A105501 -id:A105501 - cp's OEIS frontend

A105511 Number of times 1 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

0, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 7, 7, 7, 7, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 12, 13, 13, 13, 13, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18, 18, 18, 19, 20, 20, 20, 20, 21, 22, 22, 22, 22, 23, 23, 23, 23, 23, 24

Offset: 0

Reinhard Zumkeller, Apr 11 2005

Winston de Greef, Table of n, a(n) for n = 0..10000

Cf. A000030, A000045, A008963, A105501.

Cf. A105512, A105513, A105514, A105515, A105516, A105517, A105518, A105519.

Accumulate[Table[If[IntegerDigits[Fibonacci[n]][[1]] == 1, 1, 0], {n, 0, 100}]] (* Amiram Eldar, Jan 12 2023 *)

(leadingdigit(n, b=10) = n \ 10^logint(n, b));
(isok(n) = leadingdigit(fibonacci(n))==1);
(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

a(n) = #{k: A008963(k) = 1 and 0<=k<=n};
a(A105501(n)) = a(A105501(n) - 1) + 1;
n = a(n) + A105512(n) + A105513(n) + A105514(n) + A105515(n) + A105516(n) + A105517(n) + A105518(n) + A105519(n).
a(n) ~ log_10(2) * n. - Amiram Eldar, Jan 12 2023

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

Reinhard Zumkeller, Apr 11 2005

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

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

Winston de Greef, Table of n, a(n) for n = 1..10000

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

Select[Range[400],First[IntegerDigits[Fibonacci[#]]]==2&] (* Harvey P. Dale, Jul 13 2015 *)

is(n)=digits(fibonacci(n))[1]==2 \\ Charles R Greathouse IV, Oct 07 2016

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

A105505 Numbers n such that 5 is the leading digit of the n-th Fibonacci number in decimal representation.

Original entry on oeis.org

5, 10, 29, 34, 53, 58, 77, 96, 101, 120, 125, 139, 144, 163, 168, 187, 192, 206, 211, 230, 235, 254, 273, 278, 297, 302, 321, 340, 345, 364, 369, 388, 407, 412, 431, 436, 455, 474, 479, 498, 503, 522, 541, 546, 565, 570, 584, 589, 608, 613, 632, 637, 651, 656

Offset: 1

Reinhard Zumkeller, Apr 11 2005

A008963(a(n)) = 5; A105515(a(n)) = A105515(a(n) - 1) + 1.

			a(10)=120: A008963(120) = A000030(A000045(120)) =
A000030(5358359254990966640871840) = 5.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000030, A000045, A072703, A105501, A105502, A105503, A105504, A105506, A105507, A105508, A105509.

ld:= x -> floor(x/10^ilog10(x)):
select(n -> ld(combinat:-fibonacci(n))=5, [$1..1000]); # Robert Israel, Oct 26 2020

Select[Range[700],First[IntegerDigits[Fibonacci[#]]]==5&] (* Harvey P. Dale, Jul 31 2018 *)

is(n)=digits(fibonacci(n))[1]==5 \\ Charles R Greathouse IV, Oct 07 2016

a(n) ~ kn by the equidistribution theorem, where k = log(10)/(log(6) - log(5)) = 12.629253.... - Charles R Greathouse IV, Oct 07 2016

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

Reinhard Zumkeller, Apr 11 2005

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

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

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

Select[Table[{n,Fibonacci[n]},{n,450}],First[IntegerDigits[#[[2]]]]==3&][[All,1]] (* Harvey P. Dale, Apr 13 2019 *)

is(n)=digits(fibonacci(n))[1]==3 \\ Charles R Greathouse IV, Oct 07 2016

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

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

Reinhard Zumkeller, Apr 11 2005

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

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

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

is(n)=digits(fibonacci(n))[1]==4 \\ Charles R Greathouse IV, Oct 07 2016

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

Reinhard Zumkeller, Apr 11 2005

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

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

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

is(n)=digits(fibonacci(n))[1]==6 \\ Charles R Greathouse IV, Oct 07 2016

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

Reinhard Zumkeller, Apr 11 2005

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

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

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

Flatten[Position[Fibonacci[Range[1000]],?(IntegerDigits[#][[1]]==7&)]] (* _Harvey P. Dale, Nov 20 2015 *)

is(n)=digits(fibonacci(n))[1]==7 \\ Charles R Greathouse IV, Oct 07 2016

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

Reinhard Zumkeller, Apr 11 2005

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

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

Flatten[Position[Fibonacci[Range[1000]],?(First[IntegerDigits[#]]==8&)]] (* _Harvey P. Dale, Jan 04 2015  *)

isok(n) = digits(fibonacci(n))[1] == 8; \\ Michel Marcus, Jan 10 2014

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

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

Reinhard Zumkeller, Apr 11 2005

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...."

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

M. Bicknell-Johnson, A generalized magic trick from Fibonacci: Designer decimals, College Mathematics Journal 35(March):125-126, 2004.
O-Y. Chan and J. Smoak, More designer decimals: The integers and their geometric extensions College Mathematics Journal 37(November):355-363, 2006.
Ivars Peterson, Designer Decimals, Science News, Week of Nov 04 2006; Vol. 170, No. 19.
J. Smoak, and T.J. Osler, A magic trick from Fibonacci. College Mathematics Journal, 34 (2003):58-60.

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

Select[Range@ 1200, First@ IntegerDigits@ Fibonacci@ # == 9 &] (* Michael De Vlieger, Aug 21 2016 *)

is(n)=digits(fibonacci(n))[1]==9 \\ Charles R Greathouse IV, Oct 07 2016

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

A045725 Fibonacci numbers having initial digit '1'.

Original entry on oeis.org

1, 13, 144, 1597, 10946, 17711, 121393, 196418, 1346269, 14930352, 102334155, 165580141, 1134903170, 1836311903, 12586269025, 139583862445, 1548008755920, 10610209857723, 17167680177565, 117669030460994, 190392490709135, 1304969544928657, 14472334024676221, 160500643816367088

Offset: 1

Jeff Burch

Cf. A105501.

Intersection of A000045 and A131835.

[Fibonacci(n): n in [2..100] | Intseq(Fibonacci(n))[#Intseq(Fibonacci(n))] eq 1]; // Vincenzo Librandi, Jan 30 2019

Select[Fibonacci[Range[2, 100]], IntegerDigits[#, 10][[1]] == 1 &] (* T. D. Noe, Nov 01 2006 *)

select(x->(digits(x)[1] == 1), vector(85, n, fibonacci(n+1))) \\ Michel Marcus, Jan 30 2019

def fibonacci(n: BigInt): BigInt = {
  val zero = BigInt(0)
  def fibTail(n: BigInt, a: BigInt, b: BigInt): BigInt = n match {
    case `zero` => a
    case _ => fibTail(n - 1, b, a + b)
  }
  fibTail(n, 0, 1)
} // Tail recursion by Dario Carrrasquel
((2 to 100).map(fibonacci())).filter(.toString.startsWith("1")) // Alonso del Arte, Apr 22 2019

Corrected by T. D. Noe, Nov 01 2006

cp's OEIS Frontend

A105511 Number of times 1 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

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

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A105505 Numbers n such that 5 is the leading digit of the n-th Fibonacci number in decimal representation.

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

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

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

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

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