A073833 -id:A073833 - cp's OEIS frontend

A233770 Decimal expansion of lim_{n -> infinity} b(n)^2 - 2n - (log n)/2 where b(i) = b(i-1) + 1/b(i-1) for i >= 2, b(1) = 1 (see A073833).

2, 7, 6, 8, 5, 7, 6, 2, 4, 8, 6, 2, 5, 7, 6, 5, 3, 8, 9, 3, 6, 4, 3, 7, 2, 5, 0, 8, 2, 3, 5, 7, 3, 3, 9, 6, 3, 1, 7, 9, 7, 9, 7, 3, 7, 5, 2, 7, 5, 1, 3, 7, 3, 9, 1, 5, 9, 7, 7, 3, 1, 6, 4, 3, 5, 4, 8, 5, 0, 1, 4, 1, 8, 0, 8, 2, 9, 7, 1, 2, 4, 3, 1, 1, 8, 9, 8

Offset: 0

Jon E. Schoenfield and N. J. A. Sloane, Dec 15 2013

b(n)^2 = t/2 + u + (u - 1/2)/t + (-u^2 + 2*u - 11/12)/t^2 + (4*u^3/3 - 5*u^2 + 17*u/3 - 65/36)/t^3 + ... where t=4*n, u=(log n)/2+c, and c=-0.27685762486257653893643725082....

c = (log c1)/2 where c1 is a constant described in the comments in A073833; its digits are in A232975.

			-0.27685762486257653893643725082357339631797973752751373915977316435485014180...

Cf. A073833, A232975.

A232975 Decimal expansion of lim_{n -> infinity} (1/n)exp(2(b(n)^2-2n)) where b(i) = b(i-1) + 1/b(i-1) for i >= 2, b(1) = 1 (see A073833).

Original entry on oeis.org

5, 7, 4, 8, 1, 0, 2, 7, 4, 6, 7, 1, 7, 8, 5, 0, 6, 5, 2, 9, 6, 3, 4, 7, 8, 4, 6, 3, 5, 3, 0, 9, 2, 8, 0, 0, 7, 6, 0, 9, 8, 7, 8, 4, 5, 5, 2, 0, 8, 9, 0, 6, 1, 9, 0, 1, 5, 1, 5, 5, 4, 7, 8, 6, 6, 6, 4, 5, 5, 7, 3, 4, 1, 9, 9, 1, 6, 2, 5, 9, 1, 2, 6, 9, 5, 8, 7

Offset: 0

Jon E. Schoenfield and N. J. A. Sloane, Dec 07 2013

A161500 Primes dividing some member of A073833.

Original entry on oeis.org

2, 5, 29, 41, 89, 101, 109, 269, 421, 509, 521, 709, 929, 941, 1549, 1861, 2281, 2521, 2749, 2801, 2909, 3121, 3169, 3469, 5821, 5881, 7109, 8069, 8969, 9041, 9181, 10061, 10601, 11549, 15121, 16061, 16889, 16981, 21929, 30089, 30169, 32561, 41149

Offset: 1

Franklin T. Adams-Watters, Jun 11 2009

nonn

Primes that divide A073833(n) will divide A073834(m) for any m > n, and this is all the prime divisors of A073834(m).
Iterating f(x) = x + 1/x modulo p will eventually either produce a zero (in which case p is in this sequence), or it will loop to an earlier term (in which case it is not). Since f(-x) = -f(x), encountering the negation of an earlier term means that the iteration is looping.
Note that A073833(6) = 969581 = 521 * 1861 is the first composite member of that sequence.

Cf. A073833, A073834, A000057.

ina(p)=local(m,k,v);m=Mod(1,p);v=vector(p\2);while(m!=0,k=lift(m);if(2*k>p,k=p-k);if(v[k],return(0));v[k]=1;m+=1/m);1

A147987 Coefficients of numerator polynomials P(n,x) associated with reciprocation.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 1, 0, 7, 0, 13, 0, 7, 0, 1, 1, 0, 15, 0, 83, 0, 220, 0, 303, 0, 220, 0, 83, 0, 15, 0, 1, 1, 0, 31, 0, 413, 0, 3141, 0, 15261, 0, 50187, 0, 115410, 0, 189036, 0, 222621, 0, 189036, 0, 115410, 0, 50187, 0, 15261, 0, 3141, 0, 413, 0, 31, 0, 1, 1, 0, 63

Offset: 1

Clark Kimberling, Nov 24 2008

P(n,1)=A073833(n) for n>=1; P(n,2)=A073833(n+1) for n>=0.
P(n)=P(n-1)^2+P(n-1)*P(n-2)^2-P(n-2)^4 for n>=3.
For n>=3, P(n)=P(n,x)=S(n,i*x), where S(n) is the polynomial at A147985.
Thus all the zeros of P(n,x), for n>=2, are nonreal.

