A007955 Product of divisors of n.

1, 2, 3, 8, 5, 36, 7, 64, 27, 100, 11, 1728, 13, 196, 225, 1024, 17, 5832, 19, 8000, 441, 484, 23, 331776, 125, 676, 729, 21952, 29, 810000, 31, 32768, 1089, 1156, 1225, 10077696, 37, 1444, 1521, 2560000, 41, 3111696, 43, 85184, 91125, 2116, 47, 254803968, 343

Offset

1,2

Comments

All terms of this sequence occur only once. See the second T. D. Noe link for a proof. - T. D. Noe, Jul 07 2008

Every natural number has a unique representation in terms of divisor products. See the W. Lang link. - Wolfdieter Lang, Feb 08 2011

a(n) = n only if n is prime or 1 (or, if n is in A008578). - Alonso del Arte, Apr 18 2011

Sometimes called the "divisorial" of n. - Daniel Forgues, Aug 03 2012

a(n) divides EulerPhi(x^n-y^n) (see A. Rotkiewicz link). - Michel Marcus, Dec 15 2012

The proof that all the terms of this sequence occur only once (mentioned above) was given by Niven in 1984. - Amiram Eldar, Aug 16 2020

References

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, p. 57.

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 83.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Wolfdieter Lang, Divisor Product Representation for Natural Numbers.

M. Le, On Smarandache Divisor Products, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, 144-145.

F. Luca, On the product of divisors of n and sigma(n), J. Ineq. Pure Appl. Math. 4 (2) 2003, Article 46.

T. D. Noe, The Divisor Product is Unique

A. Rotkiewicz, On the numbers Phi(a^n +/- b^n), Proc. Amer. Math. Soc. 12 (1961), 419-421.

Rodica Simon and Frank W. Schmid, Problem E 2946, The American Mathematical Monthly, Vol. 89, No. 5 (1982), p. 333, Ivan Niven, Product  of all Positive Divisors of n, solution to problem E 2946, ibid., Vol. 91, No. 10 (1984), p. 650.

F. Smarandache, Only Problems, Not Solutions!.

Eric Weisstein's World of Mathematics, Divisor Product.

Zhu Weiyi, On the divisor product sequences, Smarandache Notions J., Vol. 14 (2004), pp. 144-146.

OEIS Wiki, Divisorial.

Formula

a(n) = n^(d(n)/2) = n^(A000005(n)/2). Since a(n) = Product_(d|n) d = Product_(d|n) n/d, we have a(n)*a(n) = Product_(d|n) d*(n/d) = Product_(d|n) n = n^(tau(n)), whence a(n) = n^(tau(n)/2).

a(p^k) = p^A000217(k). - Enrique Pérez Herrero, Jul 22 2011

a(n) = A078599(n) * A178649(n). - Reinhard Zumkeller, Feb 06 2012

a(n) = A240694(n, A000005(n)). - Reinhard Zumkeller, Apr 10 2014

From Antti Karttunen, Mar 22 2017: (Start)

a(n) = A000196(n^A000005(n)). [From the original formula.]

A001222(a(n)) = A069264(n). [See Geoffrey Critzer's Feb 03 2015 comment in the latter sequence.]

A046523(a(n)) = A283995(n).

(End)

a(n) = Product_{k=1..n} gcd(n,k)^(1/phi(n/gcd(n,k))) = Product_{k=1..n} (n/gcd(n,k))^(1/phi(n/gcd(n,k))) where phi = A000010. - Richard L. Ollerton, Nov 07 2021

From Bernard Schott, Jan 11 2022: (Start)

a(n) = n^2 iff n is in A007422.

a(n) = n^3 iff n is in A162947.

a(n) = n^4 iff n is in A111398.

a(n) = n^5 iff n is in A030628.

a(n) = n^(3/2) iff n is in A280076. (End)

From Amiram Eldar, Oct 29 2022: (Start)

a(n) = n * A007956(n).

Sum_{k=1..n} 1/a(k) ~ log(log(n)) + c + O(1/log(n)), where c is a constant (Weiyi, 2004; Sandor and Crstici, 2004). (End)

a(n) = Product_{k=1..n} (n * (1 - ceiling(n/k - floor(n/k))))/k + ceiling(n/k - floor(n/k)). - Adriano Steffler, Feb 08 2024

Example

Divisors of 10 = [1, 2, 5, 10]. So, a(10) = 2*5*10 = 100. - Indranil Ghosh, Mar 22 2017

Maple

A007955 := proc(n) mul(d, d=numtheory[divisors](n)) ; end proc: # R. J. Mathar, Mar 17 2011
seq(isqrt(n^numtheory[tau](n)), n=1..50); # Gary Detlefs, Feb 15 2019

Mathematica

Array [ Times @@ Divisors[ # ]&, 100 ]
a[n_] := n^(DivisorSigma[0, n]/2); Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 21 2013 *)

Prog

(Magma) f := function(n); t1 := &*[d : d in Divisors(n) ]; return t1; end function;
(PARI) a(n)=if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2)) \\ Charles R Greathouse IV, Feb 11 2011
(Haskell)
a007955 = product . a027750_row  -- Reinhard Zumkeller, Feb 06 2012
(Sage) [prod(divisors(n)) for n in (1..100)] # Giuseppe Coppoletta, Dec 16 2014
(Scheme)
;; A naive stand-alone implementation:
(define (A007955 n) (let loop ((d n) (m 1)) (cond ((zero? d) m) ((zero? (modulo n d)) (loop (- d 1) (* m d))) (else (loop (- d 1) m)))))
;; Faster, if A000005 and A000196 are available:
(define (A007955 n) (A000196 (expt n (A000005 n))))
;; Antti Karttunen, Mar 22 2017
(Python)
from sympy import prod, divisors
print([prod(divisors(n)) for n in range(1, 51)]) # Indranil Ghosh, Mar 22 2017
(Python)
from math import isqrt
from sympy import divisor_count
def A007955(n):
    d = divisor_count(n)
    return isqrt(n)**d if d % 2 else n**(d//2) # Chai Wah Wu, Jan 05 2022
(GAP) List(List([1..50], n->DivisorsInt(n)), Product); # Muniru A Asiru, Feb 17 2019

Crossrefs

Cf. A000005, A000196, A001222, A007422, A007956, A027750, A030628, A046523, A069264, A072046, A111398, A162947, A224381, A243103, A280076, A283995.

Cf. A000203 (sums of divisors).

Cf. A000010 (comments on product formulas).

Sequence in context: A140651 A190997 A184392 * A324502 A170826 A162537

Adjacent sequences: A007952 A007953 A007954 * A007956 A007957 A007958

Keyword

nonn,nice

Author

R. Muller

Status

approved