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.

Showing 1-9 of 9 results.

A105501 Numbers n such that 1 is the leading digit of the n-th Fibonacci number in decimal representation.

Original entry on oeis.org

1, 2, 7, 12, 17, 21, 22, 26, 27, 31, 36, 40, 41, 45, 46, 50, 55, 60, 64, 65, 69, 70, 74, 79, 84, 88, 89, 93, 94, 98, 103, 107, 108, 112, 113, 117, 122, 127, 131, 132, 136, 137, 141, 146, 151, 155, 156, 160, 161, 165, 170, 174, 175, 179, 180, 184, 189, 194, 198, 199
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 11 2005

Keywords

Comments

A008963(a(n)) = 1; A105511(a(n)) = A105511(a(n) - 1) + 1.

Examples

			a(10)=31: A008963(31) = A000030(A000045(31)) =
A000030(1346269) = 1.

Links

Crossrefs

Cf. A000030, A000045, A072675, A105502, A105503, A105504, A105505, A105506, A105507, A105508, A105509.

Programs

  • Maple
    filter:= proc(n) local t;
  t:= combinat:-fibonacci(n);
  t < 2*10^ilog10(t)
end proc:
select(filter, [$1..200]); # Robert Israel, May 02 2018
  • Mathematica
    fQ[n_] := IntegerDigits[Fibonacci[n]][[1]] == 1; Select[Range@200, fQ] (* Robert G. Wilson v, May 02 2018 *)
  • PARI
    is(n)=digits(fibonacci(n))[1]==1 \\ Charles R Greathouse IV, Oct 07 2016

Formula

a(n) ~ kn by the equidistribution theorem, where k = log(10)/log(2) = 3.321928.... - Charles R Greathouse IV, Oct 07 2016

A105502 Numbers m such that 2 is the leading digit of the m-th Fibonacci number in decimal representation.

Original entry on oeis.org

3, 8, 13, 18, 23, 32, 37, 42, 47, 51, 56, 61, 66, 75, 80, 85, 90, 99, 104, 109, 114, 118, 123, 128, 133, 142, 147, 152, 157, 166, 171, 176, 185, 190, 195, 200, 209, 214, 219, 224, 233, 238, 243, 252, 257, 262, 267, 276, 281, 286, 291, 295, 300, 305, 310, 319
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 11 2005

Keywords

Comments

A008963(a(n)) = 2; A105512(a(n)) = A105512(a(n) - 1) + 1.

Examples

			a(10)=51: A008963(51) = A000030(A000045(51)) = A000030(20365011074) = 2.

Links

Crossrefs

Cf. A000030, A000045, A072682, A105501, A105503, A105504, A105505, A105506, A105507, A105508, A105509.

Programs

Formula

a(n) ~ kn by the equidistribution theorem, where k = log(10)/(log(3) - log(2)) = 5.67887.... - Charles R Greathouse IV, Oct 07 2016

A105515 Number of times 5 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, Apr 11 2005

Keywords

Links

Crossrefs

Cf. A000030, A000045, A008963, A105505.
Cf. A105511, A105512, A105513, A105514, A105516, A105517, A105518, A105519.

Programs

  • Mathematica
    Accumulate[If[First[IntegerDigits[#]]==5,1,0]&/@Fibonacci[Range[0,110]]] (* Harvey P. Dale, Nov 02 2014 *)
  • PARI
    (leadingdigit(n, b=10) = n \ 10^logint(n, b));
(isok(n) = leadingdigit(fibonacci(n))==5);
(lista(n)=my(a=vector(1+n), r=0); for (i=1, n, r+=isok(i); a[1+i]=r); a) \\ Winston de Greef, Mar 17 2023

Formula

a(n) = #{k: A008963(k) = 5 and 0<=k<=n};
a(A105505(n)) = a(A105505(n) - 1) + 1;
n = A105511(n) + A105512(n) + A105513(n) + A105514(n) + a(n) + A105516(n) + A105517(n) + A105518(n) + A105519(n).
a(n) ~ log_10(6/5) * n. - Amiram Eldar, Jan 12 2023

A105503 Numbers n such that 3 is the leading digit of the n-th Fibonacci number in decimal representation.

