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A007955 Product of divisors of n. 141
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 (list; graph; refs; listen; history; text; internal format)
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

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

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

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 #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

F. Smarandache, Only Problems, Not Solutions!.

Eric Weisstein's World of Mathematics, Divisor Product

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

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)

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

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 operator import mul

from sympy.ntheory import divisors

print [reduce(mul, divisors(n)) for n in xrange(1, 101)] # Indranil Ghosh, Mar 22 2017

CROSSREFS

Cf. A000005, A000196, A001222, A007956, A027750, A046523, A069264, A072046, A243103, A283995.

Cf. A000203 (sums of divisors).

Sequence in context: A140651 A190997 A184392 * A170826 A162537 A109844

Adjacent sequences:  A007952 A007953 A007954 * A007956 A007957 A007958

KEYWORD

nonn,nice,changed

AUTHOR

R. Muller

STATUS

approved

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Last modified March 27 02:08 EDT 2017. Contains 284143 sequences.