A000030 -id:A000030 - cp's OEIS frontend

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

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

A105512 Number of times 2 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16

Offset: 0

Reinhard Zumkeller, Apr 11 2005

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

Cf. A000030, A000045, A008963, A105502.

Cf. A105511, A105513, A105514, A105515, A105516, A105517, A105518, A105519.

Accumulate[Table[If[IntegerDigits[Fibonacci[n]][[1]] == 2, 1, 0], {n, 0, 100}]] (* Amiram Eldar, Jan 12 2023 *)

(leadingdigit(n, b=10) = n \ 10^logint(n, b));
(isok(n) = leadingdigit(fibonacci(n))==2);
(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) = 2 and 0<=k<=n};
a(A105502(n)) = a(A105502(n) - 1) + 1;
n = A105511(n) + a(n) + A105513(n) + A105514(n) + A105515(n) + A105516(n) + A105517(n) + A105518(n) + A105519(n).
a(n) ~ log_10(3/2) * n. - Amiram Eldar, Jan 12 2023

A105513 Number of times 3 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 0

Reinhard Zumkeller, Apr 11 2005

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

Cf. A000030, A000045, A008963, A105503.

Cf. A105511, A105512, A105514, A105515, A105516, A105517, A105518, A105519.

Accumulate[Table[If[IntegerDigits[Fibonacci[n]][[1]] == 3, 1, 0], {n, 0, 100}]] (* Amiram Eldar, Jan 12 2023 *)

(leadingdigit(n, b=10) = n \ 10^logint(n, b));
(isok(n) = leadingdigit(fibonacci(n))==3);
(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) = 3 and 0<=k<=n};
a(A105503(n)) = a(A105503(n) - 1) + 1;
n = A105511(n) + A105512(n) + a(n) + A105514(n) + A105515(n) + A105516(n) + A105517(n) + A105518(n) + A105519(n).
a(n) ~ log_10(4/3) * n. - Amiram Eldar, Jan 12 2023

A105514 Number of times 4 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 0

Reinhard Zumkeller, Apr 11 2005

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

Cf. A000030, A000045, A008963, A105504.

Cf. A105511, A105512, A105513, A105515, A105516, A105517, A105518, A105519.

Accumulate[Table[If[IntegerDigits[Fibonacci[n]][[1]] == 4, 1, 0], {n, 0, 100}]] (* Amiram Eldar, Jan 12 2023 *)

(leadingdigit(n, b=10) = n \ 10^logint(n, b));
(isok(n) = leadingdigit(fibonacci(n))==4);
(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) = 4 and 0<=k<=n};
a(A105504(n)) = a(A105504(n) - 1) + 1;
n = A105511(n) + A105512(n) + A105513(n) + a(n) + A105515(n) + A105516(n) + A105517(n) + A105518(n) + A105519(n).
a(n) ~ log_10(5/4) * n. - Amiram Eldar, Jan 12 2023

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

Reinhard Zumkeller, Apr 11 2005

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

Cf. A000030, A000045, A008963, A105505.

Cf. A105511, A105512, A105513, A105514, A105516, A105517, A105518, A105519.

Accumulate[If[First[IntegerDigits[#]]==5,1,0]&/@Fibonacci[Range[0,110]]] (* Harvey P. Dale, Nov 02 2014 *)

(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

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

A105516 Number of times 6 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 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

Offset: 0

Reinhard Zumkeller, Apr 11 2005

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

Cf. A000030, A000045, A008963, A105506.

Cf. A105511, A105512, A105513, A105514, A105515, A105517, A105518, A105519.

Prepend[Accumulate[If[First[IntegerDigits[#]]==6,1,0]&/@Fibonacci[ Range[ 110]]],0] (* Harvey P. Dale, Feb 18 2011 *)

(leadingdigit(n, b=10) = n \ 10^logint(n, b));
(isok(n) = leadingdigit(fibonacci(n))==6);
(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) = 6 and 0<=k<=n};
a(A105506(n)) = a(A105506(n) - 1) + 1;
n = A105511(n) + A105512(n) + A105513(n) + A105514(n) + A105515(n) + a(n) + A105517(n) + A105518(n) + A105519(n).
a(n) ~ log_10(7/6) * n. - Amiram Eldar, Jan 12 2023

A105517 Number of times 7 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 0

Reinhard Zumkeller, Apr 11 2005

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

Cf. A000030, A000045, A008963, A105507.

Cf. A105511, A105512, A105513, A105514, A105515, A105516, A105518, A105519.

