A157488 -id:A157488 - cp's OEIS frontend

A332622 Numbers where records occur for the product of exponential divisors function (A157488).

1, 2, 3, 4, 8, 9, 12, 16, 20, 28, 36, 72, 100, 144, 324, 400, 576, 900, 1764, 2700, 3528, 3600, 7056, 10800, 14400, 28224, 32400, 44100, 88200, 108900, 129600, 176400, 396900, 435600, 608400, 705600, 1587600, 3920400, 5336100, 5475600, 6350400, 14288400, 15681600

Offset: 1

Amiram Eldar, Jun 05 2020

nonn

The corresponding record values are 1, 2, 3, 8, 16, 27, 72, 128, 200, 392, 46656, 186624, 1000000, ...

Amiram Eldar, Table of n, a(n) for n = 1..68 (terms below 10^10)

Cf. A034287, A061742, A157488, A328134, A318278, A322791.

f[p_, e_] := p^(DivisorSigma[1, e]/DivisorSigma[0, e]); exprod[n_] := (Times @@ (f @@@ (fct = FactorInteger[n])))^(Times @@ DivisorSigma[0, Last /@ fct]); em = 0; s = {}; Do[If[(e = exprod[n]) > em, em = e; AppendTo[s, n]], {n, 1, 10^6}]; s

The first 8 terms of A157488 are 1, 2, 3, 8, 5, 6, 7 and 16. The record values occur at 1, 2, 3, 4 and 8 - the first 5 terms of this sequence.

A160199 Product of non-exponential divisors of n.

Original entry on oeis.org

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

Jaroslav Krizek, May 04 2009

nonn

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

			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.

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

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

[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

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 *)

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

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

Edited by Klaus Brockhaus, May 26 2009

cp's OEIS Frontend

A332622 Numbers where records occur for the product of exponential divisors function (A157488).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula