cp's OEIS Frontend

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

%I A001543 M4091 N1699 #59 Jul 08 2025 16:26:46
%S A001543 1,6,11,71,4691,21982031,483209576974811,
%T A001543 233491495280173380882643611671,
%U A001543 54518278368171228201482876236565907627201914279213829353891
%N A001543 a(0) = 1, a(n) = 5 + Product_{i=0..n-1} a(i) for n > 0.
%C A001543 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
%D A001543 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001543 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001543 Alois P. Heinz, <a href="/A001543/b001543.txt">Table of n, a(n) for n = 0..12</a>
%H A001543 A. V. Aho and N. J. A. Sloane, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/11-4/aho-a.pdf">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
%H A001543 A. V. Aho and N. J. A. Sloane, <a href="/A000058/a000058.pdf">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
%H A001543 S. W. Golomb, <a href="http://www.jstor.org/stable/2311857">On certain nonlinear recurring sequences</a>, Amer. Math. Monthly 70 (1963), 403-405.
%H A001543 R. Mestrovic, <a href="http://arxiv.org/abs/1202.3670">Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof</a>, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - _N. J. A. Sloane_, Jun 13 2012
%H A001543 S. Mustonen, <a href="http://www.survo.fi/papers/resseq.pdf">On integer sequences with mutual k-residues</a>
%H A001543 Seppo Mustonen, <a href="/A000215/a000215.pdf">On integer sequences with mutual k-residues</a> [Local copy]
%H A001543 <a href="/index/Aa#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 + ...</a>
%F A001543 a(n) = a(n-1) * (a(n-1) - 5) + 5. - _Charles R Greathouse IV_, Dec 09 2011
%F A001543 a(n) ~ c^(2^n), where c = 1.696053774403103324180661918166106455311376345474042496749974632237971081462... . - _Vaclav Kotesovec_, Dec 17 2014
%t A001543 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 *)
%t A001543 Join[{1},NestList[#(#-5)+5&,6,10]] (* _Harvey P. Dale_, Oct 10 2016 *)
%o A001543 (PARI) {
%o A001543   print1("1, 6");
%o A001543   n=6;
%o A001543   m=Mod(5,6);
%o A001543   for(i=2,9,
%o A001543     n=m.mod+lift(m);
%o A001543     m=chinese(m,Mod(5,n));
%o A001543     print1(", "n)
%o A001543   )
%o A001543 } \\ _Charles R Greathouse IV_, Dec 09 2011
%Y A001543 Column k=5 of A177888.
%K A001543 nonn
%O A001543 0,2
%A A001543 _N. J. A. Sloane_
%E A001543 New name from _Alonso del Arte_, Dec 09 2011