A089951 -id:A089951 - cp's OEIS frontend

A000030 Initial digit of n.

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

Offset: 0

N. J. A. Sloane, Simon Plouffe

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 a(n) and A217657(n). - Reinhard Zumkeller, Oct 10 2012, improved by M. F. Hasler, Nov 17 2018, and corrected by Glen Whitney, Jul 01 2022

Equivalent definition: The initial a(0) = 0 is followed by each digit in S = {1,...,9} once. Thereafter, repeat 10 times each digit in S. Then, repeat 100 times each digit in S, etc.

			23 begins with a 2, so a(23) = 2.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David W. Wilson, Table of n, a(n) for n = 0..10000
A. Cobham, Uniform Tag Sequences, Mathematical Systems Theory, 6 (1972), 164-192.

Cf. A010879 (final digit of n).
Cf. A061681, A130571, A109453, A134010, A052038.
Cf. A002993, A089951, A002994, A143464, A098174, A098175, A072543, A072544, A073600, A073601, A037904. - Reinhard Zumkeller, Aug 17 2008

a000030 = until (< 10) (`div` 10) -- Reinhard Zumkeller, Feb 20 2012, Feb 11 2011

[Intseq(n)[#Intseq(n)]: n in [1..100]]; // Vincenzo Librandi, Nov 17 2018

A000030 := proc(n)
    if n = 0 then
        0;
    else
        convert(n,base,10) ;
        %[-1] ;
    end if;
end proc:
seq(A000030(n),n=0..200) ;# N. J. A. Sloane, Feb 10 2017

Join[{0},First[IntegerDigits[#]]&/@Range[90]] (* Harvey P. Dale, Mar 01 2011 *)
Table[Floor[n/10^(Floor[Log10[n]])], {n, 1, 50}] (* G. C. Greubel, May 16 2017 *)
Table[NumberDigit[n,IntegerLength[n]-1],{n,0,100}] (* Harvey P. Dale, Aug 29 2021 *)

a(n)=if(n<10,n,a(n\10)) \\ Mainly for illustration.

A000030(n)=n\10^logint(n+!n,10) \\ Twice as fast as a(n)=digits(n)[1]. Before digits() was added in PARI v.2.6.0 (2013), one could use, e.g., Vecsmall(Str(n))[1]-48. - M. F. Hasler, Nov 17 2018

def a(n): return int(str(n)[0])
print([a(n) for n in range(85)]) # Michael S. Branicky, Jul 01 2022

a(n) = [n / 10^([log_10(n)])] where [] denotes floor and log_10(n) is the logarithm is base 10. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001

a(n) = k for k*10^j <= n < (k+1)*10^j for some j. - M. F. Hasler, Mar 23 2015

A002993 Initial digits of squares.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Gelfand's Question

Cf. A000030, A000290, A089951, A002994.

a002993 = a000030 . a000290  -- Reinhard Zumkeller, Apr 01 2015

a:= n-> parse(""||(n^2)[1]):
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 06 2021

a[n_] := First[IntegerDigits[n^2]]; Table[a[n], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Dec 21 2008 *)
First[IntegerDigits[#]]&/@(Range[70]^2) (* Harvey P. Dale, Feb 09 2011 *)

a(n) = A000030(A000290(n)). - Reinhard Zumkeller, Aug 17 2008

A086457 Both n and n^2 have the same initial digit and also n and n^2 have the same final digit when expressed in base 10.

Original entry on oeis.org

0, 1, 10, 11, 95, 96, 100, 101, 105, 106, 110, 111, 115, 116, 120, 121, 125, 126, 130, 131, 135, 136, 140, 141, 895, 896, 950, 951, 955, 956, 960, 961, 965, 966, 970, 971, 975, 976, 980, 981, 985, 986, 990, 991, 995, 996, 1000, 1001, 1005, 1006, 1010, 1011

Offset: 1

Jeremy Gardiner, Jul 20 2003

All terms of A045953 appear in this sequence.
Subsequence of A008851; A045953 and A046851 are subsequences. [Reinhard Zumkeller, Jul 27 2011]
Intersection of A008851 and A089951. - Michel Marcus, Mar 19 2015

			a(12) = 115 appears in the sequence because 115*115 = 13225.

