0,3

Also known as Gobel's (or Goebel's) Sequence. Asymptotically, a(n) ~ n*C^(2^n) where C=1.0478... (A115632). A more precise asymptotic formula is given in A116603. - M. F. Hasler, Dec 12 2007

Let s(n) = (n-1)*a(n). By considering the p-adic representation of s(n) for primes p=2,3,...,43, one finds that a(44) is the first nonintegral value in this sequence. Furthermore, for n>44, the valuation of s(n) w.r.t. 43 is -2^(n-44), implying that both s(n) and a(n) are nonintegral. (M. F. Hasler and Max A. Alekseyev, Mar 03 2009)

R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.

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

T. D. Noe, Table of n, a(n) for n=0..16

N. Lygeros & M. Mizony, Study of primality of terms of a_k(n)=(1+(sum from 1 to n-1)(a_k(i)^k))/(n-1)

D. Rusin, Law of small numbers

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

D. Zagier, Problems posed at the St Andrews Colloquium, 1996

D. Zagier, Solution: Day 5, problem 3

a(n+1) = ((n-1)*a(n)+a(n)^2)/n

(PARI) A003504(n, s=2)=if(n-->0, for(k=1, n-1, s+=(s/k)^2); s/n, 1) \\ M. F. Hasler, Dec 12 2007

Cf. A005166, A005167, A108394, A115632, A116603 (asymptotic formula).

Sequence in context: A088938 A000617 A132183 this_sequence A003182 A134294 A154956

Adjacent sequences: A003501 A003502 A003503 this_sequence A003505 A003506 A003507

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy

a(0)..a(43) are integral, but from a(44) onwards every term is nonintegral - H. W. Lenstra, Jr.

Corrected and extended by M. F. Hasler (maximilian.hasler(AT)gmail.com), Dec 12 2007

Further corrections from Max Alekseyev (maxale(AT)gmail.com), Mar 04 2009