A094353 Smallest integer not yet used such that 1 + Product_{k=1..n} a(k) is a square.

3, 1, 5, 8, 7, 2, 17, 44, 53, 1011, 7262969, 27755899054, 10713771825916682198, 1451983503000530523834049701901973110, 5317619734003376302895262678416297955761358855419919105266696681033714

Offset

1,1

Comments

This sequence does not include every natural number. Suppose it contains b^2. Then letting A be the product up to b^2, we have A + 1 = i^2 and b^2 A + 1 = j^2. Multiplying the first equation by b^2 and subtracting, we get (bi)^2 = j^2 + b^2 - 1, which puts an upper bound on i and j (and hence on A). Probably the sequence contains no squares other than 1. - Franklin T. Adams-Watters, Aug 29 2006

a(n) <= 2 + Product_{k=1..n-1} a(k). - Max Alekseyev, Apr 26 2010

Links

Table of n, a(n) for n=1..15.

Example

3 + 1 = 4, 3*1*5 + 1 = 16, 3*1*5*8 + 1 = 121 etc. are squares.

Prog

(PARI) v=[1]; n=1; while(n<1100, s=1+n*prod(i=1, #v, v[i]); if(issquare(s)&&!vecsearch(vecsort(v), n), v=concat(v, n); n=0); n++); v \\ Derek Orr, Jun 16 2015

Crossrefs

Cf. A094354.

Sequence in context: A280384 A124420 A176105 * A298662 A306780 A209996

Adjacent sequences: A094350 A094351 A094352 * A094354 A094355 A094356

Keyword

nonn

Author

Amarnath Murthy, May 22 2004

Extensions

More terms from Franklin T. Adams-Watters, Aug 29 2006

a(11) from D. S. McNeil, Mar 20 2009

a(12) from Sean A. Irvine, Nov 18 2009

a(13)-a(15) from Max Alekseyev, Apr 26 2010

Status

approved