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A007955
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Product of divisors of n.
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218
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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)
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 83.
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LINKS
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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 #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
OEIS Wiki, Divisorial.
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FORMULA
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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)
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EXAMPLE
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Divisors of 10 = [1, 2, 5, 10]. So, a(10) = 2*5*10 = 100. - Indranil Ghosh, Mar 22 2017
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MAPLE
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A007955 := proc(n) mul(d, d=numtheory[divisors](n)) ; end proc: # R. J. Mathar, Mar 17 2011
with(numtheory):seq( simplify (n^(tau(n)/2)), n=1..50) # Gary Detlefs, Feb 15 2019
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MATHEMATICA
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Array [ Times @@ Divisors[ # ]&, 100 ]
a[n_] := n^(DivisorSigma[0, n]/2); Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 21 2013 *)
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PROG
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(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
(GAP) List(List([1..50], n->DivisorsInt(n)), Product); # Muniru A Asiru, Feb 17 2019
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CROSSREFS
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Cf. A000005, A000196, A001222, A007956, A027750, A046523, A069264, A072046, A224381, A243103, A283995.
Cf. A000203 (sums of divisors).
Sequence in context: A140651 A190997 A184392 * A324502 A170826 A162537
Adjacent sequences: A007952 A007953 A007954 * A007956 A007957 A007958
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KEYWORD
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nonn,nice
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AUTHOR
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R. Muller
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STATUS
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approved
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