A001685 a(0) = 1, a(1) = 2, a(2) = 3; for n >= 3, a(n) = a(n-2) + a(n-1)*Product_{i=1..n-3} a(i).

1, 2, 3, 5, 13, 83, 2503, 976253, 31601312113, 2560404986164794683, 202523113189037952478722304798003, 506227391211661106785411233681995783881012463859772443053

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

From a continued fraction.

Every term is relatively prime to all others. - Michael Somos, Feb 01 2004

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• John Cerkan, Table of n, a(n) for n = 0..16
• V. C. Harris, Another proof of the infinitude of primes, Amer. Math. Monthly, 63 (1956), 711.
• R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - From _N. J. A. Sloane_, Jun 13 2012

Crossrefs

Cf. A003686, A064526, A109134.

Programs

Mathematica

Clear[a]; a[0]=1; a[1]=2; a[2]=3; a[n_]:=a[n] = a[n-2] + a[n-1]*Product[a[j],{j,1,n-3}]; Table[a[n],{n,0,15}] (* Vaclav Kotesovec, May 21 2015 *)
Clear[a];RecurrenceTable[{a[n]==a[n-2]+a[n-1]*a[n-3]*(a[n-1]-a[n-3])/a[n-2],a[0]==1,a[1]==2,a[2]==3},a,{n,0,15}] (* Vaclav Kotesovec, May 21 2015 *)

PARI

a(n)=if(n<3,max(0,n+1),a(n-2)+a(n-1)*prod(i=1,n-3,a(i))) /* Michael Somos, Feb 01 2004 */

Formula

a(n) = a(n-2) + a(n-1)*a(n-3)*(a(n-1)-a(n-3))/a(n-2). - Vaclav Kotesovec, May 21 2015

a(n) ~ c^(d^n), where d = A109134 = 1.754877666246692760049508896358528691894606617772793143989283970646... is the root of the equation d*(d-1)^2 = 1, c = 1.3081335128180696870655208993764956995000211962454918672885690026423582299... . - Vaclav Kotesovec, May 21 2015

Extensions

Edited by N. J. A. Sloane, Jun 12 2006