A048740 Product of divisors of n-th composite number.

8, 36, 64, 27, 100, 1728, 196, 225, 1024, 5832, 8000, 441, 484, 331776, 125, 676, 729, 21952, 810000, 32768, 1089, 1156, 1225, 10077696, 1444, 1521, 2560000, 3111696, 85184, 91125, 2116, 254803968, 343, 125000, 2601, 140608, 8503056, 3025, 9834496

Offset: 1

Author

Enoch Haga

Keywords

• easy
• nonn

Examples

			The third composite number is 8. The product of all divisors of 8 is 8*4*2*1 = 64.
Divisors(48) = {1,2,3,4,6,8,12,16,24,48} => product {1,2,3,4,6,8,12,16,24,48} = 254803968.
Divisors(49) = {1,7,49} => product {1,7,49} = 343.
Divisors(50) = {1,2,5,10,25,50} => product {1,2,5,10,25,50} = 125000.
		

References

• Albert H. Beiler, Recreations in the Theory of Numbers, 2nd ed., pages 10, 23. New York: Dover, 1966. ISBN 0-486-21096-0.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002808, A007955, A048741.

Programs

Mathematica

Rest[Times@@Divisors[#]&/@Complement[Range[100], Prime[ Range[ PrimePi[ 100]]]]] (* Harvey P. Dale, Jan 08 2011 *)
pd[n_] := n^(DivisorSigma[0, n]/2); pd /@ Select[Range[100], CompositeQ] (* Amiram Eldar, Sep 07 2019 *)

Python

from math import isqrt
from sympy import divisor_count, composite
def A048740(n): return (lambda m:isqrt(m)**c if (c:=divisor_count(m)) & 1 else m**(c//2))(composite(n)) # Chai Wah Wu, Jun 25 2022

Formula

a(n) = A007955(A002808(n)). - Michel Marcus, Sep 07 2019

Extensions

Corrected by Neven Juric (neven.juric(AT)apis-it.hr), May 25 2006