A183091 a(n) is the product of the divisors p^k of n where p is prime and k >= 1.

1, 2, 3, 8, 5, 6, 7, 64, 27, 10, 11, 24, 13, 14, 15, 1024, 17, 54, 19, 40, 21, 22, 23, 192, 125, 26, 729, 56, 29, 30, 31, 32768, 33, 34, 35, 216, 37, 38, 39, 320, 41, 42, 43, 88, 135, 46, 47, 3072, 343, 250, 51, 104, 53, 1458

Offset: 1

Author

Jaroslav Krizek, Dec 25 2010

Keywords

• nonn
• easy
• mult

Comments

Product of n-th row of triangle A210208. - Reinhard Zumkeller, Mar 18 2012

Examples

			For n = 12, set of such divisors is {1, 2, 3, 4}; a(12) = 1*2*3*4 = 24.
		

Links

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

Crossrefs

Cf. A007955, A023888, A153038, A183092, A210208.

Programs

Haskell

a183091 = product . a210208_row  -- Reinhard Zumkeller, Mar 18 2012

Maple

A183091 := proc(n) local a,d; a := 1 ;for d in numtheory[divisors](n) minus {1} do  if nops( numtheory[factorset](d)) = 1 then a := a*d; end if; end do: a ; end proc: # R. J. Mathar, Apr 14 2011

Mathematica

Table[Product[d, {d, Select[Divisors[n], Length[FactorInteger[#]] == 1 &]}], {n,1, 54}] (* Geoffrey Critzer, Mar 18 2015 *)
f[p_, e_] := p^(e*(e+1)/2); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 31 2023 *)

PARI

a(n)=my(f=factor(n)); prod(i=1,#f~, f[i,1]^binomial(f[i,2]+1,2)) \\ Charles R Greathouse IV, Nov 11 2014

Formula

a(n) = A007955(n) / A183092(n).

Multiplicative with a(p^k) = p^(k*(k+1)/2).

The Dirichlet g.f. of a(n) / abs(A153038(n)) is Product_{k >= 0} zeta(s+k). - Álvar Ibeas, Nov 10 2014