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

1,6

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..25000

Eric Weisstein's World of Mathematics, e-Divisor

Formula

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

Example

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.

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A354857 A371348 A358981 * A178646 A144540 A292107

Adjacent sequences: A160196 A160197 A160198 * A160200 A160201 A160202

Keyword

nonn

Author

Jaroslav Krizek, May 04 2009

Extensions

Edited by Klaus Brockhaus, May 26 2009

Status

approved