A138335 -id:A138335 - cp's OEIS frontend

A138336 First occurrence of exactly n consecutive numbers after A138335(n) (A138335(n) excluded).

18, 27, 127, 1111

Offset: 1

Artur Jasinski, Mar 15 2008

Such number digits after decimal point of number Pi where the approximation to the number Pi by a root of a polynomial of 2 degree does not improve the accuracy exactly n next digits.

It seems that sequence A138335, and therefore also this sequence, are ill defined. - M. F. Hasler, May 21 2025

			a(1)=18 because 3.141592653589793238 (18 digits) is root of -3061495 + 674903*x + 95366*x^2 and 3.1415926535897932385 (19 digits) also is root of that same polynomial -3061495 + 674903*x + 95366*x^2.

Cf. A138335, A138337.

There are comments in the entry for A138335 which may cast doubt on the validity of this and related sequences. - N. J. A. Sloane, Mar 19 2008

A138339 Smallest index in A138343 which starts a plateau of n consecutive integers.

Original entry on oeis.org

0, 7, 4, 12

Offset: 1

Artur Jasinski, Mar 16 2008

The sequence provides an indication of regions of nearly no improvements of the sequence of convergents to Pi when measured by the count of correct digits after rounding. A "plateau" of n consecutive integers is defined by a(n) = j such that A138343(j) = A138343(j+n-1) and such that none of the intermediate A138343 with index j+1 up to j+n-2 drop below this level.

			a(3)=4 because the A138343(4) to A138343(6) are the earliest plateau of length 3.
The fact that there is a local maximum at A138343(5)=9 above the plateau is consistent with the definition.

Cf. A138335, A138336, A138337, A138343.

Definition and values adapted to redefined A138343 - R. J. Mathar, Oct 01 2009

A138343 Count of post-period decimal digits up to which the rounded n-th convergent to Pi agrees with the exact value.

Original entry on oeis.org

0, 2, 3, 6, 8, 9, 8, 10, 10, 11, 11, 13, 15, 15, 16, 15, 17, 17, 18, 19, 20, 23, 24, 23, 26, 27, 29, 30, 29, 31, 33, 34, 37, 39, 39, 40, 42, 43, 44, 45, 45, 47, 46, 49, 49, 51, 52, 52, 54, 55, 56, 55, 56, 57, 59, 58, 59, 60, 61, 61, 63, 64, 64, 65, 65, 66, 67, 67, 68, 69, 70, 71, 72, 72

Offset: 0

Artur Jasinski, Mar 16 2008

This is a measure of the quality of the n-th convergent to A000796 if the convergent and the exact value are compared rounded to an increasing number of digits. (This is similar to A084407 which compares the truncated/floored values).
The sequence of rounded values of Pi is 3, 3.1, 3.14, 3.142, 3.1416, 3.14159, 3.141593, 3.1415927 etc, and the n-th convergent (provided by A002485 and A002486) is to be represented by its equivalent sequence.
a(n) represents the maximum number of post-period digits of the two sequences if compared at the same level of rounding. Counting only post-period digits (which is one less than the full number of decimal digits) is just a convention taken from A084407.

			For n=3, the 3rd convergent is 355/113 = 3.141592920353.., with a sequence of rounded representations 3, 3.1, 3.14, 3.142, 3.1416, 3.141593, 3.1415929, 3.14159292 etc.
Rounded to 1, 2, 3, 4, 5 or 6 post-period decimal digits, this is the same as the rounded version of the exact Pi, but disagrees if both are rounded to 7 decimal digits, where 3.1415927 <> 3.1415929.
So a(3) = 6 (digits), the maximum rounding level of agreement.

Cf. A138335, A138336, A138337, A138339.

Definition and values replaced as defined via continued fractions by R. J. Mathar, Oct 01 2009

A138337 Positions of digits after decimal point of number Pi where the approximation to the number Pi by a root of a polynomial of 3 degree does not improve the accuracy.