Accumulate[Table[If[IntegerDigits[Fibonacci[n]][[1]]==7,1,0],{n,0,120}]] (* Harvey P. Dale, Apr 29 2018 *)

(leadingdigit(n, b=10) = n \ 10^logint(n, b));
(isok(n) = leadingdigit(fibonacci(n))==7);
(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) = 7 and 0<=k<=n};
a(A105507(n)) = a(A105507(n) - 1) + 1;
n = A105511(n) + A105512(n) + A105513(n) + A105514(n) + A105515(n) + A105516(n) + a(n) + A105518(n) + A105519(n).
a(n) ~ log_10(8/7) * n. - Amiram Eldar, Jan 12 2023

A105518 Number of times 8 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

Original entry on oeis.org

0, 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, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 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

Offset: 0

Reinhard Zumkeller, Apr 11 2005

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

Cf. A000030, A000045, A008963, A105508.

Cf. A105511, A105512, A105513, A105514, A105515, A105516, A105517, A105519.

Accumulate[Table[If[IntegerDigits[Fibonacci[n]][[1]] == 8, 1, 0], {n, 0, 100}]] (* Amiram Eldar, Jan 12 2023 *)

(leadingdigit(n, b=10) = n \ 10^logint(n, b));
(isok(n) = leadingdigit(fibonacci(n))==8);
(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) = 8 and 0<=k<=n};
a(A105508(n)) = a(A105508(n) - 1) + 1;
n = A105511(n) + A105512(n) + A105513(n) + A105514(n) + A105515(n) + A105516(n) + A105517(n) + a(n) + A105519(n).
a(n) ~ log_10(9/8) * n. - Amiram Eldar, Jan 12 2023

A062332 Primes starting and ending with 1.

Original entry on oeis.org

11, 101, 131, 151, 181, 191, 1021, 1031, 1051, 1061, 1091, 1151, 1171, 1181, 1201, 1231, 1291, 1301, 1321, 1361, 1381, 1451, 1471, 1481, 1511, 1531, 1571, 1601, 1621, 1721, 1741, 1801, 1811, 1831, 1861, 1871, 1901, 1931, 1951, 10061, 10091, 10111, 10141

Offset: 1

Amarnath Murthy, Jun 21 2001

Complement of A208261 (nonprime numbers with all divisors starting and ending with digit 1) with respect to A208262 (numbers with all divisors starting and ending with digit 1). - Jaroslav Krizek, Mar 04 2012

Intersection of A030430 and A045707. - Michel Marcus, Jun 08 2013

			102701 is a member as it is a prime and the first and the last digits are both 1.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A208259 (Numbers starting and ending with digit 1).

Cf. A010051, A017281, A131835, A000030.

a062332 n = a062332_list !! (n-1)
a062332_list = filter ((== 1) . a010051') a208259_list
-- Reinhard Zumkeller, Jul 16 2014

fl1Q[n_]:=Module[{idn=IntegerDigits[n]},First[idn]==Last[idn]==1]; Select[ Prime[Range[1300]],fl1Q] (* Harvey P. Dale, Apr 30 2012 *)

{ n=-1; t=log(10); forprime (p=2, 5*10^5, if ((p-10*(p\10)) == 1 && (p\10^(log(p)\t)) == 1, write("b062332.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 05 2009

A010051(a(n)) * A000030(a(n)) * (a(n) mod 10) = 1. - Reinhard Zumkeller, Jul 16 2014

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jun 29 2001

Missing term a(36)=1901 added by Harry J. Smith, Aug 05 2009

A088133 Sum of first and last digits of n. Different from A115299.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 8, 9, 10, 11, 12, 13, 14

Offset: 0

Zak Seidov, Sep 20 2003

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

Cf. A046704, A007953, A088134, A088135, A088136.

Total[{First[IntegerDigits[#]],Last[IntegerDigits[#]]}]&/@Range[90] (* Harvey P. Dale, Aug 21 2018 *)

apply( {A088133(n)=n\10^logint(n+!n, 10)+n%10}, [0..99]) \\ M. F. Hasler, Apr 22 2024

list(map(A088133 := lambda n: int(str(n)[0])+n%10, range(99))) # M. F. Hasler, Apr 22 2024

a(n) = A000030(n) + A010879(n). - M. F. Hasler, Apr 22 2024

Extended to a(0) = 0 by M. F. Hasler, Apr 22 2024

cp's OEIS Frontend

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

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A105512 Number of times 2 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

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A105513 Number of times 3 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

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A105514 Number of times 4 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

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

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A105516 Number of times 6 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

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A105517 Number of times 7 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

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A105518 Number of times 8 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

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