cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A163580 Primes of the form floor(k+A000217(k-1)*Pi), Pi = A000796, k integer.

Original entry on oeis.org

5, 53, 151, 967, 1129, 2129, 2617, 2879, 4217, 4549, 6397, 6599, 7013, 7877, 8101, 8329, 9029, 10007, 10259, 11839, 12391, 26881, 30707, 35257, 35729, 36683, 37649, 41131, 41641, 49667, 50227, 56597, 63347, 71143, 75211, 79393, 82963, 102797
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 31 2009

Keywords

Links

Crossrefs

Cf. A163579

Programs

  • Mathematica
    s=0;lst={};Do[s+=n;p=IntegerPart[s];If[PrimeQ[p],AppendTo[lst,p]],{n, 1,7!,Pi}];lst
  • PARI
    for(n=1,500, m=floor(n + n*(n-1)*Pi/2); if(isprime(m), print1(m, ", "))) \\ G. C. Greubel, Jul 28 2017

Extensions

Definition clarified by R. J. Mathar, Aug 01 2009

A171059 a(n) is the lexically first sequence of distinct nonzero integers such that if S(n) is the string formed from the digits of a(1)a(2)...a(n), then dividing S(n) into substrings with lengths equal to the successive digits of S(n) (treating 0 as 10) results in substrings beginning with the successive digits of Pi (A000796).

Original entry on oeis.org

3, 1, 2, 14, 4, 15, 5, 6, 7, 9, 8, 10, 26, 11, 12, 50, 13, 23, 16, 17, 25, 18, 19, 20, 80, 21, 22, 24, 29, 27, 28, 30, 37, 90, 31, 32, 33, 34, 35, 200, 36, 43, 84, 60, 201, 38, 61, 39, 40, 41, 42, 430, 53, 48, 44, 320, 45, 46, 79, 47, 49, 51, 52
Offset: 1

Views

Author

N. J. A. Sloane, Sep 04 2010, based on a posting to the Sequence Fans Mailing List by Eric Angelini, Aug 24 2010

Keywords

Comments

Erase the punctuation:
S(Pi) = 312144155679810261112501323161725181920802122242927283037903132333435...
Divide into chunks -- the size of each chunk is given by the successive DIGITS of S(Pi):
312.1.44.1.5567.9810.2.61112.50132.316172.5181920.802122242.92728303.7.9031323334.35
(the "0" digits produce a 10-digit chunk)
Replace all dots (.) with carriage returns:
312
1
44
1
5567
9810
2
61112
50132
316172
5181920
802122242
92728303
7
9031323334
35
...
The first column shows Pi!
a(63) = 52 is the last term, a(64) would have to begin with a 0. - Charlie Neder, Jun 24 2018

Links

Extensions

a(40)-a(63) from Charlie Neder, Jun 24 2018

A181052 The sequence of numbers where the n-th term is (Pi^n - e^n) rounded down to the nearest integer, where Pi is the ratio of a circle's circumference to its diameter (A000796) and e is Euler's constant (A001113).

Original entry on oeis.org

0, 0, 2, 10, 42, 157, 557, 1923, 6507, 21706, 71621, 234329, 761514, 2461263, 7919566, 25389128, 81146110, 258689610, 822922433, 2613081648, 8284791600, 26232816108, 82971091345, 262178903447, 827784397784, 2611774514980, 8235612082447, 25955792878501
Offset: 0

Views

Author

Jonathan D. B. Hodgson, Oct 01 2010

Keywords

Examples

			A(0)=0, A(1)=0, A(2)=2 etc...

Programs

  • Mathematica
    Table[Floor[Pi^n-E^n],{n,0,40}] (* Harvey P. Dale, Jun 02 2015 *)

Formula

A(n) = floor(Pi^n - e^n)

Extensions

More terms from Harvey P. Dale, Jun 02 2015

A105790 Number of bisections to an inscribed triangle to approximate Pi (A000796) to n decimal digits of accuracy.

Original entry on oeis.org

1, 4, 4, 6, 8, 9, 11, 13, 14, 16, 17, 19, 21, 23, 25, 26, 27, 30, 31, 33, 34, 36, 38, 40, 41, 43, 45, 46, 47, 49, 53, 53, 54, 56, 58, 60, 61, 62, 65, 66, 67, 70, 71, 72, 75, 76, 78, 80, 83, 83, 84, 87, 89, 89, 91, 93, 94, 96, 98, 99, 100, 103, 105, 107, 107, 109, 112, 112
Offset: 1

Views

Author

Robert G. Wilson v, Apr 20 2005

Keywords

References

  • Howard Anton, Irl C. Bivens and Stephen L. Davis, Calculus, Early Transcendentals, 7th Edition, John Wiley & Sons, Inc., NY, Section 6.1 An Overview of the Area Problem, page 372-377.
  • William Dunham, The Calculus Gallery, Masterpieces from Newton to Lebesgue, Princeton University Press, Princeton, NJ 2005, page 56-57.

Crossrefs

Cf. A000796.

Programs

  • Mathematica
    $MaxExtraPrecision =128; p=RealDigits[ Pi, 10, 100][[1]]; f[n_] := 3*2^(n)*Sqrt[2 - Nest[ Sqrt[2 + # ] &, Sqrt[3], n - 1]]; g[n_] := Block[{k = 1, q = Take[p, n + 1]}, While[ Take[ RealDigits[ f[k], 10, 100][[1]], n + 1] != q, k++ ]; k]; Table[ g[n], {n, 69}]

Formula

a(n) = 3*2^n*sqrt(2- sqrt(2+ sqrt(2+ ... sqrt(2+ sqrt(3))...))).
A(n) in Table 6.1.1 = Sin( 2Pi/n )*n/2. - Anton.

