A001203 -id:A001203 - cp's OEIS frontend

A032523 Index of first occurrence of n as a term in A001203, the continued fraction for Pi.

4, 9, 1, 30, 40, 32, 2, 44, 130, 100, 276, 55, 28, 13, 3, 78, 647, 137, 140, 180, 214, 83, 203, 91, 791, 112, 574, 175, 243, 147, 878, 455, 531, 421, 1008, 594, 784, 3041, 721, 1872, 754, 119, 492, 429, 81, 3200, 825, 283, 3027, 465, 1437, 3384, 1547, 1864, 446

Offset: 1

Wouter Meeussen

Incorrectly indexed version of A225802 (assuming the c.f. is [a_1; a_2, a_3, ...] instead of [a_0; a_1, a_2, ...]).
Until it is proved that every integer n>0 does occur in A001203, we should tacitly understand a convention like "A032523(n) = 0 if n does not occur in A001203". - M. F. Hasler, Mar 31 2008
All positive integers <= 33674 occur in the first 5,821,569,425 terms of the c.f. - Eric W. Weisstein, Sep 19 2011
All positive integers <= 47086 occur in the first 10,672,905,501 terms of the c.f. (the first that do not are 47087, 49004, 50465, 50471, ...) - Eric W. Weisstein, Jul 18 2013

M. F. Hasler and Eric W. Weisstein, Table of n, a(n) for n = 1..47086 (terms n = 1..3131 from _M. F. Hasler_, using data from H. Havermann)
H. Havermann, 24345 terms of a trivial variation (first term of Pi excluded) of A032523
Eric Weisstein's World of Mathematics, Pi Continued Fraction

Cf. A225802 (= a(n) - 1).

Cf. A001203, A107892, A138758, A138759.

With[{cfp=ContinuedFraction[Pi,5000]},Flatten[Table[Position[cfp,n,1,1],{n,60}]]] (* Harvey P. Dale, Dec 11 2012 *)

default( realprecision, 15000); v=contfrac(Pi); a(n) = for( i=1,#v, v[i]==n && return(i)) \\ - W. Meeussen, simplified by M. F. Hasler, Mar 31 2008

a(n) = A225802(n) + 1.

A032523(n) = min { k | A001203(k)=n }. - M. F. Hasler, Mar 31 2008

Edited by M. F. Hasler, Mar 31 2008

A138759 Indices for which A001203 (continued fraction for Pi) is prime.

Original entry on oeis.org

1, 2, 9, 11, 14, 17, 18, 19, 20, 23, 27, 28, 31, 36, 37, 39, 40, 46, 48, 49, 50, 52, 59, 65, 70, 71, 72, 73, 75, 85, 86, 90, 93, 95, 97, 101, 102, 105, 106, 109, 110, 111, 118, 120, 122, 123, 124, 127, 128, 131, 132, 133, 140, 142, 145, 146, 151, 152, 153, 155, 159

Offset: 1

M. F. Hasler, Mar 31 2008

nonn

			This sequence starts 1,2,9,11,... since the first, 2nd, 9th, 11th...
term of sequence A001203 = (3, 7, 15, 1, 292, 1, 1, 1, 2, ...) are primes.

Cf. A001203, A005042, A107892, A138758.

Position[ContinuedFraction[Pi,200],?PrimeQ]//Flatten(* _Harvey P. Dale, Aug 07 2019 *)

default(realprecision,1000); t=contfrac(Pi); for( k=1,#t, isprime(t[k]) & print1(k","))

k is in A138759 <=> A001203(k) is in A000040

A080406 Boustrophedon transform of the continued fraction of Pi (cf. A001203).

Original entry on oeis.org

3, 10, 32, 73, 457, 1994, 6407, 29489, 148253, 852592, 5420543, 37975111, 290066507, 2400720769, 21396506651, 204322668174, 2081209926313, 22523982873141, 258105780607144, 3121989826825492, 39750408190737416

Offset: 0

Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 17 2003

			We simply apply the Boustrophedon transform to [3,7,15,1,292,1,1,1,...]

J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J.Combin. Theory, 17A (1996) 44-54 (Abstract, pdf, ps).
N. J. A. Sloane, Transforms
Index entries for sequences related to boustrophedon transform

Cf. A001203, A000796.

a(n) appears to be asymptotic to C*n!*(2/Pi)^n where C=136.651536367325329682973604897976758877614262731284965133228708820... - Benoit Cloitre and Mark Hudson (mrmarkhudson(AT)hotmail.com)

A107892 Index of first occurrence of n-th prime in A001203, the continued fraction for Pi.

