A007956 -id:A007956 - cp's OEIS frontend

A385349 Product of odd proper divisors of n.

1, 1, 1, 1, 1, 3, 1, 1, 3, 5, 1, 3, 1, 7, 15, 1, 1, 27, 1, 5, 21, 11, 1, 3, 5, 13, 27, 7, 1, 225, 1, 1, 33, 17, 35, 27, 1, 19, 39, 5, 1, 441, 1, 11, 2025, 23, 1, 3, 7, 125, 51, 13, 1, 729, 55, 7, 57, 29, 1, 225, 1, 31, 3969, 1, 65, 1089, 1, 17, 69, 1225, 1, 27, 1, 37, 5625

Offset: 1

Ilya Gutkovskiy, Jun 26 2025

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Product.

Cf. A007955, A007956, A091570 (similar for sum), A136655, A385350 (fixed points).

a:= n-> mul(`if`(d::odd, d, 1), d=numtheory[divisors](n) minus {n}):
seq(a(n), n=1..75);  # Alois P. Heinz, Jun 27 2025

a[n_] := Times @@ Select[Divisors[n], # < n && OddQ[#] &]; Table[a[n], {n, 75}]

a(n) = my(m = n >> valuation(n,2), d = numdiv(m)); if(d % 2, sqrtint(m)^d, m^(d/2)) / if(m < n, 1, n); \\ Amiram Eldar, Jun 27 2025

from math import isqrt
from sympy import divisor_count
def A385349(n):
    d = divisor_count(m:=n>>(~n&n-1).bit_length())
    k = isqrt(m)**d if d&1 else m**(d>>1)
    return k//n if n&1 else k # Chai Wah Wu, Jun 27 2025

a(n) = Product_{d|n, d < n, d odd} d.

A089748 Numbers k that divide (sum of proper divisors of k + product of proper divisors of k).

Original entry on oeis.org

2, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240

Offset: 1

Joseph L. Pe, Jan 08 2004

All perfect numbers belong to this sequence.
Every term of A007691 is in this sequence. - T. D. Noe, Sep 29 2005
There are two sets of candidates of k: (i) k|A001065(k) and k|A007956(k) individually, or (ii) neither k|A001065(k) nor k|A007956(k) but the remainders of A001065(k)/k and A007956(k)/k sum up to k. If k has at least 4 divisors, the product of the second and penultimate divisor (in the sorted divisors list) is k, so k|A007956(k). This means for all k in A080257 we have k|A007956(k), and the k that do not divide A007956(k) are in A000430, which means k=p or k=p^2 for some prime p. If k=p, A001065(k)+A007956(k) = 1+1 =2, and the requirement here reduces to k|2 and only k=2 is left. If k=p^2, A001065(k) +A007956(k) = 1+p+p = 1+2*p, and the requirement here reduces to p^2 | (1+2*p), which has no solutions. This means case (ii) does not generate any solutions besides k=2. And this means all other solutions are from case (i), and therefore elements A007691 > 1 are the only remaining candidates. - R. J. Mathar, Oct 15 2021

Cf. A001065, A007956, A007691, A080257 (k which divide A007691(k)).

Cf. A219544.

isA087948 := proc(n)
    if modp( A001065(n)+A007956(n),n) = 0 then
        true;
    else
        false;
    end if;
end proc:
for n from 2 do
    if isA087948(n) then
        printf("%d\n",n) ;
    end if;
end do: # R. J. Mathar, Oct 15 2021

l = {}; Do[d = Drop[Divisors[n], -1]; p = Apply[Plus, d]; t = Apply[Times, d]; m = Mod[p + t, n]; If[m == 0, l = Append[l, n]], {n, 2, 10^6}]; l
Select[Range[2,22*10^5],Mod[Total[Most[Divisors[#]]]+Times@@Most[Divisors[#]],#]==0&] (* The program generates the first 11 terms of the sequence. *) (* Harvey P. Dale, Jun 05 2024 *)

from math import prod
from sympy import divisors
def ok(n): d = divisors(n)[:-1]; return n > 1 and (sum(d) + prod(d))%n == 0
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Oct 15 2021

a(11)-a(16) from Michael S. Branicky, Oct 16 2021

A188902 Numerator of the base n logarithm of the product of the divisors of n.

