A007956 -id:A007956 - cp's OEIS frontend

A066729 a(n) = Product_{d|n, d

1, 2, 3, 2, 5, 6, 7, 8, 3, 10, 11, 144, 13, 14, 15, 64, 17, 324, 19, 400, 21, 22, 23, 13824, 5, 26, 27, 784, 29, 27000, 31, 1024, 33, 34, 35, 279936, 37, 38, 39, 64000, 41, 74088, 43, 1936, 2025, 46, 47, 5308416, 7, 2500, 51, 2704, 53, 157464, 55, 175616, 57

Offset: 1

Reinhard Zumkeller, Jan 15 2002

a(n) = n if n is prime, otherwise a(n) = A007956(n);

a(A084116(n)) = A084116(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A002033, A006881, A030514, A000040.

Cf. A027751, A007956, A084116, A010051.

a066729 n = if pds == [1] then n else product pds
            where pds = a027751_row n
-- Reinhard Zumkeller, Jul 31 2014

a[1] = 1; a[n_ /; PrimeQ[n]] := n; a[n_] := Times @@ Most[Divisors[n]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, May 28 2015 *)
Table[If[CompositeQ[n],Times@@Most[Divisors[n]],n],{n,60}] (* Harvey P. Dale, Jun 24 2016 *)

a(n) = my(pd = vecprod(divisors(n))); if (isprime(n), pd, pd/n); \\ Michel Marcus, Jan 09 2021

a(n) = n^c(n) * ( Product_{d|n, dA010051). - Wesley Ivan Hurt, Jan 10 2021

Revised and data corrected by Reinhard Zumkeller, Jul 31 2014

A048753 Composite numbers k whose product of aliquot divisors divided by number of aliquot divisors is an integer.

Original entry on oeis.org

4, 6, 15, 16, 20, 21, 27, 33, 36, 39, 42, 45, 48, 50, 51, 56, 57, 69, 70, 75, 87, 93, 100, 105, 111, 120, 123, 129, 132, 141, 154, 159, 162, 175, 177, 182, 183, 189, 196, 198, 201, 210, 213, 219, 220, 231, 237, 238, 245, 249, 256, 266, 267, 270, 273, 275, 291

Offset: 1

Enoch Haga

			For k=6, the product of aliquot divisors is 3*2*1=6; the number of aliquot divisors is 3; 6/3 = 2 (an integer), so 6 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007956, A032741, A048752, A048754.

padQ[n_]:=Module[{ad=Most[Divisors[n]]},!PrimeQ[n]&&Divisible[Times@@ad, Length[ad]]]; Select[Range[2,300],padQ] (* Harvey P. Dale, May 07 2012 *)

A048754 Mean integral quotients associated with A048753.

Original entry on oeis.org

1, 2, 5, 16, 80, 7, 9, 11, 34992, 13, 10584, 405, 589824, 500, 17, 25088, 19, 23, 49000, 1125, 29, 31, 1250000, 165375, 37, 23887872000000, 41, 43, 3643149312, 47, 521752, 53, 76527504, 6125, 59, 861224, 61, 964467, 13176688, 27665165088, 67

Offset: 1

Enoch Haga

			a(6)=21, n=21; quotient of product of aliquot divisors divided by number of aliquot divisors is 7 (aliquot divisors of 21 = 1*3*7 and 21/3 divisors = 7). Since 7 is integral, add to sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007956, A032741, A048753.

a(n) = A007956(A048753(n))/A032741(A048753(n)). - Amiram Eldar, Sep 06 2019

Offset corrected by Amiram Eldar, Sep 06 2019

A064499 Composite numbers n such that product of aliquot divisors of n is a perfect square.

Original entry on oeis.org

12, 16, 18, 20, 28, 32, 44, 45, 48, 50, 52, 63, 68, 75, 76, 80, 81, 92, 98, 99, 112, 116, 117, 124, 147, 148, 153, 162, 164, 171, 172, 175, 176, 180, 188, 192, 207, 208, 212, 236, 242, 243, 244, 245, 252, 256, 261, 268, 272, 275, 279, 284, 288, 292, 300, 304

Offset: 1

Robert G. Wilson v, Oct 05 2001

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A048699, A007956.

