A069264 -id:A069264 - 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

A318366 a(n) = Sum_{d|n} bigomega(d)*bigomega(n/d).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 4, 1, 2, 0, 8, 0, 2, 2, 10, 0, 8, 0, 8, 2, 2, 0, 20, 1, 2, 4, 8, 0, 12, 0, 20, 2, 2, 2, 24, 0, 2, 2, 20, 0, 12, 0, 8, 8, 2, 0, 40, 1, 8, 2, 8, 0, 20, 2, 20, 2, 2, 0, 34, 0, 2, 8, 35, 2, 12, 0, 8, 2, 12, 0, 52, 0, 2, 8, 8, 2, 12, 0, 40, 10, 2, 0, 34, 2, 2, 2, 20, 0, 34, 2, 8, 2, 2, 2

Offset: 1

Ilya Gutkovskiy, Aug 24 2018

nonn

Dirichlet convolution of A001222 with itself.

			24 has 8 divisors, namely 1, 2, 3, 4, 6, 8, 12, 24, and four prime factors counted with multiplicity. The divisors have 0, 1, 1, 2, 2, 3, 3, 4 divisors respectively. So a(24) = 0 * (4 - 0) + 1 * (4 - 1) + 1 * (4 - 1) + 2 * (4 - 2) + 2 * (4 - 2) + 3 * (4 - 3) + 4 * (4 - 4) = 0 + 3 + 3 + 4 + 4 + 3 + 3 + 0 = 20. - _David A. Corneth_, Jan 12 2019

Antti Karttunen, Table of n, a(n) for n = 1..20160
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A001222, A008578 (positions of 0's), A069264, A070288, A112967, A317938, A322375.

f:= proc(n) local F,G,t,x;
   F:= map(t -> t[2], ifactors(n)[2]);
   G:= unapply(normal(mul((1-x^(t+1))/(1-x), t = F)),x);
  (convert(F,`+`)-1)*D(G)(1) - (D@@2)(G)(1);
end proc:
map(f, [$1..100]); # Robert Israel, Jan 17 2019

Table[Sum[PrimeOmega[d] PrimeOmega[n/d], {d, Divisors[n]}], {n, 95}]

a(n) = sumdiv(n, d, bigomega(d)*bigomega(n/d)); \\ Michel Marcus, Aug 25 2018

a(n) = bn = bigomega(n); sumdiv(n, d, bd = bigomega(d); bd * (bn - bd)) \\ David A. Corneth, Jan 12 2019

a(A025487(n)) = A322375(n). - David A. Corneth, Jan 12 2019
From Robert Israel, Jan 17 2019: (Start)
If x and y are coprime, a(x*y) = a(x)*A000005(y) + A000005(x)*a(y) + A000005(x*y)*A001222(x)*A001222(y).
If p is prime, a(p^k) = (k^3-k)/6 = A000292(k-1). (End)

A283995 Least number with same prime signature as the n-th divisorial: a(n) = A046523(A007955(n)).

Original entry on oeis.org

1, 2, 2, 8, 2, 36, 2, 64, 8, 36, 2, 1728, 2, 36, 36, 1024, 2, 1728, 2, 1728, 36, 36, 2, 331776, 8, 36, 64, 1728, 2, 810000, 2, 32768, 36, 36, 36, 10077696, 2, 36, 36, 331776, 2, 810000, 2, 1728, 1728, 36, 2, 254803968, 8, 1728, 36, 1728, 2, 331776, 36, 331776, 36, 36, 2, 46656000000, 2, 36, 1728, 2097152, 36, 810000, 2, 1728, 36, 810000, 2, 139314069504, 2, 36

Offset: 1

Antti Karttunen, Mar 22 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1025

Cf. A007955, A046523, A069264, A283990.

Table[If[n == 1, 1, Times @@ MapIndexed[Prime[First@ #2]^#1 &, FactorInteger[Times @@ Divisors@ n][[All, -1]]]], {n, 74}] (* Michael De Vlieger, Mar 22 2017 *)

A007955(n) = if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2)); \\ From Charles R Greathouse IV, Feb 11 2011
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From Charles R Greathouse IV, Aug 17 2011
A283995(n) = A046523(A007955(n));

from math import prod, isqrt
from sympy import prime, factorint, divisor_count
def A283995(n): return (lambda n:prod(prime(i+1)**e for i, e in enumerate(sorted(factorint(n).values(), reverse=True))))((isqrt(n) if (c:=divisor_count(n)) & 1 else 1)*n**(c//2)) # Chai Wah Wu, Jun 25 2022

(define (A283995 n) (A046523 (A007955 n)))

a(n) = A046523(A007955(n)).

A330018 a(n) = Sum_{d|n} (bigomega(d) - omega(d)).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 3, 1, 0, 0, 2, 0, 0, 0, 6, 0, 2, 0, 2, 0, 0, 0, 6, 1, 0, 3, 2, 0, 0, 0, 10, 0, 0, 0, 6, 0, 0, 0, 6, 0, 0, 0, 2, 2, 0, 0, 12, 1, 2, 0, 2, 0, 6, 0, 6, 0, 0, 0, 4, 0, 0, 2, 15, 0, 0, 0, 2, 0, 0, 0, 13, 0, 0, 2, 2, 0, 0, 0, 12, 6, 0, 0, 4, 0, 0, 0, 6, 0, 4

Offset: 1

Ilya Gutkovskiy, Nov 27 2019

nonn

Inverse Moebius transform of A046660.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A005117 (positions of 0's), A046660, A062799, A069264, A268340.

N:= 100: # for a(1)..a(N)
V:= Vector(N):
for d from 1 to N do
  v:= add(t[2]-1, t=ifactors(d)[2]);
  L:= [seq(i,i=d..N,d)]:
  V[L]:= map(`+`,V[L],v);
od:
convert(V,list); # Robert Israel, Jun 12 2020

a[n_] := Sum[PrimeOmega[d] - PrimeNu[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 90}]

a(n) = sumdiv(n, d, bigomega(d) - omega(d)); \\ Michel Marcus, Jun 12 2020

G.f.: Sum_{k>=1} A046660(k) * x^k / (1 - x^k).
a(n) = A069264(n) - A062799(n).
If m and n are coprime, a(m*n) = tau(m)*a(n) + tau(n)*a(m), where tau = A000005. - Robert Israel, Jun 12 2020

cp's OEIS Frontend

A007955 Product of divisors of n.

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A318366 a(n) = Sum_{d|n} bigomega(d)*bigomega(n/d).

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A283995 Least number with same prime signature as the n-th divisorial: a(n) = A046523(A007955(n)).

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A330018 a(n) = Sum_{d|n} (bigomega(d) - omega(d)).

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