A078358 Non-oblong numbers: Complement of A002378.

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

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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
(Python)
from operator import sub
from sympy import integer_nthroot
def A078358(n): return n+sub(*integer_nthroot(n, 2)) # Chai Wah Wu, Oct 01 2024

Crossrefs

a(n)=(A077425(n)-1)/4.

Cf. A049068 (subsequence), A144786.

Sequence in context: A039177 A058986 A184431 * A175968 A152012 A173153

Adjacent sequences: A078355 A078356 A078357 * A078359 A078360 A078361

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 29 2002

Status

approved