Do[ If[ !PrimeQ[n] && IntegerQ[ Sqrt[ Apply[ Times, Delete[ Divisors[n], -1]]]], Print[n]], {n, 2, 500} ]

pad(n)=my(d=divisors(n), p=1); for (i=1, #d-1, p*=d[i]); p
n=0; for (m=2, 10^9, if (!isprime(m) && issquare(pad(m)), write("b064499.txt", n++, " ", m); if (n==1000, break))) \\ Harry J. Smith, Sep 16 2009

is(n)=!isprime(n) && (ispower(n,4) || numdiv(n)%4==2) && n>1 \\ Charles R Greathouse IV, Oct 17 2015

A229970 Numbers n such that the product of their proper divisors is a palindrome > 1 and not equal to n.

Original entry on oeis.org

4, 9, 25, 49, 121, 212, 1001, 2636, 10201, 17161, 22801, 32761, 36481, 97969, 110011, 124609, 139129, 146689, 528529, 573049, 619369, 635209, 844561, 863041, 1100011, 10100101, 11000011, 101000101, 106110601, 110000011, 110271001, 112381201, 127938721, 130210921

Offset: 1

Derek Orr, Oct 04 2013

Since the product of proper divisors must be > 1, these terms are necessarily composite. - Derek Orr, Apr 05 2015

			The product of the proper divisors of 2636 is 6948496 (a palindrome). So, 2636 is a member of this sequence.
The product of the proper divisors of 8 is 8 (a palindrome) but equal to 8. So 8 is not in this sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..62

Cf. A007956.

isA002113 := proc(n)
    dgs := convert(n,base,10) ;
    for i from 1 to nops(dgs)/2 do
        if op(i,dgs) <> op(-i,dgs) then
            return false;
        end if;
    end do:
    true ;
end proc:
for n from 4 do
    if not isprime(n) then
        ppd := A007956(n) ;
        if n <> ppd and isA002113(ppd) then
            printf("%d,",n);
        end if;
    end if;
end do: # R. J. Mathar, Oct 09 2013

palQ[n_] := Block[{d = IntegerDigits@ n}, d == Reverse@ d]; fQ[n_] := Block[{s = Times @@ Most@ Divisors@ n}, And[palQ@s, s > 1, s != n]]; Select[Range@ 1000000, CompositeQ@ # && fQ@ # &] (* Michael De Vlieger, Apr 06 2015 *)

ispal(n)=Vecrev(n=digits(n))==n
is(n)=my(k=if(issquare(n,&k),k^numdiv(n)/n,n^(numdiv(n)/2-1))); k!=n && k>1 && ispal(k) \\ Charles R Greathouse IV, Oct 09 2013

pal(n)=d=digits(n);Vecrev(d)==d
for(n=1,10^6,D=divisors(n);p=prod(i=1,#D-1,D[i]);if(pal(p)&&p-1&&p-n,print1(n,", "))) \\ Derek Orr, Apr 05 2015

from sympy import divisors
def PD(n):
  p = 1
  for i in divisors(n):
    if i != n:
      p *= i
  return p
def pal(n):
  r = ''
  for i in str(n):
    r = i + r
  return r == str(n)
{print(n, end=', ') for n in range(1, 10**4) if pal(PD(n)) and (PD(n)-1) and PD(n)-n}
## Simplified by Derek Orr, Apr 05 2015

a(14)-a(18) from R. J. Mathar, Oct 09 2013
a(19)-a(34) from Charles R Greathouse IV, Oct 09 2013
Definition edited by Derek Orr, Apr 05 2015

A229972 Nonprime numbers whose product of proper divisors is a perfect cube.