			P(1) = x
P(2) = x^2+1
P(3) = x^4+3*x^2+1
P(4) = x^8+7*x^6+13*x^4+7x^2+1
so that, as an array, the sequence begins with:
1 0
1 0 1
1 0 3 0 1
1 0 7 0 13 0 7 0 1

Clark Kimberling, Polynomials associated with reciprocation, Journal of Integer Sequences 12 (2009, Article 09.3.4) 1-11.

Cf. A147985, A147986, A147988, A147989, A147990, A147991, A147992, A147993.

p[1] = x; q[1] = 1; p[n_] := p[n] = p[n-1]^2 + q[n-1]^2; q[n_] := q[n] = p[n-1]*q[n-1]; row[n_] := CoefficientList[p[n], x] // Reverse; Table[row[n], {n, 1, 7}] // Flatten (* Jean-François Alcover, Apr 22 2013 *)

The basic idea is to iterate the reciprocation-sum mapping x/y -> x/y+y/x.

Let x be an indeterminate, P(1)=x, Q(1)=1 and for n>1, define P(n)=P(n-1)^2+Q(n-1)^2 and Q(n)=P(n-1)*Q(n-1), so that P(n)/Q(n)=P(n-1)/Q(n-1)-Q(n-1)/P(n-1).

A073834 Denominators of b(n) where b(1) = 1, b(i) = b(i-1) + 1/b(i-1).

Original entry on oeis.org

1, 1, 2, 10, 290, 272890, 264588959090, 268440386798659418988490, 295105036840595214385430531020664149472669868290, 377908709746050392481071609609580527436122569261424131112048023467330784739529329885668846964890

Offset: 1

Alex Fink, Aug 12 2002

a(n) is also the denominator of the fractional chromatic number of the Mycielski graph M_n - Eric W. Weisstein, Mar 05 2011

H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 187.
D. J. Newman, A Problem Seminar, Springer; see Problem #60.
J. H. Silverman, The arithmetic of dynamical systems, Springer, 2007, see p. 113 Table 3.1

Steven Finch, Popa's "Recurrent Sequences" and Reciprocity, arXiv:2412.11806 [math.CA], 2024. See p. 16.
Eric Weisstein's World of Mathematics, Fractional Chromatic Number
Eric Weisstein's World of Mathematics, Mycielski Graph

See A073833 for numerators.

f[n_]:=n+1/n; Prepend[Denominator[NestList[f,2,9]],1] (* Vladimir Joseph Stephan Orlovsky, Nov 19 2010 *)
Denominator[NestList[# + 1/# &, 1, 10]] (* Eric W. Weisstein, Mar 05 2011 *)

{a(n) = local(x, y); if( n<1, 0, if( n<3, n, x = a(n-2)^2; y = a(n-1); y*y + x * (y - x)))} /* Michael Somos, Mar 05 2012 */

A367787 Let b(0) = 1, b(n) = Sum_{k=0..n-1} b(k) / b(n-k-1), then a(n) is the numerator of b(n).

Original entry on oeis.org

1, 1, 2, 7, 44, 3459, 21398845, 204701870532176, 47683439994850565666251869149, 203292005443961363023193564438853229653319486912062841397

Offset: 0

Ilya Gutkovskiy, Nov 30 2023

The next term is too large to include.

			1, 1, 2, 7/2, 44/7, 3459/308, 21398845/1065372, 204701870532176/5699432573835, ...

Cf. A000108, A022857, A022858, A073833, A367788 (denominators).

b[0] = 1; b[n_] := b[n] = Sum[b[k]/b[n - k - 1], {k, 0, n - 1}]; a[n_] := Numerator[b[n]]; Table[a[n], {n, 0, 9}]

G.f. for fractions satisfies: 1 / Sum_{n>=0} b(n) * x^n = 1 - x * Sum_{n>=0} x^n / b(n).