Original entry on oeis.org

9, 1, 40, 2, 276, 28, 647, 140, 203, 243, 878, 784, 754, 492, 825, 1547, 907, 868, 1789, 9215, 898, 6222, 9131, 4829, 1516, 6700, 22640, 872, 11170, 3204, 223, 10387, 8299, 30086, 31079, 12637, 8486, 20644, 8451, 53069, 32093, 16297, 20276, 1002, 21264

Offset: 1

Zak Seidov, May 25 2005

nonn

Until it is proved that every prime does indeed occur in A001203, we should tacitly understand a convention like "A107892(n) = 0 if A000040(n) does not occur in A001203". - M. F. Hasler, Mar 31 2008

Among first 1000000 terms of the continued fraction for Pi, the first absent primes have indices 129, 132, 137, 146, 147, 158, 160, 165, 170, 172, 175, 180, 182, 184, 189, 193, 197, 198, 199. The 200th prime is in the 947040th place, thus A107892(200)=947040.

M. F. Hasler (using data from H. Havermann), Table of n, a(n) for n=1,...,445.

Cf. A032523: first occurrence of n in A001203.

Cf. A138758, A138759.

A107892(n) = A032523(A000040(n)) = min { k | A001203(k)=A000040(n) }. - M. F. Hasler, Mar 31 2008

Edited by M. F. Hasler, Mar 31 2008

A138758 Index of A001203(n) (continued fraction for Pi) in A000040 (primes), or 0 if A001203(n) is not prime.

Original entry on oeis.org

2, 4, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 2, 6, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 2, 3, 0, 0, 0, 0, 0, 4, 0, 1, 2, 4, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 1, 1, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 3, 0, 0, 1

Offset: 1

M. F. Hasler, Mar 31 2008

nonn

			This sequence starts 2,4,0,0,... since the 1st and 2nd terms of the continued fraction expansion of Pi, A001203 = (3, 7, 15, 1, ...) are the 2nd resp. 4th primes, while the next two terms are not primes.

Cf. A001203, A138757, A138759, A005042.

default(realprecision,1000); t=contfrac(Pi); vector(#t,i,isprime(t[i])*primepi(t[i]))

a(n) = A000720(A001203(n)) * A010051(A001203(n)).

A007541 Incrementally largest terms in the continued fraction for Pi-2 (cf. A001203).

Original entry on oeis.org

1, 7, 15, 292, 436, 20776, 78629, 179136, 528210, 12996958, 878783625, 5408240597, 5916686112, 9448623833, 9787547328, 52662113289

Offset: 1

N. J. A. Sloane

nonn

No larger term in the first 10,672,905,501 terms of the c.f. - Eric W. Weisstein, Jul 18 2013

R. W. Gosper, Jr., Table of the simple continued fraction for pi and the derived decimal approximation, Math. Comp., 31 (1977), 1044.
R. W. Gosper, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
See A001203 for many additional references and links.

H. Havermann, Simple Continued Fraction for Pi [a 483 MB file containing 180 million terms]
Eric Weisstein's World of Mathematics, Pi
Eric Weisstein's World of Mathematics, Pi Continued Fraction
Index entries for sequences related to the number Pi

Cf. A001203, A000796, A033090.

Apart from initial term, same as A033089.

upto=10^7;a={};r=0;f=ContinuedFraction[Pi-2,upto];Do[If[f[[i]]>r,AppendTo[a,r=f[[i]]]],{i, upto}];a (* Paolo Xausa, Nov 28 2021 *)

allocatemem(4096*10^6);
default(realprecision, 50000);
v = contfrac(Pi-2);
m = 0;
for (i=1, #v, if (v[i] > m, m = v[i]; print1(m, ", "))); \\ Michel Marcus, Aug 05 2017; to get 7 terms

Corrected (missing a(9) added) by Eric W. Weisstein, Dec 08 2010
a(12) from Eric W. Weisstein, Dec 08 2010
a(13) from Eric W. Weisstein, Sep 16 2011
a(14) from Eric W. Weisstein, Sep 17 2011
a(15) from Eric W. Weisstein, Jul 18 2013
a(6) corrected by Bobby Jacobs, Aug 05 2017
a(16) = A033089(16) from Jeppe Stig Nielsen, Nov 28 2021

A185591 Primes in continued fraction expansion of Pi (A001203).

Original entry on oeis.org

3, 7, 2, 3, 2, 2, 2, 2, 2, 2, 3, 13, 2, 2, 2, 3, 5, 7, 2, 3, 7, 2, 3, 2, 5, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 5, 2, 2, 5, 2, 2, 2, 2, 7, 3, 3, 7, 2, 7, 2, 3, 19, 2, 3, 7, 3, 3, 3, 2, 2, 2, 13, 2, 3, 3, 3, 2, 5, 3, 2, 2, 3, 23, 3, 7, 2, 2, 2, 127, 5, 3, 13, 7, 5, 3, 29, 3, 2, 2, 3, 3, 3, 3, 2, 7, 2, 11, 3, 7, 5, 2, 3, 2, 3, 2, 2, 2

Offset: 1

Jani Melik, Feb 04 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A185809.