Original entry on oeis.org

4, 9, 14, 28, 33, 38, 52, 57, 71, 76, 81, 95, 100, 105, 119, 124, 138, 143, 148, 162, 167, 172, 181, 186, 191, 205, 210, 215, 229, 234, 239, 248, 253, 258, 272, 277, 282, 296, 301, 306, 315, 320, 325, 339, 344, 349, 363, 368, 382, 387, 392, 406, 411, 416, 430
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 11 2005

Keywords

Comments

A008963(a(n)) = 3; A105513(a(n)) = A105513(a(n) - 1) + 1.

Examples

			a(10)=76: A008963(76) = A000030(A000045(76)) =
A000030(3416454622906707) = 3.

Crossrefs

Cf. A000030, A000045, A072683, A105501, A105502, A105504, A105505, A105506, A105507, A105508, A105509.

Programs

  • Mathematica
    Select[Table[{n,Fibonacci[n]},{n,450}],First[IntegerDigits[#[[2]]]]==3&][[All,1]] (* Harvey P. Dale, Apr 13 2019 *)
  • PARI
    is(n)=digits(fibonacci(n))[1]==3 \\ Charles R Greathouse IV, Oct 07 2016

Formula

a(n) ~ kn by the equidistribution theorem, where k = log(10)/(log(4) - log(3)) = 8.00392.... - Charles R Greathouse IV, Oct 07 2016

Extensions

Definition clarified by Harvey P. Dale, Apr 13 2019

A105504 Numbers m such that 4 is the leading digit of the n-th Fibonacci number in decimal representation.

Original entry on oeis.org

19, 24, 43, 48, 62, 67, 72, 86, 91, 110, 115, 129, 134, 153, 158, 177, 182, 196, 201, 220, 225, 244, 249, 263, 268, 287, 292, 311, 316, 330, 335, 354, 359, 373, 378, 383, 397, 402, 421, 426, 440, 445, 450, 464, 469, 488, 493, 507, 512, 517, 531, 536, 555, 560
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 11 2005

Keywords

Comments

A008963(a(n)) = 4; A105514(a(n)) = A105514(a(n) - 1) + 1.

Examples

			a(10)=110: A008963(110) = A000030(A000045(110)) =
A000030(43566776258854844738105) = 4.

Crossrefs

Cf. A000030, A000045, A072702, A105501, A105502, A105503, A105505, A105506, A105507, A105508, A105509.

Programs

Formula

a(n) ~ kn by the equidistribution theorem, where k = log(10)/(log(5) - log(4)) = 10.318851.... - Charles R Greathouse IV, Oct 07 2016

A105506 Numbers m such that 6 is the leading digit of the n-th Fibonacci number in decimal representation.

Original entry on oeis.org

15, 20, 39, 63, 82, 87, 106, 130, 149, 154, 173, 197, 216, 221, 240, 259, 264, 283, 288, 307, 326, 331, 350, 355, 374, 393, 398, 417, 422, 441, 460, 465, 484, 508, 527, 532, 551, 575, 594, 599, 618, 642, 661, 666, 685, 709, 728, 733, 752, 771, 776, 795, 800
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 11 2005

Keywords

Comments

A008963(a(n)) = 6; A105516(a(n)) = A105516(a(n) - 1) + 1.

Examples

			a(10)=154: A008963(154) = A000030(A000045(154)) =
A000030(68330027629092351019822533679447) = 6.

Crossrefs

Cf. A000030, A000045, A072708, A105501, A105502, A105503, A105504, A105505, A105507, A105508, A105509.

Programs

Formula

a(n) ~ kn by the equidistribution theorem, where k = log(10)/(log(7) - log(6)) = 14.9372.... - Charles R Greathouse IV, Oct 07 2016

A105507 Numbers m such that 7 is the leading digit of the n-th Fibonacci number in decimal representation.