Original entry on oeis.org

1, 1, 3, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 2, 5, 1, 3, 1, 3, 2, 2, 1, 4, 3, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, 9, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 5, 3, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 6, 1, 2, 3, 7, 2, 4, 1, 3, 2, 4, 1, 6, 1, 2, 3, 3, 2

Offset: 2

Alonso del Arte, Apr 19 2011

Obviously the product of divisors of n (see A007955) is a multiple of n. But often it is also a perfect power of n, a number of the form n^m with m an integer. But if n is a perfect square (A000290), then the logarithm is a rational number but not an integer.
a(1) is of course indeterminate since it can be any value desired, whether real, imaginary or complex.
The denominator is A010052(n) + 1.

Antti Karttunen, Table of n, a(n) for n = 2..10000

Cf. A000005, A007955, A007956, A038548, A056924, A010052.

Numerator[Table[FullSimplify[Log[n, Times@@Divisors[n]]], {n, 2, 75}]]

A188902(n) = numerator(numdiv(n)/2); \\ Antti Karttunen, May 27 2017

from sympy import divisor_count, Integer
def a(n): return (divisor_count(n) / 2).numerator
print([a(n) for n in range(2, 51)])  # Indranil Ghosh, May 27 2017

a(n) = numerator(A000005(n)/2).

a(n) = (A038548(n) + A056924(n)) / 2 for n > 1.

A191905 Composite deficient numbers k such that (product of proper divisors of k) mod (sum of proper divisors of k) is a prime number.

Original entry on oeis.org

4, 9, 10, 25, 33, 39, 49, 57, 91, 93, 98, 105, 111, 119, 121, 145, 155, 169, 183, 185, 187, 189, 201, 205, 209, 215, 225, 235, 237, 242, 245, 265, 289, 291, 299, 305, 327, 335, 351, 355, 361, 371, 403, 413, 415, 417, 425, 427, 437, 469, 471, 475, 485, 493, 497, 515, 527, 529, 535, 543, 549, 553

Offset: 1

Juri-Stepan Gerasimov, Jun 19 2011

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001065, A005100, A007956, A125493, A191906.

isA191905 := proc(n) if not isA125493(n) then false; else isprime( A191906(n)) ; end if; end proc:
for n from 3 to 710 do if isA191905(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Jun 27 2011

fQ[n_]:=Module[{pd=Most[Divisors[n]]},!PerfectNumberQ[n]&&CompositeQ[n] && DivisorSigma[ 1,n]<2n&& PrimeQ[Mod[Times@@pd,Total[pd]]]] Select[Range[2,600],fQ] (* Harvey P. Dale, Jul 14 2024 *)

Corrected by R. J. Mathar, Jun 27 2011

A192034 Least k such that (product of proper divisors of k) mod (sum of proper divisors of k) equals n.

Original entry on oeis.org

2, 8, 4, 9, 14, 25, 15, 49, 22, 18, 21, 57, 45, 169, 34, 69, 38, 205, 143, 119, 46, 87, 217, 93, 130, 133, 58, 323, 62, 111, 160, 553, 319, 63, 74, 129, 30, 305, 82, 75, 86, 36, 68, 335, 48, 159, 301, 355, 369, 171, 106, 177

Offset: 0

Juri-Stepan Gerasimov, Jun 21 2011

nonn

Greedy inverse of A191906.

			a(0)=2 because A007956(2) mod A001065(2) = 1 mod 1 = 0, and 2 is the smallest number for which this is the case;
a(1)=8 because A007956(8) mod A001065(8) = 8 mod 7 = 1, and 8 is the smallest number for which this is the case;
a(2)=4 because A007956(4) mod A001065(4) = 2 mod 3 = 2, and 4 is the smallest number for which this is the case.

Cf. A001065, A007956, A191906.

A192034 := proc(n) local k ; for k from 2 do if A191906(k) = n then return k ; end if; end do: end proc: # R. J. Mathar, Jul 01 2011

ds[n_]:=Module[{divs=Most[Divisors[n]]},Mod[Times@@divs,Total[divs]]]; Join[ {2},Transpose[Table[SelectFirst[Table[{n,ds[n]},{n,2,2000}],#[[2]] == i&],{i,60}]][[1]]] (* Harvey P. Dale, Apr 11 2015 *)

Corrected by R. J. Mathar, Jul 01 2011

Example section corrected by Jon E. Schoenfield, Feb 24 2019

A192035 Numbers k with equal remainders of (product of divisors of k) mod (sum of divisors of k) and (product of proper divisors of k) mod (sum of proper divisors of k).