Digits := 600: p1 := convert(Pi,confrac,300): select(isprime, p1);

Select[ContinuedFraction[Pi,500],PrimeQ] (* Harvey P. Dale, Jan 22 2012 *)

A185809 Composite numbers in continued fraction expansion of Pi (A001203).

Original entry on oeis.org

15, 292, 14, 84, 15, 4, 6, 6, 99, 6, 6, 8, 12, 8, 6, 4, 4, 16, 161, 45, 22, 4, 24, 10, 4, 8, 26, 4, 8, 42, 4, 9, 57, 18, 9, 18, 30, 8, 15, 4, 12, 28, 10, 20, 4, 6, 4, 120, 15, 16, 21, 9, 6, 4, 14, 9, 4, 10, 12, 4, 4, 48, 16, 4, 4, 20, 4, 436, 8, 6, 4, 4, 6, 9, 15, 24, 4, 4, 4, 6, 4, 58, 15, 4, 8, 4, 9, 4, 15, 24, 4, 10, 12, 21, 34, 4, 15, 4, 44, 4, 20776, 94, 55, 32, 14, 50, 16, 4, 6, 28, 4, 4

Offset: 1

Jani Melik, Feb 04 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A185591.

ts_composite := proc(n)
`if`(isprime(n) or n=1, true, false);
end proc:
Digits := 1000: p1 := convert(Pi,confrac,500): remove(ts_composite, p1);

Select[ContinuedFraction[\[Pi],500],#!=1&&!PrimeQ[#]&] (* Harvey P. Dale, May 02 2011 *)

A080407 Decimal expansion of the number which results when the Boustrophedon transform of the continued fraction of Pi (A080406, A001203) is interpreted as a continued fraction.

Original entry on oeis.org

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

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 17 2003

			3.0996886064030483425267288917220886641288...

J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (1996) 44-54 (Abstract, pdf, ps).
N. J. A. Sloane, Transforms
Index entries for sequences related to boustrophedon transform

Cf. A001203, A000796, A080406.

Offset corrected by R. J. Mathar, Feb 05 2009

A059101 Number of terms of the fractional part of A001203 for which the geometric mean produces increasingly better approximations to Khinchin's constant.

Original entry on oeis.org

1, 3, 7, 8, 9, 10, 11, 15, 16, 17, 97, 100, 103, 117, 976, 32307, 32760, 32787, 60508, 60601, 60663, 187154, 230084, 1120375, 1146529, 2211732, 4497058, 1434927965, 1434935064, 1434935232, 1434935281, 1471575921, 1471636101, 1490844937, 1491643951, 1498931686

Offset: 1

Hans Havermann, Feb 13 2001

nonn

Next term > 3*10^10. - Hans Havermann, Jul 29 2024

The geometric mean of 1498931686 terms is Khinchin + 1.00240496*10^-13.

			The geometric mean of 17 terms (Khinchin + 0.00752006) is not bettered until we calculate the geometric mean of 97 terms (Khinchin - 0.00326655).

Hans Havermann, Simple Continued Fraction for Pi

Cf. A001203, A048613.

p = Rest[{A001203}]; q = N[1, 100]; r = p[[1]] + 1; t = {}; Do[q = q*p[[i]]; g = q^(1/i) - Khinchin; If[Abs[g] < r, r = Abs[g]; t = Append[t, i]], {i, 1, Length[p]}]; t

a(28)-a(36) from Hans Havermann, Dec 27 2012

cp's OEIS Frontend

A032523 Index of first occurrence of n as a term in A001203, the continued fraction for Pi.

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A138759 Indices for which A001203 (continued fraction for Pi) is prime.

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A080406 Boustrophedon transform of the continued fraction of Pi (cf. A001203).

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A107892 Index of first occurrence of n-th prime in A001203, the continued fraction for Pi.

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A138758 Index of A001203(n) (continued fraction for Pi) in A000040 (primes), or 0 if A001203(n) is not prime.

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A007541 Incrementally largest terms in the continued fraction for Pi-2 (cf. A001203).

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Mathematica

PARI

Extensions

A185591 Primes in continued fraction expansion of Pi (A001203).

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Maple

Mathematica

A185809 Composite numbers in continued fraction expansion of Pi (A001203).

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Maple

Mathematica

A080407 Decimal expansion of the number which results when the Boustrophedon transform of the continued fraction of Pi (A080406, A001203) is interpreted as a continued fraction.