Original entry on oeis.org

7, 13, 17, 30, 37, 48, 62, 63, 77, 81, 86, 92, 97, 114, 117, 125, 129, 143, 148, 152, 156, 159, 168, 174, 180, 185, 196, 200, 204, 211, 227, 235, 244, 247, 259, 266, 267, 282

Offset: 1

Artur Jasinski, Mar 15 2008

dead

If there is a set of consecutive numbers in this sequence starting at k, this means that k-1 is a good approximation to Pi.
If the set of successive integers is longer that approximation k-1 better (see A138338)
This sequence appears to be ill defined: There are many different polynomials of degree 3 that give an approximation of Pi with the same precision, and any such approximation to n+1 digits is also an approximation of Pi to n digits, so the sequence should be empty. - M. F. Hasler, May 21 2025

			a(1)=7 because 3.141593 (6 digits) is root of cubic 2 + 29 x - 22 x^2 + 4 x^3 and 3.1415927 (7 digits) also is root of that same polynomial -3061495+674903*x+95366*x^2

Cf. A138335, A138336, A138338.

b = {}; a = {}; Do[k = Recognize[N[Pi,n + 1], 3, x]; If[MemberQ[a, k], AppendTo[b, n], AppendTo[a, k]], {n, 2, 300}]; b

A138369 Count of post-period decimal digits up to which the rounded n-th convergent to 4sin(4Pi/5) agrees with the exact value.

Original entry on oeis.org

0, 2, 2, 3, 4, 4, 6, 6, 7, 8, 10, 12, 13, 14, 14, 16, 17, 18, 19, 19, 23, 25, 26, 28, 27, 29, 31, 31, 33, 35, 37, 38, 38, 39, 40, 41, 41, 42, 42, 45, 45, 48, 50, 51, 51, 52, 54, 54, 55, 56, 57, 57, 61, 65, 66, 67, 68, 69, 70, 71, 71, 72, 73, 72, 75, 75, 76, 77, 77, 78, 79, 80, 81, 81, 83

Offset: 2

Artur Jasinski, Mar 17 2008

This is a measure of the quality of the n-th convergent to 4*sin(4*Pi/5) = sqrt(2)*sqrt(5-sqrt(5)) = 2.351141009169892... if the convergent and the exact value are compared rounded to an increasing number of digits.
The sequence of rounded values of the sine (or square root) is 2, 2.4, 2.35, 2.351, 2.3511, 2.35114, 2.351141, 2.3511410 etc. The n-th convergents are 5/2 (n=1), 7/3 (n=2), 40/17 (n=3), 47/20, 87/37, 221/94, 308/131 etc. and are represented by their equivalent rounding sequence.
a(n) is the maximum number of post-period digits of the two rounding sequences if compared at the same level of rounding. Counting only post-period digits (which is one less than the total number of decimal digits) is just a convention taken from A084407.

			For n=4, the 4th convergent is 47/20 = 2.350000000..., with a sequence of rounded representations 2, 2.4, 2.35, 2.350, 2.3500, 2.35000, etc.
Rounded to 1 or 2 post-period decimal digits, this is the same as the rounded version of the exact square root, but disagrees if both are rounded to 3 decimal digits, where 2.351 <> 2.350.
So a(4) = 2 (digits), the maximum rounding level of agreement.

Cf. A138335, A138336, A138337, A138339, A138343, A138366, A138367, A138369, A138370.

Definition and values replaced as defined via continued fractions by R. J. Mathar, Oct 01 2009

A138366 Count of post-period decimal digits up to which the rounded n-th convergent to exp(1) agrees with the exact value.

Original entry on oeis.org

0, 1, 0, 1, 2, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 12, 12, 13, 14, 16, 15, 16, 19, 18, 20, 22, 22, 24, 25, 25, 26, 27, 28, 30, 32, 32, 32, 35, 36, 36, 39, 39, 41, 43, 43, 44, 46, 46, 48, 50, 50, 52, 52, 54, 56, 57, 58, 59, 61, 61, 63, 65, 64, 67, 69, 69, 71, 72, 73, 74, 77, 77, 79, 80, 81, 83

Offset: 1

Artur Jasinski, Mar 17 2008

This is a measure of the quality of the n-th convergent to E = A001113 if the convergent and the exact value are compared rounded to an increasing number of digits.
The sequence of rounded values of exp(1) is 3, 2.7, 2.72, 2.718, 2.7183, 2.71828, 2.718282, 2.7182818 etc, and the n-th convergent (provided by A007676 and A007677) is to be represented by its equivalent sequence.
a(n) represents the maximum number of post-period digits of the two sequences if compared at the same level of rounding. Counting only post-period digits (which is one less than the full number of decimal digits) is just a convention taken from A084407.

