A000030 -id:A000030 - cp's OEIS frontend

A088136 Primes such that sum of first and last digits is prime.

11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 131, 151, 181, 191, 211, 223, 229, 233, 239, 241, 251, 263, 269, 271, 281, 283, 293, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 601, 607, 617, 631, 641, 647, 661, 677, 691

Offset: 1

Zak Seidov, Sep 20 2003

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A008040 (primes), A010051 (isprime), A000030 (first digit of n), A010879 (last digit of n).

Cf. A046704, A007953, A088133, A088134, A088135.

Select[Prime[Range[400]],PrimeQ[First[IntegerDigits[#]]+ Last[ IntegerDigits[ #]]]&] (* Harvey P. Dale, Jun 23 2017 *)

select( {is_A088136(p)=isprime(p\10^logint(p,10)+p%10)&&isprime(p)}, primes(99)) \\ M. F. Hasler, Apr 23 2024

from sympy import isprime, primerange
def ok(p): s = str(p); return isprime(int(s[0]) + int(s[-1]))
def aupto(limit): return [p for p in primerange(1, limit+1) if ok(p)]
print(aupto(691)) # Michael S. Branicky, Nov 23 2021

A111395 First digit of powers of 5.

Original entry on oeis.org

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

Offset: 0

Almerio A. Castro (almerio.castro(AT)gmail.com), Nov 11 2005

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A008952, A060956.

Cf. A000030, A000351, A008566.

First[IntegerDigits[#]]&/@(5^Range[0,100]) (* Harvey P. Dale, Jan 13 2015 *)

a(n) = digits(5^n)[1]; \\ Michel Marcus, Jan 07 2014

a(n) = A000030(A000351(n)). - Seiichi Manyama, Jul 15 2023

a(0)=1 prepended, and more terms from Michel Marcus, Jan 07 2014

A130571 Lexicographically earliest permutation of the natural numbers such that in decimal representation the final digit of each term is distinct from the initial digit of the succeeding term.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 30, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 40, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 50, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 60, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 70, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 80, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 90, 89, 100, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 200, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 201, 202, 112, 113, 114

Offset: 0

Reinhard Zumkeller, Jun 05 2007

More than the usual number of terms are displayed, in order to show the difference between this and some closely related sequences. - N. J. A. Sloane, Mar 13 2014
A010879(a(n)) <> A000030(a(n+1));
A130572 is the inverse permutation; A130573(n) = a(a(n));
a(A130575(n)) = A130575(n);
see A130576 and A130577 for record values and where they occur.

For a closely related family of sequences see A239083-A239086, A239136-A239139, A239087-A239090, A239215-A239218, A239235.

A217657 Delete the initial digit in decimal representation of n.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Oct 10 2012

When n - a(n)*10^[log_10 n] >= 10^[(log_10 n) - 1], where [] denotes floor, or when n < 100 and 10|n, n is the concatenation of A000030(n) and a(n) - corrected by Glen Whitney, Jul 01 2022

a(110) = 10 is the first term > 9. The sequence consists of 10 repetitions of 0 (n = 0..9), then 9 repetitions of {0, ..., 9} (n = 10..99), then 9 repetitions of {0, ..., 99} (n = 100..999), and so on. - M. F. Hasler, Oct 18 2017

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A059995 (drop final digit of n), A000030 (initial digit of n), A202262.

a217657 n | n <= 9    = 0
          | otherwise = 10 * a217657 n' + m where (n', m) = divMod n 10

Array[FromDigits@ Rest@ IntegerDigits@ # &, 121, 0] (* Michael De Vlieger, Dec 22 2019 *)

apply( A217657(n)=n%10^logint(n+!n,10), [0..199]) \\ M. F. Hasler, Oct 18 2017, edited Dec 22 2019

def a(n): return 0 if n < 10 else int(str(n)[1:])
print([a(n) for n in range(121)]) # Michael S. Branicky, Jul 01 2022

a(n) = 0 if n <= 9, otherwise 10*a(floor(n/10)) + n mod 10.

a(n) = n mod 10^floor(log_10(n)), a(0) = 0. - M. F. Hasler, Oct 18 2017

Data extended to include the first terms larger than 9, by M. F. Hasler, Dec 22 2019

A073729 Concatenation of initial and final digits of n in decimal representation.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset: 1

Reinhard Zumkeller, Aug 05 2002

10 <= a(n) <= 99; a(a(n))=a(n).

			a(12321)=11; a(3210123)=33; a(59387)=57; a(8923)=83.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A073731, A073730.

a073729 n = 10 * a000030 n + a010879 n  -- Reinhard Zumkeller, Dec 31 2012

Table[FromDigits[{First[x = IntegerDigits[n]], Last[x]}], {n, 69}] (* Jayanta Basu, Jul 08 2013 *)

a(n)=10*digits(n)[1]+n%10 \\ Charles R Greathouse IV, Dec 28 2012

a(n) = A000030(n)*10 + A010879(n).