Original entry on oeis.org

1, 8, 16, 24, 27, 30, 40, 42, 54, 56, 64, 66, 70, 78, 81, 88, 102, 104, 105, 110, 114, 125, 128, 130, 135, 136, 138, 152, 154, 165, 170, 174, 182, 184, 186, 189, 190, 192, 195, 216, 222, 230, 231, 232, 238, 240, 246, 248, 250, 255, 258, 266, 273, 282, 285

Offset: 1

Derek Orr, Oct 04 2013

A nonprime number m is a term if and only if m is a cube or the number of divisors of m is of the form 3k+2. - Chai Wah Wu, Mar 09 2016

			The set of proper divisors of 8 is {1,2,4} and 1*2*4 = 2^3 so 8 is in the sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A007956.

Union of A000578 and (intersection of A002808 and A211338).

Select[Range[343],!PrimeQ[#]&&IntegerQ[(Apply[Times,Divisors[#]]/#)^(1/3)]&] (* Farideh Firoozbakht Oct 10 2013 *)
Select[Range[300],!PrimeQ[#]&&IntegerQ[Surd[Times@@Most[Divisors[ #]],3]]&] (* Harvey P. Dale, Oct 24 2017 *)
m = 7; Union[Range[m]^3, Select[Range[m^3], !PrimeQ[#] && Mod[DivisorSigma[0, #], 3] == 2 &]] (* Amiram Eldar, Jul 07 2022 *)

for(n=1,10^3,d=divisors(n);p=prod(i=1,#d-1,d[i]);if(p!=1&&ispower(p,3),print1(n,", ")))

from gmpy2 import iroot
from sympy import divisor_count, isprime
A229972_list = [i for i in range(1,10**3) if not isprime(i) and (iroot(i,3)[1] or divisor_count(i) % 3 == 2)] # Chai Wah Wu, Mar 10 2016

Corrected and edited by Farideh Firoozbakht Oct 10 2013

A290480 Product of proper unitary divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 10, 1, 12, 1, 14, 15, 1, 1, 18, 1, 20, 21, 22, 1, 24, 1, 26, 1, 28, 1, 27000, 1, 1, 33, 34, 35, 36, 1, 38, 39, 40, 1, 74088, 1, 44, 45, 46, 1, 48, 1, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 216000, 1, 62, 63, 1, 65, 287496, 1, 68, 69, 343000, 1, 72, 1, 74, 75, 76, 77, 474552, 1, 80

Offset: 1

Ilya Gutkovskiy, Aug 03 2017

nonn

			a(12) = 12 because 12 has 6 divisors {1, 2, 3, 4, 6, 12} among which 3 are proper unitary {1, 3, 4} and 1*3*4 = 12.

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

Cf. A001221, A007774 (fixed points), A007955, A007956, A034460, A061537, A062758, A063919, A078599, A087652, A126192, A136655, A183091.

with(numtheory):
a:= n-> mul(d, d=select(x-> igcd(x, n/x)=1, divisors(n) minus {n})):
seq(a(n), n=1..80);  # Alois P. Heinz, Aug 03 2017

Table[Product[d, {d, Select[Divisors[n], GCD[#, n/#] == 1 &]}]/n, {n, 80}]
Table[n^(2^(PrimeNu[n] - 1) - 1), {n, 80}]

A290480(n) = if(1==n,n,n^(2^(omega(n)-1)-1)); \\ Antti Karttunen, Aug 06 2018

from sympy import divisors, gcd, prod
def a(n): return prod(d for d in divisors(n) if gcd(d, n//d) == 1)//n
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Aug 04 2017

a(n) = A061537(n)/n.
a(n) = n^(2^(omega(n)-1)-1), where omega() is the number of distinct primes dividing n (A001221).
a(n) = 1 if n is a prime power.

A339793 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier that is a multiple of s(a(n-1)), the sum of the proper divisors of a(n-1).

Original entry on oeis.org

1, 2, 3, 4, 6, 12, 16, 15, 9, 8, 7, 5, 10, 24, 36, 55, 17, 11, 13, 14, 20, 22, 28, 56, 64, 63, 41, 18, 21, 33, 30, 42, 54, 66, 78, 90, 144, 259, 45, 99, 57, 23, 19, 25, 48, 76, 128, 127, 26, 32, 31, 27, 39, 34, 40, 50, 43, 29, 35, 52, 46, 104, 106, 112, 136, 134, 70, 74, 80, 212, 166, 86, 92, 152

Offset: 1

Scott R. Shannon, Dec 17 2020

nonn

The sequence is possibly a permutation of the positive integers as when a(n-1) is prime a(n) will be the next smallest number that has not previously occurred. However this will depend on the likelihood of a(n) being a prime as n goes to infinity. For the first 478 terms the last prime is a(144) = 59, while a(478) = 19140499834691254267668, indicating prime values become increasingly rare, and could potentially have a finite number as n->infinity.

