A130064 a(n) = (n / SmallestPrimeFactor(n)) * GreatestPrimeFactor(n).

1, 2, 3, 4, 5, 9, 7, 8, 9, 25, 11, 18, 13, 49, 25, 16, 17, 27, 19, 50, 49, 121, 23, 36, 25, 169, 27, 98, 29, 75, 31, 32, 121, 289, 49, 54, 37, 361, 169, 100, 41, 147, 43, 242, 75, 529, 47, 72, 49, 125, 289, 338, 53, 81, 121, 196, 361, 841, 59, 150, 61, 961, 147, 64, 169, 363

Offset: 1

Author

Reinhard Zumkeller, May 05 2007

Keywords

• nonn

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Paul Erdős and J. H. van Lint, On the average ratio of the smallest and largest prime divisor of n, Indagationes Mathematicae (Proceedings), Vol. 85, No. 2 (1982), pp. 127-132.

Crossrefs

Cf. A006530, A020639, A032742, A000961.

Cf. A001221, A001222.

Cf. A000290, A046665, A074320, A066048, A130065

Programs

Mathematica

a[n_] := With[{pp = FactorInteger[n][[All, 1]]}, n*pp[[-1]]/pp[[1]]];
Array[a, 100] (* Jean-François Alcover, Nov 18 2021 *)

PARI

a(n) = if (n==1, 1, my(f=factor(n)[, 1]~); n*vecmax(f)/vecmin(f)); \\ Michel Marcus, Sep 24 2022

Python

from sympy import factorint
def a(n): f = factorint(n); return 1 if n == 1 else n//min(f)*max(f)
print([a(n) for n in range(1, 67)]) # Michael S. Branicky, Sep 24 2022

Formula

a(n) = n*A006530(n)/A020639(n) = A032742(n)*A006530(n);

a(n) >= n.

a(n) = n iff n is a prime power: a(A000961(n)) = A000961(n);

a(A001221(n)) <= A001221(n); a(A001222(n)) = A001222(n);

a(n) = A130065(n)+n*A046665(n)*A074320(n)/A066048(n) = A000290(n)/A130065(n).

Sum_{k=1..n} k/a(k) ~ n/log(n) + 3*n/log(n)^2 + o(n/log(n)^2) (Erdős and van Lint, 1982). - Amiram Eldar, Oct 14 2022