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

Reinhard Zumkeller, Apr 11 2005

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

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

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

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

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

fQ[n_] := IntegerDigits[Fibonacci[n]][[1]] == 1; Select[Range@200, fQ] (* Robert G. Wilson v, May 02 2018 *)

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

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

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

Original entry on oeis.org

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

A105519 Number of times 9 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 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

Offset: 0

Reinhard Zumkeller, Apr 11 2005

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

Cf. A000030, A000045, A008963, A105509.

Cf. A105511, A105512, A105513, A105514, A105515, A105516, A105517, A105518.

Table[If[First[IntegerDigits[Fibonacci[n]]]==9,1,0],{n,0,110}]// Accumulate (* Harvey P. Dale, Nov 27 2018 *)

(leadingdigit(n, b=10) = n \ 10^logint(n, b));
(isok(n) = leadingdigit(fibonacci(n))==9);
(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 18 2023

a(n) = #{k: A008963(k) = 9 and 0<=k<=n};
a(A105509(n)) = a(A105509(n) - 1) + 1;
n = A105511(n) + A105512(n) + A105513(n) + A105514(n) + A105515(n) + A105516(n) + A105517(n) + A105518(n) + a(n).
a(n) ~ (1 - log_10(9)) * n. - Amiram Eldar, Jan 12 2023

A076654 Smallest natural number not a multiple of 10, not occurring earlier and starting with the end of the previous term.

Original entry on oeis.org

1, 11, 12, 2, 21, 13, 3, 31, 14, 4, 41, 15, 5, 51, 16, 6, 61, 17, 7, 71, 18, 8, 81, 19, 9, 91, 101, 102, 22, 23, 32, 24, 42, 25, 52, 26, 62, 27, 72, 28, 82, 29, 92, 201, 103, 33, 34, 43, 35, 53, 36, 63, 37, 73, 38, 83, 39, 93, 301, 104, 44, 45, 54, 46, 64, 47, 74, 48, 84, 49, 94

Offset: 1

Amarnath Murthy, Oct 28 2002

Reinhard Zumkeller, Table of n, a(n) for n = 1..8936, all terms < 10000

Cf. A076652, A076653.

Cf. A000030, A010879, A067251.

import Data.List (delete)
a076654 n = a076654_list !! (n-1)
a076654_list = f a067251_list 1 where
  f xs z = g xs where
    g (y:ys) = if a000030 y == mod z 10 then y : f (delete y xs) y else g ys
-- Reinhard Zumkeller, Aug 15 2015

startsWith := proc(n,dig) local nshft ; nshft := n ; while nshft >= 10 do nshft := floor(nshft/10) ; od ; if dig = nshft then RETURN(true) ; else RETURN(false) ; fi ; end: A076654 := proc(nmax) local candid,a; a := [1] ; while nops(a) < nmax do candid := 2 ; while not startsWith(candid,op(-1,a) mod 10) or candid mod 10 = 0 or candid in a do candid := candid+1 ; od ; a := [op(a),candid] ; od ; RETURN(a) ; end: a := A076654(200) : for n from 1 to nops(a) do printf("%d,",op(n,a)) ; od ; # R. J. Mathar, Nov 12 2006

A000030(a(n+1)) = A010879(a(n)). - Reinhard Zumkeller, Aug 15 2015

More terms from R. J. Mathar, Nov 12 2006

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

cp's OEIS Frontend

A088136 Primes such that sum of first and last digits is prime.

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A111395 First digit of powers of 5.

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A130571 Lexicographically earliest permutation of the natural numbers such that in decimal representation the final digit of each term is distinct from the initial digit of the succeeding term.

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A217657 Delete the initial digit in decimal representation of n.

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A073729 Concatenation of initial and final digits of n in decimal representation.

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

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

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A105519 Number of times 9 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.