A162947 -id:A162947 - cp's OEIS frontend

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

R. Muller

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

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

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.

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.

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).

List(List([1..50],n->DivisorsInt(n)),Product); # Muniru A Asiru, Feb 17 2019

a007955 = product . a027750_row  -- Reinhard Zumkeller, Feb 06 2012

f := function(n); t1 := &*[d : d in Divisors(n) ]; return t1; end function;

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

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

a(n)=if(issquare(n,&n),n^numdiv(n^2),n^(numdiv(n)/2)) \\ Charles R Greathouse IV, Feb 11 2011

from sympy import prod, divisors
print([prod(divisors(n)) for n in range(1, 51)]) # Indranil Ghosh, Mar 22 2017

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

[prod(divisors(n)) for n in (1..100)] # Giuseppe Coppoletta, Dec 16 2014

[n^(sigma(n,0)/2) for n in (1..49)] # Stefano Spezia, Jul 14 2025

;; 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

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

A003680 Smallest number with 2n divisors.

Original entry on oeis.org

2, 6, 12, 24, 48, 60, 192, 120, 180, 240, 3072, 360, 12288, 960, 720, 840, 196608, 1260, 786432, 1680, 2880, 15360, 12582912, 2520, 6480, 61440, 6300, 6720, 805306368, 5040, 3221225472, 7560, 46080, 983040, 25920, 10080, 206158430208, 3932160, 184320, 15120

Offset: 1

N. J. A. Sloane, Mira Bernstein

Refers to the least number which is multiplicatively n-perfect, i.e. least number m the product of whose divisors equals m^n. - Lekraj Beedassy, Sep 18 2004

For n=1 to 5, a(n) equals second term of A008578, A007422, A162947, A048945, A030628. - Michel Marcus, Feb 04 2014

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 23.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 1..3300 (terms 1..1000 from T. D. Noe using A005179)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A005179 (n), A061283 (2n-1), A118224 (at least 2n).

A005179 = Cases[Import["https://oeis.org/A005179/b005179.txt", "Table"], {, }][[All, 2]];
A = {#, DivisorSigma[0, #]}& /@ A005179;
a[n_] := SelectFirst[A, #[[2]] == 2n&][[1]];
a /@ Range[1000] (* Jean-François Alcover, Nov 10 2019 *)
mp[1, m_] := {{}}; mp[n_, 1] := {{}}; mp[n_?PrimeQ, m_] := If[m < n, {}, {{n}}]; mp[n_, m_] := Join @@ Table[Map[Prepend[#, d] &, mp[n/d, d]], {d, Select[Rest[Divisors[n]], # <= m &]}]; mp[n_] := mp[n, n]; Table[mulpar = mp[2*n] - 1; Min[Table[Product[Prime[s]^mulpar[[j, s]], {s, 1, Length[mulpar[[j]]]}], {j, 1, Length[mulpar]}]], {n, 1, 100}] (* Vaclav Kotesovec, Apr 04 2021 *)
With[{tbl=Table[{n,DivisorSigma[0,n]},{n,800000}]},Table[SelectFirst[tbl,#[[2]]==2k&],{k,20}]][[;;,1]] (* The program generates the first 20 terms of the sequence. *) (* Harvey P. Dale, Jul 06 2025 *)

a(n)=my(k=2*n); while(numdiv(k)!=2*n, k++); k \\ Charles R Greathouse IV, Jun 23 2017

from sympy import divisors
def a(n):
  m = 4*n - 2
  while len(divisors(m)) != 2*n: m += 1
  return m
print([a(n) for n in range(1, 19)]) # Michael S. Branicky, Feb 06 2021

Bisection of A005179(n). - Lekraj Beedassy, Sep 21 2004

More terms from Jud McCranie Oct 15 1997

A292286 a(n) = k if the product of the divisors of n is n^k for some integer k, or -1 if no such k exists. For the ambiguous case, define a(1) = 0.

Original entry on oeis.org

0, 1, 1, -1, 1, 2, 1, 2, -1, 2, 1, 3, 1, 2, 2, -1, 1, 3, 1, 3, 2, 2, 1, 4, -1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, -1, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 5, -1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 6, 1, 2, 3, -1, 2, 4, 1, 3, 2, 4, 1, 6, 1, 2, 3, 3, 2, 4, 1, 5, -1, 2, 1, 6, 2, 2, 2, 4, 1, 6, 2, 3, 2, 2, 2, 6, 1, 3, 3, -1

Offset: 1

Juri-Stepan Gerasimov, Sep 13 2017

sign

If the number of divisors (nd) of n > 1 is odd, then a(n) = -1, else a(n) = nd/2. - Michel Marcus, Sep 14 2017
First occurrence of k beginning with -1 is A293570(r). - Robert G. Wilson v, Oct 10 2017
Records occur for A293570(r): 4, 6, 12, 24, 48, 60, 192, 240, 3072, 12288, 196608, 786432, 12582912, 805306368, etc. - Robert G. Wilson v, Oct 10 2017

			a(10) = 2 because divisors of 10 are 1,2,5,10 with product 100 = 10^2.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from exponents in factorization of n

Numbers n such that the product of divisors of n is n^k: A000040 (k = 1), A007422 (k = 2), A162947 (k = 3), A111398 (k = 4), A030628 (k = 5), A030630 (k = 6).

Cf. A000037, A000290, A056924, A007955, A293570.

Table[Boole[n == 1] + If[OddQ@ #, -1, #/2] &@ DivisorSigma[0, n], {n, 100}] (* Michael De Vlieger, Sep 15 2017 *)

a(n) = if (n==1, 0, my(nd = numdiv(n)); if (nd % 2, -1, nd/2)); \\ Michel Marcus, Sep 14 2017

a(n)=my(k=numdiv(n)); if(k%2, if(n>1, -1, 0), k/2) \\ Charles R Greathouse IV, Sep 19 2017

a(1) = 0, a(A000290(n+1)) = -1, a(A000037(n+1)) = A056924(A000037(n+1)), where A000290 = the squares and A000037 = the nonsquares.

Definition corrected by Charles R Greathouse IV, Sep 13 2017

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A007955 Product of divisors of n.

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A003680 Smallest number with 2n divisors.

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A292286 a(n) = k if the product of the divisors of n is n^k for some integer k, or -1 if no such k exists. For the ambiguous case, define a(1) = 0.

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