028233 n))) ;; Needs also code from A020639 and A028233. - Antti Karttunen, May 29 2017

Formula

a(n) = n / A028233(n).

A001221(a(n)) = A001221(n)-1; A001222(a(n)) = A001222(n)-A067029(n). - Reinhard Zumkeller, May 13 2006

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Sum_{k>=0} A005867(k)/(prime(k+1)*(prime(k+1)+1)*A002110(k)) = 0.114813... . - Amiram Eldar, Nov 19 2022

Extensions

Edited name to include a(1) = 1 by Franklin T. Adams-Watters, Jan 27 2018