Original entry on oeis.org

6, 14, 28, 51, 120, 260, 270, 496, 672, 679, 752, 924, 1260, 1320, 1540, 1960, 2055, 2262, 2651, 3808, 3948, 4381, 6413, 6435, 6944, 7900, 7980, 8010, 8128, 9809, 9945, 10242, 10920, 12690, 15456, 16830, 18018, 21728, 21970, 22320, 25296, 27930, 29190, 29792

Offset: 1

Juri-Stepan Gerasimov, Jun 21 2011

nonn

The even perfect numbers (A000396) are a subsequence.

The deficient numbers (A005100) in the sequence are 14, 51, 679, 752, 2055, 2651, 4381, 6413, 9809, 9945, 21970, ... - Juri-Stepan Gerasimov, Jul 07 2011

			14 is in this sequence because (1*2*7*14) mod (1+2+7+14) = 196 mod 24 = 4 and (1*2*7) mod (1+2+7) = 14 mod 10 = 4.

Amiram Eldar, Table of n, a(n) for n = 1..600

Cf. A001065, A007956, A187680, A191906.

erQ[n_]:=Module[{divs=Divisors[n],ds=DivisorSigma[1,n]},Mod[ Times@@ divs,ds] == Mod[ Times@@Most[divs],ds-n]]; Select[Range[2,30000],erQ] (* Harvey P. Dale, Jun 13 2015 *)
Select[Range[2, 30000], Mod[(p = #^(DivisorSigma[0, #]/2)), (s = DivisorSigma[1, #])] == Mod[p/#, s - #] &] (* Amiram Eldar, Jul 21 2019 *)

{ k : A187680(k) = A191906(k) }.

Values from a(4) onwards from R. J. Mathar, Jul 05 2011

A277169 Product of squares of proper divisors of n.

Original entry on oeis.org

1, 1, 1, 4, 1, 36, 1, 64, 9, 100, 1, 20736, 1, 196, 225, 4096, 1, 104976, 1, 160000, 441, 484, 1, 191102976, 25, 676, 729, 614656, 1, 729000000, 1, 1048576, 1089, 1156, 1225, 78364164096, 1, 1444, 1521, 4096000000, 1, 5489031744, 1, 3748096, 4100625, 2116, 1, 28179280429056, 49, 6250000

Offset: 1

Ilya Gutkovskiy, Oct 19 2016

			a(6) = 36 because 6 has 3 proper divisors {1,2,3} and 1^2*2^2*3^2 = 36.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Product
Eric Weisstein's World of Mathematics, Proper divisors

Cf. A000290, A001248, A007422, A007955, A007956, A008578, A062758.

seq(n^(numtheory:-tau(n)-2), n=1..50); # Robert Israel, Nov 13 2016

Table[n^(DivisorSigma[0, n] - 2), {n, 1, 50}]

a(n) = n^(sigma_0(n)-2).
a(n) = n^A000005(n)/A000290(n).
a(n) = A000290(A007956(n))/A000290(n).
a(n) = A000290(A007955(n)/n)/A000290(n).
a(n) = A062758(n)/A000290(n).
a(n) = 1 if n is prime or n = 1 (A008578).
a(n) = n if n is square of prime (A001248).
a(n) = n^2 if n is multiplicatively perfect number (A007422).

A290479 Product of nonprime squarefree divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 10, 1, 6, 1, 14, 15, 1, 1, 6, 1, 10, 21, 22, 1, 6, 1, 26, 1, 14, 1, 27000, 1, 1, 33, 34, 35, 6, 1, 38, 39, 10, 1, 74088, 1, 22, 15, 46, 1, 6, 1, 10, 51, 26, 1, 6, 55, 14, 57, 58, 1, 27000, 1, 62, 21, 1, 65, 287496, 1, 34, 69, 343000, 1, 6, 1, 74, 15, 38, 77, 474552, 1, 10

Offset: 1

Ilya Gutkovskiy, Aug 03 2017

nonn

			a(30) = 27000 because 30 has 8 divisors {1, 2, 3, 5, 6, 10, 15, 30} among which 5 are nonprime squarefree {1, 6, 10, 15, 30} and 1*6*10*15*30 = 27000.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000005, A000469, A006881 (fixed points), A007947, A007955, A007956, A061537, A062758, A078599, A087652, A126192, A136655, A183091, A284118.

Table[Product[d, {d, Select[Divisors[n], !PrimeQ[#] && SquareFreeQ[#] &]}], {n, 80}]
Table[Last[Select[Divisors[n], SquareFreeQ]]^(DivisorSigma[0, Last[Select[Divisors[n], SquareFreeQ]]]/2 - 1), {n, 80}]

A290479(n) = if(1==n, n, my(r=factorback(factorint(n)[, 1])); (r^((numdiv(r)/2)-1))); \\ Antti Karttunen, Aug 06 2018

a(n) = A078599(n)/A007947(n).
a(n) = rad(n)^(d(rad(n))/2-1), where d() is the number of divisors of n (A000005) and rad() is the squarefree kernel of n (A007947).
a(n) = 1 if n is a prime power.

