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A078358 Non-oblong numbers: Complement of A002378. 18
1, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The (primitive) period length k(n)=A077427(n) of the (regular) continued fraction of (sqrt(4*a(n)+1)+1)/2 determines whether or not the Diophantine equation (2*x-y)^2 - (1+4*a(n))*y^2 = +4 or -4 is solvable and the approximants of this continued fraction give all solutions. See A077057.
The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446. - Jeremy Gardiner, Mar 16 2003
Infinite series 1/A078358(n) is divergent. Proof: Harmonic series 1/A000027(n) is divergent and can be distributed on two subseries 1/A002378(k+1) and 1/A078358(m). The infinite subseries 1/A002378(k+1) is convergent to 1, so Sum_{n>=1} 1/A078358(n) is divergent. - Artur Jasinski, Sep 28 2008
REFERENCES
O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).
LINKS
Oskar Perron, Die Lehre von den Kettenbrüchen, Teubner, Leipzig, 1913.
FORMULA
4*a(n)+1 is not a square number.
a(n) = ceiling(sqrt(n)) + n -1. - Leroy Quet, Jul 06 2007
A005369(a(n)) = 0. - Reinhard Zumkeller, Jul 05 2014
MATHEMATICA
Complement[Range[930], Table[n (n + 1), {n, 0, 30}]] (* and *) Table[Ceiling[Sqrt[n]] + n - 1, {n, 900}] (* Vladimir Joseph Stephan Orlovsky, Jul 20 2011 *)
PROG
(Haskell)
a078358 n = a078358_list !! (n-1)
a078358_list = filter ((== 0) . a005369) [0..]
-- Reinhard Zumkeller, Jul 04 2014, May 08 2012
(PARI) a(n)=sqrtint(n-1)+n \\ Charles R Greathouse IV, Jan 17 2013
CROSSREFS
a(n)=(A077425(n)-1)/4.
Cf. A049068 (subsequence), A144786.
Sequence in context: A039177 A058986 A184431 * A175968 A152012 A173153
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 29 2002
STATUS
approved

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Last modified February 26 11:39 EST 2024. Contains 370352 sequences. (Running on oeis4.)