A060796 Upper central divisor of n-th primorial.

2, 3, 6, 15, 55, 182, 715, 3135, 15015, 81345, 448630, 2733549, 17490603, 114388729, 785147363, 5708795638, 43850489690, 342503171205, 2803419704514, 23622001517543, 201817933409378, 1793779635410490, 16342166369958702, 154171363634898185, 1518410187442699518, 15259831781575946565

Offset: 1

Author

Labos Elemer, Apr 27 2001

Keywords

• nonn

Comments

Also: Write product of first n primes as x*y with x < y and x maximal; sequence gives value of y. Indeed, p(n)# = primorial(n) = A002110(n) is never a square for n >= 1; all exponents in the prime factorization are 1. Therefore primorial(n) has N = 2^n distinct divisors. Since this is an even number, the N divisors can be grouped in N/2 pairs {d(k), d(N+1-k)} with product equal to p(n)#. One of the two is always smaller and one is larger than sqrt(p(n)#). This sequence gives the (2^(n-1)+1)-th divisor, which is the smallest one larger than sqrt(p(n)#). - M. F. Hasler, Sep 20 2011

Examples

			n = 8, q(8) = 2*3*5*7*11*13*17*19 = 9699690. Its 128th and 129th divisors are {3094, 3135}: a(8) = 3135, and 3094 < A000196(9699690) = 3114 < 3135. [Corrected by _M. F. Hasler_, Sep 20 2011]
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..70

Crossrefs

Cf. A060755, A000196, A002110, A033677, A060776, A060777, A061057, A060795 (x), A061060 (y-x), A182987 (x+y), A061030, A061031, A061032, A061033, A200744.

Programs

Mathematica

k = 1; Do[k *= Prime[n]; l = Divisors[k]; x = Length[l]; Print[l[[x/2 + 1]]], {n, 1, 24}] (* Ryan Propper, Jul 25 2005 *)

PARI

A060796(n) = divisors(prod(k=1,n,prime(k)))[2^(n-1)+1] \\ Requires stack size > 2^(n+5). - M. F. Hasler, Sep 20 2011

Formula

a(n) = A033677(A002110(n)).

a(n) = A002110(n)/A060795(n). - M. F. Hasler, Mar 21 2022

Extensions

More terms from Ryan Propper, Jul 25 2005

a(24)-a(37) in b-file calculated from A182987 by M. F. Hasler, Sep 20 2011

a(38) from David A. Corneth, Mar 21 2022

a(39)-a(70) in b-file from Max Alekseyev, Apr 20 2022