The sum of the proper divisors of n is given by A001065(n).

			a(3) = 3 as s(a(2)) = s(2) = 1, and 3 is the smallest multiple of 1 that has not previously occurred.
a(5) = 6 as s(a(4)) = s(4) = 3, and as 3 has already occurred the next lowest multiple is used, being 6.
a(12) = 5 as s(a(11)) = s(7) = 1, and 5 is the smallest multiple of 1 that has not previously occurred.

Scott R. Shannon, Table of n, a(n) for n = 1..478
Wikipedia, Aliquot sum.

Cf. A001065, A027751, A000203, A032741, A032742, A007956.

from sympy import divisors
def s(k): return sum(d for d in divisors(k)[:-1])
def aupto(n):
  alst, aset = [1, 2], {1, 2}
  for k in range(2, n):
    ak = sanm1 = s(alst[-1])
    while ak in aset: ak += sanm1
    alst.append(ak); aset.add(ak)
  return alst     # use alst[n-1] for a(n)
print(aupto(478)) # Michael S. Branicky, Dec 29 2020

A375960 Numbers whose product of proper divisors is a cube.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 13, 16, 17, 19, 23, 24, 27, 29, 30, 31, 37, 40, 41, 42, 43, 47, 53, 54, 56, 59, 61, 64, 66, 67, 70, 71, 73, 78, 79, 81, 83, 88, 89, 97, 101, 102, 103, 104, 105, 107, 109, 110, 113, 114, 125, 127, 128, 130, 131, 135, 136, 137, 138, 139, 149

Offset: 1

Stefano Spezia, Sep 04 2024

Wells erroneously writes that the smallest number on this list should be 24.

All the primes are in this list since they have the only proper divisor 1 which is trivially a cube.

			16 is a term since 1*2*4*8 = 64 = 4^3.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 101.

Stefano Spezia, Table of n, a(n) for n = 1..10000

Cf. A000040 (subsequence), A000578, A007956, A027751, A032741.

Complement of A375962.

Select[Range[150],IntegerQ[Product[Part[Divisors[#],i],{i,DivisorSigma[0,#]-1}]^(1/3)] &]

isok(k) = my(d=divisors(k)); ispower(vecprod(Vec(d, #d-1)), 3); \\ Michel Marcus, Sep 04 2024

A375962 Numbers whose product of proper divisors is not a cube.

Original entry on oeis.org

4, 6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 25, 26, 28, 32, 33, 34, 35, 36, 38, 39, 44, 45, 46, 48, 49, 50, 51, 52, 55, 57, 58, 60, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 106, 108, 111, 112, 115, 116, 117, 118, 119, 120

Offset: 1

Stefano Spezia, Sep 04 2024

All the terms are composites.

			28 is a term since 1*2*4*7*14 = 784 is not a cube.

Stefano Spezia, Table of n, a(n) for n = 1..10000

Cf. A000578, A002808 (supersequence), A007956, A027751, A032741.

Complement of A375960.

Select[Range[120], !IntegerQ[Product[Part[Divisors[#], i], {i, DivisorSigma[0, #]-1}]^(1/3)] &]

cp's OEIS Frontend

A066729 a(n) = Product_{d|n, d

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A048753 Composite numbers k whose product of aliquot divisors divided by number of aliquot divisors is an integer.

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A048754 Mean integral quotients associated with A048753.

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A064499 Composite numbers n such that product of aliquot divisors of n is a perfect square.

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A229970 Numbers n such that the product of their proper divisors is a palindrome > 1 and not equal to n.

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A229972 Nonprime numbers whose product of proper divisors is a perfect cube.

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A290480 Product of proper unitary divisors of n.

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