A160199 Product of non-exponential divisors of n.

1, 1, 1, 1, 1, 6, 1, 4, 1, 10, 1, 24, 1, 14, 15, 8, 1, 54, 1, 40, 21, 22, 1, 2304, 1, 26, 9, 56, 1, 27000, 1, 512, 33, 34, 35, 216, 1, 38, 39, 6400, 1, 74088, 1, 88, 135, 46, 1, 73728, 1, 250, 51, 104, 1, 26244, 55, 12544, 57, 58, 1, 25920000, 1, 62, 189, 512, 65, 287496, 1

Offset: 1

Author

Jaroslav Krizek, May 04 2009

Keywords

• nonn

Comments

The non-exponential divisors of n are those divisors of n that are not exponential divisors of n.

There are only a few duplicates > 1. For example a(32) = a(64) = 512, a(243) = a(729) = 19683, a(3125) = a(15625) = 1953125. Antti Karttunen, Jan 24 2025

Examples

			The divisors of 6 are 1, 2, 3, 6. The only exponential divisor of 6 is 6, hence a(6) = 1*2*3 = 6.
The divisors of 16 are 1, 2, 4, 8, 16. The exponential divisors of 16 are 2, 4, 16, hence a(16) = 1*8 = 8.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..25000
• Eric Weisstein's World of Mathematics, e-Divisor

Crossrefs

Cf. A007955 (product of divisors of n), A157488 (product of exponential divisors of n), A049419.

Programs

Magma

[1] cat [ &*[ d: d in Divisors(n) | exists(t) { p: p in P | v eq 0 or e mod v gt 0 where v is Valuation(d, p) where e is Valuation(n, p) } where P is PrimeDivisors(n) ]: n in [2..67] ]; // Klaus Brockhaus, May 26 2009

Mathematica

f[p_, e_] := p^(DivisorSigma[1, e]/DivisorSigma[0, e]); a[n_] := Module[{fct = FactorInteger[n], e}, e = fct[[;; , 2]]; n^(Times @@ (e + 1)/2)/(Times @@ (f @@@ fct))^(Times @@ DivisorSigma[0, e])]; Array[a, 100] (* Amiram Eldar, Jan 25 2025 *)

PARI

A007955(n) = if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2));
A049419(n) = factorback(apply(numdiv,factor(n)[,2]));
A157488(n) = { my(f=factor(n), dexp=A049419(n)); prod(i=1, #f~, f[i, 1]^((sigma(f[i, 2]) * dexp / numdiv(f[i, 2])))); };
A160199(n) = (A007955(n) / A157488(n)); \\ Antti Karttunen, Jan 24 2025

Formula

a(n) = A007955(n) / A157488(n).

Extensions

Edited by Klaus Brockhaus, May 26 2009