A132995 a(n) = gcd(Sum_{k=1..n} prime(k), Product{j=1..n} prime(j)).

2, 1, 10, 1, 14, 1, 2, 77, 10, 3, 10, 1, 238, 1, 82, 3, 110, 3, 2, 213, 2, 7, 874, 3, 530, 129, 158, 3, 370, 177, 430, 3, 994, 3, 2, 3, 646, 2747, 2914, 21, 3266, 3, 3638, 3, 2014, 3, 14, 4661, 1222, 5117, 1070, 69, 5830, 3, 2, 6601, 6870, 7141, 2, 1, 26, 5

Offset

1,1

Comments

From M. F. Hasler, Mar 10 2014: (Start)

The terms a(n) must have the same parity as the sum of the first n primes, A007504(n), which is the opposite of the parity of the index n. Otherwise said, the sequence is congruent to 0,1,0,1,0,1,... (mod 2).

The since the terms of this sequence are divisors of primorials A002110, they are squarefree numbers, A005117.

Is it true, and if so, can it be proved that

* all of the squarefree numbers do appear?

* all of the squarefree numbers do appear infinitely often?

At least it seems that this is the case for the terms 1, 2 and 3. (End)

A239070(n) = position of first occurrence of n-th squarefree number in this sequence. - Reinhard Zumkeller, Mar 10 2014

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 9592 terms from M. F. Hasler)

Formula

A132995(n) = gcd(A007504(n), A002110(n)). - M. F. Hasler, Mar 10 2014

Example

The first 7 primes are 2,3,5,7,11,13,17. 2+3+5+7+11+13+17 = 58 = 2*29. So a(7) = gcd(58, 2*3*5*7*11*13*17) = 2.

Maple

seq(gcd(add(ithprime(i), i=1..n), mul(ithprime(j), j=1..n)), n=1..50); # Emeric Deutsch, Nov 24 2007
# second Maple program:
with(numtheory):
s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+ithprime(n)) end:
a:= n-> mul(`if`(i<=ithprime(n), i, 1), i=factorset(s(n))):
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 10 2014

Mathematica

nn=60; With[{prs=Prime[Range[nn]]}, Table[GCD[Total[Take[prs, n]], Times@@Take[ prs, n]], {n, nn}]] (* Harvey P. Dale, May 07 2011 *)

Prog

(PARI) c=s=0; forprime(p=2, 1e5, f=factor(s+=p, p); f[, 2]=apply(t->t<=p, f[, 1]); write("/tmp/b132995.txt", c++" "factorback(f))) \\ M. F. Hasler, Mar 09 2014
(Haskell)
a132995 n = a132995_list !! (n-1)
a132995_list = tail $ f a000040_list 0 1 where
   f (p:ps) u v = (gcd u v) : f ps (p + u) (p * v)
-- Reinhard Zumkeller, Mar 09 2014

Crossrefs

Cf. A007504, A002110.

Sequence in context: A074951 A055633 A105606 * A114692 A112691 A110169

Adjacent sequences: A132992 A132993 A132994 * A132996 A132997 A132998

Keyword

nonn,look

Author

Leroy Quet, Nov 22 2007

Extensions

More terms from Emeric Deutsch, Nov 24 2007

Status

approved