OFFSET
1,2
COMMENTS
a(1)=1; for n>1, a(n) is the smallest positive integer not occurring earlier in the sequence such that a(n) does not divide Sum_{k=1..n-1} a(k). - Leroy Quet, Nov 13 2005 (This was the original definition. A simple induction argument shows that this is the same as the present definition. - N. J. A. Sloane, Mar 12 2018)
Define b(1)=2; for n>1, b(n) is the smallest number not yet in the sequence which shares a prime factor with the sum of all preceding terms. Then a simple induction argument shows that the b(n) sequence is the same as the present sequence with the first term omitted. - David James Sycamore, Feb 26 2018
Here are the details of the two induction arguments (Start)
For a(n), let A(n) = a(1)+...+a(n). The claim is that for n>2 a(n)=n+1 if n odd, n-1 if n even.
The induction hypotheses are: for i<n, a(2i)=2i+1, a(2i)=2i-1, A(2i)=i(2i+1), A(2i) = 2i^2+3i+2. This implies that when looking for a(2i), we have seen all the numbers 1 through 2i except 2i-1, so the two smallest candidates for a(2i) are 2i-1 and 2i+1. Since 2i-1 does not divide 2i^2-i-1, a(n)=2i-1. When looking for a(2i+1), we have seen all the numbers 1 through 2i already, so the two smallest candidates for a(2i+1) are 2i+1 and 2i+2. But 2i+1 does divide A(2i) and 2i+2 does not, so a(2i+1)=2i+2. QED
For b(n), the argument is very similar, except that the missing numbers when looking for b(n) are slightly different, and (setting B(n) = b(1)+...b(n)), we have B(2i)=(i+1)(2i+1), B(2i+1)=(i+2)(2i+1). - N. J. A. Sloane, Mar 14 2018
When sequence a(n) is increasing, then the Cesàro means sequence c(n) = (a(1)+...+a(n))/n is also increasing, but the converse is false. This sequence is a such an example where c(n) is increasing, while a(n) is not increasing (Arnaudiès et al.). See proof in A354008. - Bernard Schott, May 11 2022
REFERENCES
J. M. Arnaudiès, P. Delezoide et H. Fraysse, Exercices résolus d'Analyse du cours de mathématiques - 2, Dunod, Exercice 10, pp. 14-16.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..2000.
ProofWiki, Cesàro mean.
Wikipedia, Ernesto Cesàro.
Wikipédia, Lemme de Cesàro (in French).
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
FORMULA
G.f.: x*(x^4-2*x^3+x^2+x+1)/((1-x)*(1-x^2)). - N. J. A. Sloane, Mar 12 2018
The g.f. for the b(n) sequence is x*(x^3-3*x^2+2*x+2)/((1-x)*(1-x^2)). - Conjectured (correctly) by Colin Barker, Mar 04 2018
E.g.f.: 1 - x + x^2/2 + (x - 1)*cosh(x) + (x + 1)*sinh(x). - Stefano Spezia, Sep 02 2022
MATHEMATICA
a[1] = 1; a[n_] := a[n] = Block[{k = 1, s, t = Table[ a[i], {i, n - 1}]}, s = Plus @@ t; While[ Position[t, k] != {} || Mod[s, k] == 0, k++ ]; k]; Array[a, 72] (* Robert G. Wilson v, Nov 18 2005 *)
PROG
(PARI) a(n) = if (n<=2, n, if (n%2, n+1, n-1)); \\ Michel Marcus, May 16 2022
(Python)
def A114112(n): return n + (0 if n <= 2 else -1+2*(n%2)) # Chai Wah Wu, May 24 2022
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Leroy Quet, Nov 13 2005
EXTENSIONS
More terms from Robert G. Wilson v, Nov 18 2005
Entry edited (with simpler definition) by N. J. A. Sloane, Mar 12 2018
STATUS
approved