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

Author

Wolfdieter Lang, Nov 29 2002

Keywords

• nonn
• easy

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.

Crossrefs

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

Cf. A049068 (subsequence), A144786.

Programs

Haskell

a078358 n = a078358_list !! (n-1)
a078358_list = filter ((== 0) . a005369) [0..]
-- Reinhard Zumkeller, Jul 04 2014, May 08 2012

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 *)

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

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