A112602 Erroneous version of decimal expansion of Pi (see A000796 for the correct version).

Original entry on oeis.org

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

Views

Author

Keywords

References

  • Heard on Kate Bush's November 2005 Aerial.

A120584 Distance between n-1 and n in decimal expansion of Pi A000796.

Original entry on oeis.org

33, 5, 16, 11, 6, 20, 8, 1, 2, 21, 41, 11, 46, 61, 14, 19, 82, 32, 18, 49, 105, 50, 11, 37, 105, 19, 19, 136, 113, 34, 234, 77, 66, 166, 1, 51, 109, 98, 42, 19, 20, 154, 49, 26, 25, 44, 111, 93, 46, 40, 265, 77, 69, 182, 421, 286, 154, 9, 3, 24, 64, 22, 61, 69, 85, 21, 28
Offset: 0

Views

Author

Zak Seidov, Aug 17 2006

Keywords

Comments

a(0)=33 because position of first zero is 33; after this zero, "1" is at position 5, hence a(1)=5; after this 1, "2" is at position 16, hence a(2)=16; after this 2, "3" is at position 11, hence a(3)=11 etc.

Crossrefs

Cf. A000796.

Programs

  • Mathematica
    ts=ToString[FromDigits[RealDigits[N[Pi,20000]][[1]]]]; Reap[Do[sp=StringPosition[ts,ToString[n]][[1,1]];Sow[sp];ts=StringDrop[ts,sp],{n,0,100}]][[2,1]]

A130143 Gaps between consecutive primes in decimal expansion of Pi (A000796).

Original entry on oeis.org

6, 2, 0, 2, 80, 14, 26, 28, 59, 104, 104, 14, 268, 628, 11, 128, 16, 91, 143, 165, 36, 360, 498, 12, 67, 47
Offset: 1

Views

Author

Jani Melik, Aug 01 2007

Keywords

Examples

			a(1)=6 because first prime 2 occurs after 314159,
a(2)=2 because second prime 3 occurs after 26,
a(3)=0 because third prime 5 is adjacent to 3, ...

A130854 Runs of 1's of lengths 1 for decimal expansion of Pi (A000796), separated by 0's.

Original entry on oeis.org

0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1
Offset: 1

Views

Author

Jani Melik, Jul 21 2007

Keywords

Crossrefs

Cf. A093521.

A152575 A triangle of coefficients of polynomials with roots as the Pi-digits base ten A000796(n)=d(n):d(1)=3; p(x,n)=-d(1)*Product[x-d(m),{m,2,n}].

Original entry on oeis.org

-3, 3, -3, -12, 15, -3, 12, -27, 18, -3, -60, 147, -117, 33, -3, 540, -1383, 1200, -414, 60, -3, -1080, 3306, -3783, 2028, -534, 66, -3, 6480, -20916, 26004, -15951, 5232, -930, 84, -3, -32400, 111060, -150936, 105759, -42111, 9882, -1350, 99, -3, 97200
Offset: 1

Views

Author

Roger L. Bagula, Dec 08 2008

Keywords

Comments

p(x,n)=Product[(x-i)^a(i),{i,0,9}]: a(i) is the count number of the
way the digits occur by the n-th digit.
The limiting polynomial is:
pL(x)=Product[(x-i),{i,0,9}]=
-362880 x + 1026576 x^2 - 1172700 x^3 + 723680 x^4 - 269325 x^5 + 63273 x^6 - 9450 x^7 + 870 x^8 - 45 x^9 + x^10;
since if the digits occur equally:
p(x,Infinity)=-3*Product[(x-i),{i,0,9}]^(Infinity/10).
Or at the n-th digits equality:
p(x,n)=-3*Product[(x-i),{i,0,9}]^(n/10).
The n+1 digit:
p(x,n+1)=p(x,n)*(x-d(n+1)).

Examples

			{-3},
{3, -3},
{-12,15, -3},
{12, -27, 18, -3},
{-60, 147, -117, 33, -3},
{540, -1383, 1200, -414, 60, -3},
{-1080, 3306, -3783, 2028, -534, 66, -3},
{6480, -20916, 26004, -15951, 5232, -930, 84, -3},
{-32400, 111060, -150936, 105759, -42111, 9882, -1350, 99, -3},
{97200, -365580, 563868, -468213, 232092, -71757, 13932, -1647, 108, -3}

Crossrefs

A000796

Programs

  • Mathematica
    Clear[a, p, n, m];
a = Delete[Flatten[RealDigits[Pi, 10, 100]], 100];
p[x_, n_] := If[n == 1, -a[[1]], -a[[1]]*Product[x - a[[m]], {m, 2, n}]];
Table[CoefficientList[p[x, n], x], {n, 1, 10}]
Flatten[%]

Formula

Pi-digits base ten A000796(n)=d(n):
p(x,n)=-d(1)*Product[x-d(m),{m,2,n}].

A160262 a(n)=c-th digit of A000796(n) where c=n-th composite.

Original entry on oeis.org

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

Views

Author

Vladislav-Stepan Malakhovsky and Juri-Stepan Gerasimov, May 07 2009

Keywords

Comments

A000796 is the decimal expansion of Pi.

Crossrefs

Cf. A000796, A002808, A090201.
Previous Showing 21-30 of 1164 results. Next

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