A000030 -id:A000030 - cp's OEIS frontend

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

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

A162501 Lexicographically earliest permutation of the natural numbers such that in decimal representation the initial digit for each term is equal to the last nonzero digit of its predecessor; a(1)=1.

Original entry on oeis.org

1, 10, 11, 12, 2, 20, 21, 13, 3, 30, 31, 14, 4, 40, 41, 15, 5, 50, 51, 16, 6, 60, 61, 17, 7, 70, 71, 18, 8, 80, 81, 19, 9, 90, 91, 100, 101, 102, 22, 23, 32, 24, 42, 25, 52, 26, 62, 27, 72, 28, 82, 29, 92, 200, 201, 103, 33, 34, 43, 35, 53, 36, 63, 37, 73, 38, 83, 39, 93, 300, 301

Offset: 1

Reinhard Zumkeller, Jul 05 2009

A000030(a(n+1)) = A065881(a(n));
inverse of A162502: a(A162502(n)) = A162502(a(n)) = n;
a(a(n)) = A162503(n).

A076654, A130571.

A262356 a(1) = 1; for n > 1, let s denote the digit-string of a(n-1) with the first digit omitted. Then a(n) is the smallest number not yet present which starts with s, omitting leading zeros.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 20, 13, 30, 14, 40, 15, 50, 16, 60, 17, 70, 18, 80, 19, 90, 21, 100, 22, 23, 31, 101, 102, 24, 41, 103, 32, 25, 51, 104, 42, 26, 61, 105, 52, 27, 71, 106, 62, 28, 81, 107, 72, 29, 91, 108, 82, 200, 33, 34, 43, 35, 53

Offset: 1

Reinhard Zumkeller, Sep 19 2015

A simplified variation of A262282.
A permutation of the positive integers with inverse A262358;
A262363 and A262371 give the primes and where they occur: A262363(n)=a(A262371(n)).
a(A262393(n)) = A262390(n).
It seems clear that every number will appear, but it would be nice to have a formal proof. - N. J. A. Sloane, Sep 20 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A262283, A262282, A262358 (inverse), A262360 (fixed points), A262374 (binary counterpart), A262363 (primes), A262371, A000030, A262390 (starting with 1), A262393.

import Data.List (isPrefixOf, delete, genericIndex)
import Data.Set (singleton, notMember, insert)
a262356 n = a262356_list !! (n-1)
a262356_list = 1 : f "" (singleton "1") where
   f xs s = (read ys :: Int) : f (dropWhile (== '0') ys') (insert ys s)
     where ys@(_:ys') = head
             [vs | vs <- zss, isPrefixOf xs vs, notMember vs s]
   zss = map show [2..]

a[1] = 1; a[n_] := a[n] = Module[{s, k}, s = Rest[IntegerDigits[a[n - 1]]] //. {(0).., d__} :> {d}; For[k = 2, True, k++, If[FreeQ[Array[a, n - 1], k], If[s == {0}, Return[k], If[IntegerDigits[k][[1 ;; Length[s]]] == s, Return[k]]]]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 12 2019 *)

A376270 a(n) is the product of the leading digit of n and the sum of the squares of its digits.

Original entry on oeis.org

0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1, 2, 5, 10, 17, 26, 37, 50, 65, 82, 8, 10, 16, 26, 40, 58, 80, 106, 136, 170, 27, 30, 39, 54, 75, 102, 135, 174, 219, 270, 64, 68, 80, 100, 128, 164, 208, 260, 320, 388, 125, 130, 145, 170, 205, 250, 305, 370, 445, 530, 216, 222, 240, 270, 312, 366

Offset: 0

Michel Marcus, Sep 18 2024

Alois P. Heinz, Table of n, a(n) for n = 0..10000
N. Bradley Fox et al., Elated Numbers, arXiv:2409.09863 [math.NT], 2024.

Cf. A000030, A003132.

b-elated function: A000120 (2), A376270 (10).

a:= n-> (l-> l[-1]*add(i^2, i=l))(convert(n, base, 10)):
seq(a(n), n=0..65);  # Alois P. Heinz, Sep 18 2024

a[n_]:=First[d=IntegerDigits[n]]Norm[d]^2; Array[a,66,0] (* Stefano Spezia, Sep 18 2024 *)

a(n) = if (n, my(d=digits(n)); d[1]*norml2(d), 0);

def a(n): return (d:=list(map(int, str(n))))[0] * sum(di*di for di in d)
print([a(n) for n in range(66)]) # Michael S. Branicky, Sep 18 2024

a(n) = A000030(n)*A003132(n).

A040163 a(n) is the absolute value of (the first digit of n minus the last digit of n).

Original entry on oeis.org

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

Offset: 1

Felice Russo

			a(371) = abs(3 - 1) = 2.
a(567) = abs(5 - 7) = 2.

David F. Marrs, Table of n, a(n) for n = 1..10000

Cf. A000030 (first digit of n), A010879 (last digit of n).

Cf. A037904, A040114, A040115, A040997.

Array[Abs[First@ # - Last@ #] &@ IntegerDigits@ # &, 106] (* Michael De Vlieger, Oct 15 2018 *)

a(n) = my(digs = digits(n)); abs(digs[1] - digs[#digs]); \\ Michel Marcus, Sep 27 2013

apply( {A040163(n)=abs(n\10^logint(n+!n,10)-n%10)}, [0..111]) \\ M. F. Hasler, Apr 22 2024

for n in range(1,51): print(abs(int(str(n)[0])-int(str(n)[-1]))) # David F. Marrs, Oct 14 2018

a(n) = abs(floor(n / 10 ^ floor(log_10(n))) - (n - floor(n / 10) * 10)) - David F. Marrs, Oct 14 2018

cp's OEIS Frontend

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.

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

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A162501 Lexicographically earliest permutation of the natural numbers such that in decimal representation the initial digit for each term is equal to the last nonzero digit of its predecessor; a(1)=1.

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A262356 a(1) = 1; for n > 1, let s denote the digit-string of a(n-1) with the first digit omitted. Then a(n) is the smallest number not yet present which starts with s, omitting leading zeros.

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