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 1-3 of 3 results.

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.

A157721 a(n) = 0 if n is 1 or a prime, otherwise a(n) = product of composite (nonprime) divisors of n.

Original entry on oeis.org

0, 0, 0, 4, 0, 6, 0, 32, 9, 10, 0, 288, 0, 14, 15, 512, 0, 972, 0, 800, 21, 22, 0, 55296, 25, 26, 243, 1568, 0, 27000, 0, 16384, 33, 34, 35, 1679616, 0, 38, 39, 256000, 0, 74088, 0, 3872, 6075, 46, 0, 42467328, 49, 12500, 51, 5408, 0, 1417176, 55, 702464, 57, 58, 0
Offset: 1

Views

Author

Jaroslav Krizek, Mar 04 2009

Keywords

Comments

a(n) = 0 if n = 1 or n is prime; a(n) = n if n is semiprime (A001358). a(c) = A007955(c) / A007947(c) = (c^(A000005(c)/2)) / A007947(c) = A087652(c), for c = composite numbers (A002808). a(p) = A087652(p) - 1 = 0, for p = primes (A000040).

Examples

			a(12) = 4*6*12 = 288, composite divisors = {4,6,12}.

Links

Crossrefs

Cf. A007955, A007947, A000005, A087652, A002808, A000040.

Programs

  • Maple
    f:= proc(n) if isprime(n) then 0 else convert(remove(isprime,numtheory:-divisors(n)),`*`) fi end proc;
f(1):= 0:
map(f, [$1..100]); # Robert Israel, Jul 31 2024
  • Mathematica
    f[n_] := If[n == 1 || PrimeQ@n, 0, Times @@ Select[Divisors@n, ! PrimeQ@# &]]; Array[f, 60] (* Robert G. Wilson v, May 04 2009 *)

Extensions

More terms from Robert G. Wilson v, May 04 2009

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 1-3 of 3 results.

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