A324505 a(n) = numerator of Sum_{d|n} (d/pod(d)) where pod(k) = the product of the divisors of k (A007955).

Original entry on oeis.org

1, 2, 2, 5, 2, 19, 2, 21, 7, 31, 2, 529, 2, 43, 46, 169, 2, 1135, 2, 1441, 64, 67, 2, 52513, 11, 79, 64, 2801, 2, 117001, 2, 2705, 100, 103, 106, 1122553, 2, 115, 118, 238561, 2, 317521, 2, 6865, 6886, 139, 2, 20247937, 15, 8251, 154, 9569, 2, 557443, 166

Offset: 1

Jaroslav Krizek, Mar 03 2019

Sum_{d|n} (d/pod(d)) >= 1 for all n >= 1.

Sum_{d|n} (d/pod(d)) = 2 iff n = primes (A000040).

			Sum_{d|n} (d/pod(d)) for n >= 1: 1, 2, 2, 5/2, 2, 19/6, 2, 21/8, 7/3, 31/10, 2, 529/144, 2, 43/14, 46/15, 169/64, ...
For n=4; Sum_{d|4} (d/pod(d)) = 1/pod(1) + 2/pod(2) + 4/pod(4) = 1/1 + 2/2 + 4/8 = 5/2; a(4) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000040, A007955, A007956 (denominators).

[Numerator(&+[d / &*[c: c in Divisors(d)]: d in Divisors(n)]): n in [1..100]];

Table[Numerator[Sum[k/Product[j, {j, Divisors[k]}], {k, Divisors[n]}]], {n, 1, 60}] (* G. C. Greubel, Mar 04 2019 *)

A007955(n) = if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2));
A324505(n) = numerator(sumdiv(n,d,(d/A007955(d)))); \\ Antti Karttunen, Jan 27 2025

[sum(k/product(j for j in k.divisors()) for k in n.divisors()).numerator() for n in (1..60)] # G. C. Greubel, Mar 04 2019

a(p) = 2 for p = primes.

A338576 a(n) = n * pod(n) where pod(n) = the product of divisors of n (A007955).

Original entry on oeis.org

1, 4, 9, 32, 25, 216, 49, 512, 243, 1000, 121, 20736, 169, 2744, 3375, 16384, 289, 104976, 361, 160000, 9261, 10648, 529, 7962624, 3125, 17576, 19683, 614656, 841, 24300000, 961, 1048576, 35937, 39304, 42875, 362797056, 1369, 54872, 59319, 102400000, 1681

Offset: 1

Jaroslav Krizek, Nov 03 2020

nonn

			a(6) = 6 * pod(6) = 6 * 36 = 216.

Cf. A007955 (pod(n)), A007956 (pod(n) / n).
Similar sequences: A038040 (n * tau(n)), A064987 (n * sigma(n)).
Cf. A174935 (partial sums of a(n)).

Magma
```
[n * &*Divisors(n): n in [1..100]]
```

a[n_] := n^(1 + DivisorSigma[0, n]/2); Array[a, 50] (* Amiram Eldar, Nov 03 2020 *)

a(n) = n*vecprod(divisors(n)); \\ Michel Marcus, Nov 03 2020

from math import isqrt
from sympy import divisor_count
def A338576(n): return (isqrt(n) if (c:=divisor_count(n)) & 1 else 1)*n**(c//2+1) # Chai Wah Wu, Jun 25 2022

a(n) = n * A007955(n) = n^2 * A007956(n).
a(n) = lcm(n, pod(n)) * gcd(n, pod(n)).
a(p) = p^2 for p = primes (A000040).

cp's OEIS Frontend

A385349 Product of odd proper divisors of n.

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A089748 Numbers k that divide (sum of proper divisors of k + product of proper divisors of k).

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A188902 Numerator of the base n logarithm of the product of the divisors of n.

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A191905 Composite deficient numbers k such that (product of proper divisors of k) mod (sum of proper divisors of k) is a prime number.

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A192034 Least k such that (product of proper divisors of k) mod (sum of proper divisors of k) equals n.

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A192035 Numbers k with equal remainders of (product of divisors of k) mod (sum of divisors of k) and (product of proper divisors of k) mod (sum of proper divisors of k).

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A277169 Product of squares of proper divisors of n.

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