			For n=6, the 6th convergent is 106/39 = 2.7179487.., with a sequence of rounded representations 3, 2.7, 2.72, 2.718, 2.7179, 2.71795, 2.717949, etc.
Rounded to 1, 2, or 3 post-period decimal digits, this is the same as the rounded version of the exact E, but disagrees if both are rounded to 4 decimal digits, where 2.7183 <> 2.7179.
So a(6) = 3 (digits), the maximum rounding level of agreement.

Cf. A138335, A138336, A138337, A138339, A138343, A138367, A138369, A138370.

Definition and values replaced as defined via continued fractions by R. J. Mathar, Oct 01 2009

A138367 Count of post-period decimal digits up to which the rounded n-th convergent to sqrt(5) agrees with the exact value.

Original entry on oeis.org

0, 2, 4, 5, 6, 7, 8, 10, 8, 12, 14, 14, 16, 18, 19, 20, 21, 23, 24, 24, 26, 28, 29, 30, 31, 33, 33, 34, 35, 37, 39, 40, 41, 42, 44, 44, 46, 47, 48, 49, 51, 53, 53, 55, 56, 57, 59, 60, 60, 61, 64, 65, 66, 68, 69, 70, 72, 73, 74, 75, 76, 77, 79, 80, 81, 83, 83, 85, 85, 88, 89, 90, 91, 92

Offset: 1

Artur Jasinski, Mar 17 2008

This is a measure of the quality of the n-th convergent to A002163 if the convergent and the exact value are compared rounded to an increasing number of digits.
The sequence of rounded values of sqrt(5) is 2, 2.2, 2.24, 2.236, 2.2361, 2.23607, 2.236068, 2.2360680 etc, and the n-th convergent (provided by A001077 and A001076) is to be represented by its equivalent sequence.
a(n) represents the maximum number of post-period digits of the two sequences if compared at the same level of rounding. Counting only post-period digits (which is one less than the full number of decimal digits) is just a convention taken from A084407.

			For n=3, the 3rd convergent is 161/72 = 2.236111111..., with a sequence of rounded representations 2, 2.2, 2.24, 2.236, 2.2361, 2.23611, 2.236111, 2.2361111 etc.
Rounded to 1, 2, 3, or 4 post-period decimal digits, this is the same as the rounded version of the exact sqrt(5), but disagrees if both are rounded to 5 decimal digits, where 2.23607 <> 2.23611.
So a(3) = 4 (digits), the maximum rounding level of agreement.

Cf. A138335, A138336, A138337, A138339, A138343, A138366, A138369, A138370.

Definition and values replaced as defined via continued fractions by R. J. Mathar, Oct 01 2009

A138370 Count of post-period decimal digits up to which the rounded n-th convergent to 4sin(2Pi/5) agrees with the exact value.

Original entry on oeis.org

2, 3, 4, 5, 5, 6, 6, 8, 9, 10, 11, 12, 12, 13, 14, 15, 15, 17, 18, 17, 19, 21, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 33, 34, 35, 36, 37, 38, 38, 40, 41, 42, 41, 42, 43, 45, 44, 46, 44, 47, 49, 49, 50, 52, 53, 54, 55, 57, 59, 60, 61, 63, 62, 65, 67, 67, 68, 70, 69, 70, 70, 71

Offset: 2

Artur Jasinski, Mar 17 2008

The computation of A138369 is repeated for 4*sin(2*Pi/5) = sqrt(2)*sqrt(5+sqrt(5))
= 3.80422606518061.. = 4*A019881.
The convergents are 19/5 (n=2), 175/46 (n=3), 544/143 (n=4), 719/189 (n=5), 2701/710 (n=6) etc.

			a(6)=5 because 2701/710 = 3.80422535... agrees with 3.8042260651.. if both are rounded up to 5 decimal digits (3.80423 = 3.80423), but disagrees at the level of rounding to 6 decimal digits (3.804226 <> 3.804225) or more.

Cf. A138335, A138336, A138337, A138339, A138343, A138366, A138367, A138369.

Definition and values replaced as defined via continued fractions - R. J. Mathar, Oct 01 2009

A138373 Count of post-period decimal digits up to which the rounded n-th convergent to sqrt(5)/2 agrees with the exact value.

