A078358 - OEIS

%I #40 Oct 01 2024 03:39:05

%S 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,

%T 31,32,33,34,35,36,37,38,39,40,41,43,44,45,46,47,48,49,50,51,52,53,54,

%U 55,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,73,74

%N Non-oblong numbers: Complement of A002378.

%C 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.

%C The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446. - _Jeremy Gardiner_, Mar 16 2003

%C 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

%D O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

%H Reinhard Zumkeller, <a href="/A078358/b078358.txt">Table of n, a(n) for n = 1..10000</a>

%H Oskar Perron, <a href="https://archive.org/details/dielehrevondenk00perrgoog">Die Lehre von den Kettenbrüchen</a>, Teubner, Leipzig, 1913.

%F 4*a(n)+1 is not a square number.

%F a(n) = ceiling(sqrt(n)) + n -1. - _Leroy Quet_, Jul 06 2007

%F A005369(a(n)) = 0. - _Reinhard Zumkeller_, Jul 05 2014

%t 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 *)

%o (Haskell)

%o a078358 n = a078358_list !! (n-1)

%o a078358_list = filter ((== 0) . a005369) [0..]

%o -- _Reinhard Zumkeller_, Jul 04 2014, May 08 2012

%o (PARI) a(n)=sqrtint(n-1)+n \\ _Charles R Greathouse IV_, Jan 17 2013

%o (Python)

%o from operator import sub

%o from sympy import integer_nthroot

%o def A078358(n): return n+sub(*integer_nthroot(n,2)) # _Chai Wah Wu_, Oct 01 2024

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

%Y Cf. A049068 (subsequence), A144786.

%K nonn,easy

%O 1,2

%A _Wolfdieter Lang_, Nov 29 2002