Original entry on oeis.org

25, 44, 49, 68, 92, 111, 116, 135, 159, 178, 183, 202, 226, 245, 250, 269, 293, 312, 317, 336, 360, 379, 384, 403, 427, 446, 470, 489, 494, 513, 537, 556, 561, 580, 604, 623, 628, 647, 671, 690, 695, 714, 738, 757, 762, 781, 805, 824, 829, 848, 872, 891, 915
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 11 2005

Keywords

Comments

A008963(a(n)) = 7; A105517(a(n)) = A105517(a(n) - 1) + 1.

Examples

			a(10)=178: A008963(178) = A000030(A000045(178)) =
A000030(7084593923980518516849609894969925639) = 7.

Crossrefs

Cf. A000030, A000045, A072709, A105501, A105502, A105503, A105504, A105505, A105506, A105508, A105509.

Programs

Formula

a(n) ~ kn by the equidistribution theorem, where k = log(10)/(log(8) - log(7)) = 17.24377.... - Charles R Greathouse IV, Oct 07 2016

A105508 Numbers m such that 8 is the leading digit of the m-th Fibonacci number in decimal representation.

Original entry on oeis.org

6, 11, 30, 54, 73, 78, 97, 121, 140, 145, 164, 188, 207, 231, 255, 274, 298, 322, 341, 365, 389, 408, 432, 451, 456, 475, 499, 518, 523, 542, 566, 585, 590, 609, 633, 652, 676, 700, 719, 743, 767, 786, 810, 834, 853, 877, 896, 901, 920, 944, 963, 968, 987
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 11 2005

Keywords

Examples

			a(1)=6 since the 6th Fibonacci: 8 begins with 8.
a(2)=11 since the 11th Fibonacci: 89 begins with 8.

Crossrefs

Cf. A000030, A000045, A072710, A105501, A105502, A105503, A105504, A105505, A105506, A105507, A105509.

Programs

Formula

A008963(a(n)) = A000030(A000045(a(n))) = 8.
A105518(a(n)) = A105518(a(n) - 1) + 1.
A000045(a(n)) = A045732(n).
a(n) ~ kn by the equidistribution theorem, where k = log(10)/(log(9) - log(8)) = 19.549378.... - Charles R Greathouse IV, Oct 07 2016

Extensions

Example and formulas edited by Michel Marcus, Jan 10 2014

A105509 Numbers m such that 9 is the leading digit of the m-th Fibonacci number in decimal representation.

Original entry on oeis.org

16, 35, 59, 83, 102, 126, 150, 169, 193, 212, 236, 260, 279, 303, 327, 346, 370, 394, 413, 437, 461, 480, 504, 528, 547, 571, 595, 614, 638, 657, 681, 705, 724, 748, 772, 791, 815, 839, 858, 882, 906, 925, 949, 973, 992, 1016, 1040, 1059, 1083, 1102, 1107
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 11 2005

Keywords

Comments

A008963(a(n)) = 9; A105519(a(n)) = A105519(a(n) - 1) + 1.
Comment from Jonathan Vos Post, Dec 23 2006: Peterson says: "Calculate 100/89 = 1.1235955056... This fraction generates the first five Fibonacci numbers before blurring into other digits. ... 10000/9899 = 1.0102030508132134559046368... generates the first 10 Fibonacci numbers (using two digits per number). 1000000/998999 generates the first 15 Fibonacci numbers (using three digits per number). ... in successive fractions, two 0's are appended to the numerator and a 9 to the beginning and end of the denominator...."

Examples

			a(10)=21: A008963(212) = A000030(A000045(212)) =
A000030(90343046356137747723758225621187571439538669) = 9.

Links

Crossrefs

Cf. A000030, A000045, A072711, A105501, A105502, A105503, A105504, A105505, A105506, A105507, A105508.

Programs

Formula

m such that d(m+5)-d(m) = 2 for d(m) = floor(1 + log_10(F(m))) and F(m) = m-th Fibonacci number = A000045(m). - Jonathan Vos Post, Dec 23 2006
a(n) ~ k*n by the equidistribution theorem, where k = 1/(1 - log(9)/log(10)) = 21.8543.... - Charles R Greathouse IV, Oct 07 2016
Showing 1-9 of 9 results.

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