cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A136655 Product of odd divisors of n.

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 7, 1, 27, 5, 11, 3, 13, 7, 225, 1, 17, 27, 19, 5, 441, 11, 23, 3, 125, 13, 729, 7, 29, 225, 31, 1, 1089, 17, 1225, 27, 37, 19, 1521, 5, 41, 441, 43, 11, 91125, 23, 47, 3, 343, 125, 2601, 13, 53, 729, 3025, 7, 3249, 29, 59, 225, 61, 31, 250047, 1, 4225, 1089
Offset: 1

Views

Author

Jonathan Vos Post, Jun 25 2008

Keywords

Comments

Product of rows of triangle A182469. - Reinhard Zumkeller, May 01 2012

Links

Crossrefs

Cf. A000265, A000593, A007955, A007956, A078701.
Cf. A140210, A140211, A140213, A140214, A140215.
Cf. A125911, A126192.
Cf. A001227, A183063.

Programs

  • Haskell
    a136655 = product . a182469_row  -- Reinhard Zumkeller, May 01 2012
  • Maple
    with(numtheory); f:=proc(n) local t1,i,k; t1:=divisors(n); k:=1; for i in t1 do if i mod 2 = 1 then k:=k*i; fi; od; k; end; # N. J. A. Sloane, Jul 14 2008
  • Mathematica
    Array[Times @@ Select[Divisors@ #, OddQ] &, 66] (* Michael De Vlieger, Aug 03 2017 *)
a[n_] := (oddpart = n/2^IntegerExponent[n, 2])^(DivisorSigma[0, oddpart]/2); Array[a, 100] (* Amiram Eldar, Jun 26 2022 *)
  • PARI
    a(n) = my(d=divisors(n)); prod(k=1, #d, if (d[k]%2, d[k], 1)); \\ Michel Marcus, Aug 04 2017
  • Python
    from math import isqrt
from sympy import divisor_count
def A136655(n):
    d = divisor_count(m:=n>>(~n&n-1).bit_length())
    return isqrt(m)**d if d&1 else m**(d>>1) # Chai Wah Wu, Jun 27 2025

Formula

a(p) = p if p noncomposite; a(2^n) = 1; a(pq) = p^2 * q^2 when p, q are odd primes.
a(n) = sqrt(n^od(n)/2^ed(n)), where od(n) = number of odd divisors of n = tau(2*n)-tau(n) and ed(n) = number of even divisors of n = 2*tau(n)-tau(2*n). - Vladeta Jovovic, Jun 25 2008
Also a(n) = A007955(A000265(n)). - David Wilson, Jun 26 2008
a(n) = Product_{h == 1 mod 4 and h | n}*Product_{i == 3 mod 4 and i | n}.
a(n) = Product_{j == 1 mod 6 and j | n}*Product_{k == 5 mod 6 and k | n}.
a(n) = A140210(n)*A140211(n). - R. J. Mathar, Jun 27 2008
a(n) = A007955(n) / A125911(n).

Extensions

More terms from N. J. A. Sloane, Jul 14 2008
Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

A125911 Product of the even divisors of n.

Original entry on oeis.org

1, 2, 1, 8, 1, 12, 1, 64, 1, 20, 1, 576, 1, 28, 1, 1024, 1, 216, 1, 1600, 1, 44, 1, 110592, 1, 52, 1, 3136, 1, 3600, 1, 32768, 1, 68, 1, 373248, 1, 76, 1, 512000, 1, 7056, 1, 7744, 1, 92, 1, 84934656, 1, 1000, 1, 10816, 1, 11664, 1, 1404928, 1, 116, 1, 207360000, 1, 124, 1
Offset: 1

Views

Author

N. J. A. Sloane, Jul 14 2008

Keywords

Links

Crossrefs

Cf. A136655, A126192.
Cf. A001227, A183063, A000593, A007955.

Programs

  • Mathematica
    Table[Times@@Select[Divisors[n],EvenQ],{n,70}] (* Harvey P. Dale, Jul 06 2017 *)
a[n_] := Module[{e = IntegerExponent[n, 2], o, d}, o = n/2^e; d = DivisorSigma[0, o]; n^(d*(e+1)/2)/o^(d/2)]; Array[a, 100] (* Amiram Eldar, Jun 26 2022 *)
  • PARI
    a(n) = my(d=divisors(n)); prod(k=1, #d, if (!(d[k]%2), d[k], 1)); \\ Michel Marcus, Jun 08 2020

Formula

a(n) = A007955(n) / A136655(n).
From Wesley Ivan Hurt, Jun 08 2020: (Start)
a(n) = Product_{d|n, d even} d.
If n is odd or an even squarefree number, then a(n) = floor((2*n)^(1 - ceiling(n/2) + floor(n/2)) * (d(n)/4)). (End)

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

Views

Author

Ilya Gutkovskiy, Aug 03 2017

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    Table[Product[d, {d, Select[Divisors[n], GCD[#, n/#] == 1 &]}]/n, {n, 80}]
Table[n^(2^(PrimeNu[n] - 1) - 1), {n, 80}]
  • PARI
    A290480(n) = if(1==n,n,n^(2^(omega(n)-1)-1)); \\ Antti Karttunen, Aug 06 2018
  • Python
    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

Formula

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.

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

Views

Author

Ilya Gutkovskiy, Aug 03 2017

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

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

Formula

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.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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