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

Cf. A045953, A086458.

left$(str$(n), 1) = left$(str$(n^2), 1) AND right$(str$(n), 1) = right$(str$(n^2), 1)

a086457 n = a086457_list !! (n-1)
a086457_list = filter (\x -> a000030 x == a000030 (x^2) &&
                             a010879 x == a010879 (x^2)) [0..]
-- Reinhard Zumkeller, Jul 27 2011

ldQ[n_]:=Module[{idn=IntegerDigits[n],idn2=IntegerDigits[n^2]}, First[ idn] == First[idn2]&&Last[idn]==Last[idn2]]; Select[Range[ 0,1100], ldQ]  (* Harvey P. Dale, Feb 06 2011 *)

A000030(a(n)) = A000030(a(n)^2) and A010879(a(n)) = A010879(a(n)^2).

Offset corrected by Reinhard Zumkeller, Jul 27 2011

A144582 Numbers having the same leading decimal digit as their cube.

Original entry on oeis.org

0, 1, 10, 11, 12, 28, 29, 32, 33, 34, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287

Offset: 1

Reinhard Zumkeller, Aug 17 2008

For k > 0, write k = s * 10^t, 1 <= s < 10, then k is a term if and only if s is in [1, 2^(1/3)) U (20^(1/3), 3) U (30^(1/3), 40^(1/3)) U (30^(2/3), 10). - Jianing Song, Dec 26 2022

			a(13)=99 and 970299=99^3 start both with 9.

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

Cf. A000030, A002994, A000578, A089951, A256523 (subsequence).

a144582 n = a144582_list !! (n-1)
a144582_list = [x | x <- [0..], a000030 x == a000030 (x ^ 3)]
-- Reinhard Zumkeller, Apr 01 2015

Select[Range[0,300],First[IntegerDigits[#]]== First[IntegerDigits[#^3]]&]  (* Harvey P. Dale, Mar 14 2011 *)

isok(n) = (n == 0) || (digits(n)[1] == digits(n^3)[1]); \\ Michel Marcus, Mar 18 2015

A000030(a(n)) = A002994(a(n)) = A000030(A000578(a(n))).

A256523 Numbers m such that m, m^2 and m^3 have identical initial digits in decimal representation.

Original entry on oeis.org

0, 1, 10, 11, 12, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 966, 967, 968, 969, 970, 971, 972, 973, 974, 975, 976, 977, 978, 979, 980, 981, 982, 983, 984, 985

Offset: 1

Reinhard Zumkeller, Apr 01 2015

Intersection of A089951 and A144582.
From Jianing Song, Dec 26 2022: (Start)
For k > 0, write k = s * 10^t, 1 <= s < 10, then k is a term if and only if s is in [1, 2^(1/3)) U (30^(2/3), 10).
Except for 0, terms of A144582 that start with 1 or 9. (End)

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

Cf. A000030, A089951, A144582, A000290, A000578, A002993, A002994.

a256523 n = a256523_list !! (n-1)
a256523_list = [x | x <- [0..], let i = a000030 x,
                    a000030 (x ^ 2) == i, a000030 (x ^ 3) == i]

Select[Range[0,1000],Length[Union[(IntegerDigits/@(#^Range[3]))[[;;,1]]]]==1&] (* Harvey P. Dale, Dec 17 2024 *)

initial(n)=digits(n)[1]
is(n)=if(n==0, return(1)); my(k=initial(n)); initial(n^2)==k && initial(n^3)==k \\ Charles R Greathouse IV, May 13 2015

A000030(a(n)) = A002993(a(n)) = A000030(A000290(a(n))) = A002994(a(n)) = A000030(A000578(a(n))).

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A000030 Initial digit of n.

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A002993 Initial digits of squares.

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A086457 Both n and n^2 have the same initial digit and also n and n^2 have the same final digit when expressed in base 10.

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A144582 Numbers having the same leading decimal digit as their cube.

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A256523 Numbers m such that m, m^2 and m^3 have identical initial digits in decimal representation.

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