A111398 -id:A111398 - 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

A030626 Numbers with exactly 8 divisors.

Original entry on oeis.org

24, 30, 40, 42, 54, 56, 66, 70, 78, 88, 102, 104, 105, 110, 114, 128, 130, 135, 136, 138, 152, 154, 165, 170, 174, 182, 184, 186, 189, 190, 195, 222, 230, 231, 232, 238, 246, 248, 250, 255, 258, 266, 273, 282, 285, 286, 290, 296, 297, 310, 318, 322, 328, 344, 345, 351, 354, 357, 366, 370, 374, 375, 376, 385, 399, 402

Offset: 1

Jeff Burch

nonn

Since A119479(8)=7, there are never more than 7 consecutive terms. Runs of 7 consecutive terms start at 171897, 180969, 647385, ... (subsequence of A049053). - Ivan Neretin, Feb 08 2016

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from R. J. Mathar)
Jérôme Germoni, Nombres à huit diviseurs, Images des Mathématiques, CNRS, 2017 (in French).
Eric Weisstein's World of Mathematics, Divisor Product.

Essentially the same as A111398.

Cf. A000005, A007304, A049053, A065036, A092759, A119479.

[n: n in [1..400] | DivisorSigma(0, n) eq 8]; // Vincenzo Librandi, Oct 05 2017

select(numtheory:-tau=8, [$1..1000]); # Robert Israel, Dec 17 2014

Select[Range[400], DivisorSigma[0, #]== 8 &] (* Vincenzo Librandi, Oct 05 2017 *)

Vec(select(x->x==8,vector(500, i, numdiv(i)),1)) \\ Michel Marcus, Dec 17 2014

from sympy import divisor_count
isok = lambda n: divisor_count(n) == 8
print([n for n in range(1, 400) if isok(n)]) # Darío Clavijo, Oct 17 2023

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A030626(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1),1) for b,m in enumerate(primerange(k+1,isqrt(x//k)+1),a+1))-sum(primepi(x//p**3) for p in primerange(integer_nthroot(x,3)[0]+1))+primepi(integer_nthroot(x,4)[0])-primepi(integer_nthroot(x,7)[0]))
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

A000005(a(n))=8. - Juri-Stepan Gerasimov, Oct 10 2009

Equals A065036 (p*q^3) U A007304 (p*q*r) U A092759 (p^7). - Amarnath Murthy, Apr 21 2001

A048945 Numbers whose product of divisors is a fourth power.

Original entry on oeis.org

1, 24, 30, 40, 42, 54, 56, 66, 70, 78, 88, 102, 104, 105, 110, 114, 120, 128, 130, 135, 136, 138, 152, 154, 165, 168, 170, 174, 182, 184, 186, 189, 190, 195, 210, 216, 222, 230, 231, 232, 238, 246, 248, 250, 255, 256, 258, 264, 266, 270, 273, 280, 282, 285

Offset: 1

Eric W. Weisstein

nonn

Different from sequence of numbers which are the cube root of the product of their proper divisors. Compare A111398.

Amarnath Murthy, Generalization of Partition function, Introducing Smarandache Factor partitions, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.
Amarnath Murthy, A note on Smarandache Divisor Sequence, Introducing Smarandache Factor partitions, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.
Amarnath Murthy, Some more ideas on Smarandache Factor Partitions, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Product

Cf. A111398, A111399.

for n from 2 to 1000 do it1 := sort(convert(divisors(n), list)): it2 := product(it1[j], j=1..nops(it1)-1): if it2 = n^3 then printf(`%d,`,n) fi: od:

Select[Range[300], IntegerQ[(Times @@ Divisors[#])^(1/4)] &] (* Jean-François Alcover, Nov 05 2012 *)

is(n)=my(f=factor(n)[,2]); gcd(f)*prod(i=1,#f,f[i]+1)%8==0 \\ Charles R Greathouse IV, Sep 18 2015

A162947 Numbers k such that the product of all divisors of k equals k^3.

Original entry on oeis.org

1, 12, 18, 20, 28, 32, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99, 116, 117, 124, 147, 148, 153, 164, 171, 172, 175, 188, 207, 212, 236, 242, 243, 244, 245, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 338, 356, 363, 369, 387, 388, 404, 412, 423, 425, 428

Offset: 1

Claudio Meller, Jul 18 2009

Contains the terms of A054753 (products p*q^2 of a prime p and the square of a different prime q), 1, and p^5, where p is prime.

Numbers k such that k^2 is equal to the product of proper divisors of k. - Juri-Stepan Gerasimov, May 03 2011

			18 is in the sequence because the product of its divisors is 1 * 2 * 3 * 6 * 9 * 18 = 18^3.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
David A. Corneth, PARI program

Cf. A111398, A030628. - R. J. Mathar, Jul 19 2009

Cf. A008578 (product of divisors equals n), A007422 (product of divisors equals n^2).

Select[Range[500], Surd[Times@@Divisors[#], 3] == # &] (* Harvey P. Dale, Mar 15 2017 *)
Join[{1},Select[Range[430],DivisorSigma[0,#]==6 &]] (* Stefano Spezia, Jul 14 2025 *)

isok(n) = my(d = divisors(n)); prod(i=1, #d, d[i]) == n^3; \\ Michel Marcus, Feb 04 2014

PARI
```
\\ See Corneth link
```

from itertools import chain, count, islice
from sympy import divisor_count
def A162947_gen(): # generator of terms
    return chain((1,),filter(lambda n:divisor_count(n)==6,count(2)))
A162947_list = list(islice(A162947_gen(),20)) # Chai Wah Wu, Jun 25 2022

{n: A007955(n) = A000578(n)}. - R. J. Mathar, Jul 19 2009

{1} UNION A030515. - R. J. Mathar, Jul 19 2009

Edited by R. J. Mathar, Jul 19 2009

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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A111399 Numbers in A048945 but not in A111398.

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A007955 Product of divisors of n.

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A030626 Numbers with exactly 8 divisors.

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A048945 Numbers whose product of divisors is a fourth power.

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A162947 Numbers k such that the product of all divisors of k equals k^3.

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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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