A001543 a(0) = 1, a(n) = 5 + Product_{i=0..n-1} a(i) for n > 0.

1, 6, 11, 71, 4691, 21982031, 483209576974811, 233491495280173380882643611671, 54518278368171228201482876236565907627201914279213829353891

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

This is the special case k=5 of sequences with exact mutual k-residues. In general, a(1)=k+1 and a(n)=min{m | m>a(n-1), mod(m,a(i))=k, i=1,...,n-1}. k=1 gives Sylvester's sequence A000058 and k=2 Fermat sequence A000215. - Seppo Mustonen, Sep 04 2005

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

• Alois P. Heinz, Table of n, a(n) for n = 0..12
• A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
• A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
• S. W. Golomb, On certain nonlinear recurring sequences, Amer. Math. Monthly 70 (1963), 403-405.
• 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. - _N. J. A. Sloane_, Jun 13 2012
• S. Mustonen, On integer sequences with mutual k-residues
• Seppo Mustonen, On integer sequences with mutual k-residues [Local copy]
• Index entries for sequences of form a(n+1)=a(n)^2 + ...

Crossrefs

Column k=5 of A177888.

Programs

Mathematica

Flatten[{1,RecurrenceTable[{a[1]==6, a[n]==a[n-1]*(a[n-1]-5)+5}, a, {n, 1, 10}]}] (* Vaclav Kotesovec, Dec 17 2014 *)
Join[{1},NestList[#(#-5)+5&,6,10]] (* Harvey P. Dale, Oct 10 2016 *)

PARI

{
  print1("1, 6");
  n=6;
  m=Mod(5,6);
  for(i=2,9,
    n=m.mod+lift(m);
    m=chinese(m,Mod(5,n));
    print1(", "n)
  )
} \\ Charles R Greathouse IV, Dec 09 2011

Formula

a(n) = a(n-1) * (a(n-1) - 5) + 5. - Charles R Greathouse IV, Dec 09 2011

a(n) ~ c^(2^n), where c = 1.696053774403103324180661918166106455311376345474042496749974632237971081462... . - Vaclav Kotesovec, Dec 17 2014

Extensions

New name from Alonso del Arte, Dec 09 2011