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

1,1

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

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A002808, A007955, A048741.

Sequence in context: A200713 A184294 A057345 * A139608 A321778 A009923

Adjacent sequences: A048737 A048738 A048739 * A048741 A048742 A048743

Keyword

easy,nonn

Author

Enoch Haga

Extensions

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

Status

approved