Original entry on oeis.org

1, 3, 3, 5, 6, 8, 9, 10, 9, 13, 13, 15, 15, 18, 19, 20, 22, 22, 24, 25, 26, 27, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 43, 44, 45, 47, 47, 49, 50, 52, 53, 54, 55, 56, 58, 57, 60, 61, 62, 64, 64, 67, 68, 68, 71, 72, 73, 74, 75, 76, 78, 78, 80, 82, 83, 84, 85, 86, 88, 88, 90

Offset: 1

Artur Jasinski, Mar 17 2008

Defined equivalent to A138367 considering the constant 1.1188.. = 10*A020837 and its convergents 9/8 (n=1), 19/17 (n=2), 161/144 (n=3), 341/305 (n=4), 2889/2584 (n=5).

Cf. A138335 - A138339, A138343, A138366, A138367, A138369, A138370 - A138374.

Definition and values replaced as defined via continued fractions - R. J. Mathar, Oct 01 2009

A138374 Count of post-period decimal digits up to which the rounded n-th convergent to 2^(1/3) agrees with the exact value.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 6, 6, 8, 6, 10, 10, 12, 13, 15, 16, 17, 16, 18, 19, 20, 21, 21, 22, 23, 25, 27, 28, 29, 30, 31, 32, 35, 38, 39, 40, 39, 41, 42, 45, 46, 46, 47, 49, 51, 52, 52, 54, 56, 56, 57, 58, 58, 60, 61, 62, 63, 65, 64, 66, 68, 69, 69, 70, 70, 72, 74, 74, 75, 77, 79, 81

Offset: 1

Artur Jasinski, Mar 17 2008

This is a measure of the quality of the n-th convergent to the constant A002580 if the convergent and the exact value are compared rounded to an increasing number of digits. The sequence of rounded values of A002580 is 1, 1.3, 1.26, 1.260, 1.2599, 1.25992, 1.259921, 1.2599211 etc. The n-th convergents are taken from A002352 and A002351, each with associated rounded decimal expansions.

a(n) is the maximum number of post-period digits of the two expansions if compared at the same level of rounding.

			For n=5, the 5th convergent is 63/50 = 1.26000000.., with a sequence of rounded representations 1, 1.3, 1.26, 1.260, 1.2600, 1.26000, etc.
Rounded to 1, 2, or 3 post-period decimal digits, this is the same as the rounded version of the exact value, but disagrees if both are rounded to 4 decimal digits, where 1.2599 <> 1.2600.
So a(5) = 3 (digits), the maximum rounding level with agreement.

Cf. A138335 - A138339, A138343, A138366, A138367, A138369 - A138373.

Definition and values replaced as defined via continued fractions - R. J. Mathar, Oct 01 2009

cp's OEIS Frontend

A138336 First occurrence of exactly n consecutive numbers after A138335(n) (A138335(n) excluded).

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A138339 Smallest index in A138343 which starts a plateau of n consecutive integers.

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A138343 Count of post-period decimal digits up to which the rounded n-th convergent to Pi agrees with the exact value.

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A138337 Positions of digits after decimal point of number Pi where the approximation to the number Pi by a root of a polynomial of 3 degree does not improve the accuracy.

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A138369 Count of post-period decimal digits up to which the rounded n-th convergent to 4*sin(4*Pi/5) agrees with the exact value.

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A138366 Count of post-period decimal digits up to which the rounded n-th convergent to exp(1) agrees with the exact value.

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A138367 Count of post-period decimal digits up to which the rounded n-th convergent to sqrt(5) agrees with the exact value.

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A138370 Count of post-period decimal digits up to which the rounded n-th convergent to 4*sin(2*Pi/5) agrees with the exact value.

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A138373 Count of post-period decimal digits up to which the rounded n-th convergent to sqrt(5)/2 agrees with the exact value.

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A138374 Count of post-period decimal digits up to which the rounded n-th convergent to 2^(1/3) agrees with the exact value.

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A138369 Count of post-period decimal digits up to which the rounded n-th convergent to 4sin(4Pi/5) agrees with the exact value.

A138370 Count of post-period decimal digits up to which the rounded n-th convergent to 4sin(2Pi/5) agrees with the exact value.