A332396 Decimal expansion of lim_{n->infinity} (b(n)/n - (n - log(n))/4) where {b(n)} is the real-valued sequence defined by b(1)=1 and b(n+1) = b(n) + sqrt(b(n)) for n > 0.

Original entry on oeis.org

6, 7, 5, 1, 7, 7, 4, 4, 2, 4, 5, 8, 5, 5, 7, 1, 3, 9, 8, 1, 3, 2, 8, 5, 6, 2, 5, 0, 7, 5, 8, 2, 7, 6, 3, 3, 6, 8, 4, 0, 7, 3, 1, 5, 8, 9, 8, 9, 0, 5, 1, 5, 9, 1, 4, 0, 1, 1, 3, 0, 8, 1, 0, 8, 5, 9, 1, 5, 1, 9, 7, 2, 9, 4, 9, 8, 4, 0, 2, 3, 7, 9, 3, 5, 3, 3, 7

Offset: 0

Jon E. Schoenfield, Feb 10 2020

Consider the "add the square root" sequence {b(n)} starting with 1, i.e., b(1)=1, b(n+1) = b(n) + sqrt(b(n)) for n > 0: at each step, the next term is obtained simply by adding the current term to its square root, so the sequence begins {1, 2, 2 + sqrt(2), 2 + sqrt(2) + sqrt(2 + sqrt(2)), ...}, i.e., {1, 2, 3.414213..., 5.261972..., ...}.
The 1st, 10th, 100th, and 1000th terms are
    b(10^0) = 1.0
    b(10^1) = 25.956...
    b(10^2) = 2452.850...
    b(10^3) = 248949.869...
    b(10^4) = 24983729.385...
    b(10^5) = 2499779700.562...
Since the difference between successive terms is b(n+1) - b(n) = sqrt(b(n)) and the similar differential equation (d/dx)f(x) = sqrt(f(x)) is satisified by the function f(x) = (1/4)*x^2, it is perhaps not surprising that lim_{n->infinity} b(n)/n^2 = 1/4. More precisely, it can be shown that, as n increases, b(n) approaches
   (1/4)*n^2
   + u*n
   + u^2 - u/2
   + (-(1/2)*u^2 + u/4 - 1/96)/n
   + ((1/3)*u^3 + 0*u^2 + (-5/48)*u + 7/576)/n^2
   + ...
where u = -(1/4)*log(n) + c
  and c = 0.675177442458557139813285625075...
It follows that lim_{n->infinity} (b(n)/n - (n - log(n))/4) = c.
If we were to define {b(n)} instead as b(1)=1, b(n+1) = b(n) + 1/b(n) for n > 0 (i.e., "add the reciprocal" rather than "add the square root"), we would obtain the rational-valued sequence {b(n)} = {1, 2, 5/2, 29/10, 941/290, ...} (see A073833 and, for a constant arising from that sequence, A233770).

			0.67517744245855713981328562507582763368407315898905...

Steven Finch, Popa's "Recurrent Sequences" and Reciprocity, arXiv:2412.11806 [math.CA], 2024. See p. 16.

Cf. A073833, A233770.

cp's OEIS Frontend

A233770 Decimal expansion of lim_{n -> infinity} b(n)^2 - 2n - (log n)/2 where b(i) = b(i-1) + 1/b(i-1) for i >= 2, b(1) = 1 (see A073833).

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A232975 Decimal expansion of lim_{n -> infinity} (1/n)*exp(2*(b(n)^2-2n)) where b(i) = b(i-1) + 1/b(i-1) for i >= 2, b(1) = 1 (see A073833).

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A161500 Primes dividing some member of A073833.

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A147987 Coefficients of numerator polynomials P(n,x) associated with reciprocation.

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A073834 Denominators of b(n) where b(1) = 1, b(i) = b(i-1) + 1/b(i-1).

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A367787 Let b(0) = 1, b(n) = Sum_{k=0..n-1} b(k) / b(n-k-1), then a(n) is the numerator of b(n).

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A332396 Decimal expansion of lim_{n->infinity} (b(n)/n - (n - log(n))/4) where {b(n)} is the real-valued sequence defined by b(1)=1 and b(n+1) = b(n) + sqrt(b(n)) for n > 0.

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A232975 Decimal expansion of lim_{n -> infinity} (1/n)exp(2(b(n)^2-2n)) where b(i) = b(i-1) + 1/b(i-1) for i >= 2, b(1) = 1 (see A073833).