A002994 -id:A002994 - 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

A097408 Initial decimal digit of n^4.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Aug 16 2004

			1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, ...

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Gelfand's Question

Cf. A000027, A002993, A002994, A097409, A097410, A097411, A097412, A097413, A097414.

a:= n-> parse(""||(n^4)[1]):
seq(a(n), n=1..120);  # Alois P. Heinz, Jan 28 2016

A097409 Initial decimal digit of n^5.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Aug 16 2004

			1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, ...

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Gelfand's Question

Cf. A000027, A002993, A002994, A097408, A097410, A097411, A097412, A097413, A097414.

a:= n-> parse(""||(n^5)[1]):
seq(a(n), n=1..120);  # Alois P. Heinz, Jan 28 2016

A097413 Initial decimal digit of n^9.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Aug 16 2004

			1, 512, 19683, 262144, 1953125, 10077696, 40353607, 134217728, 387420489, 1000000000, ...

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Gelfand's Question

Cf. A000027, A000030, A002993, A002994, A004216, A097408, A097409, A097410, A097411, A097412, A097414.

idd:= n -> floor(n/10^ilog10(n)):
seq(idd(n^9),n=2..100); # Robert Israel, Jan 28 2016

Table[IntegerDigits[n^9][[1]],{n,120}] (* Harvey P. Dale, Aug 25 2023 *)

a(n) = A000030(n^9) = floor(n^9/10^A004216(n^9)). - Robert Israel, Jan 28 2016

A097410 Initial decimal digit of n^6.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Aug 16 2004

			1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1000000, ...

Eric Weisstein's World of Mathematics, Gelfand's Question

Cf. A000027, A002993, A002994, A097408, A097409, A097411, A097412, A097413, A097414.

IntegerDigits[#][[1]]&/@(Range[110]^6) (* Harvey P. Dale, Mar 23 2018 *)

One prepended by Harvey P. Dale, Mar 23 2018

A097411 Initial decimal digit of n^7.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Aug 16 2004

			1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, 4782969, 10000000, ...

Eric Weisstein's World of Mathematics, Gelfand's Question

Cf. A000027, A002993, A002994, A097408, A097409, A097410, A097412, A097413, A097414.

Table[First[IntegerDigits[n^7]],{n,110}] (* Harvey P. Dale, Mar 17 2019 *)

1 prepended by Harvey P. Dale, Mar 17 2019

A097412 Initial decimal digit of n^8.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Aug 16 2004

			1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 43046721, 100000000, ...

Eric Weisstein's World of Mathematics, Gelfand's Question

Cf. A000027, A002993, A002994, A097408, A097409, A097410, A097411, A097413, A097414.

a(0) and a(1) added by Neven Juric, Sep 23 2010

A097414 Initial decimal digit of n^10.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Aug 16 2004

			1, 1024, 59049, 1048576, 9765625, 60466176, 282475249, 1073741824, 3486784401, 10000000000, ...

Eric Weisstein's World of Mathematics, Gelfand's Question

Cf. A000027, A002993, A002994, A097408, A097409, A097410, A097411, A097412, A097413.

IntegerDigits[#][[1]]&/@(Range[120]^10) (* Harvey P. Dale, Jan 12 2024 *)

a(1) inserted by Pontus von Brömssen, Jul 02 2021

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))).

cp's OEIS Frontend

A000030 Initial digit of n.

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A002993 Initial digits of squares.

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A097408 Initial decimal digit of n^4.

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A097409 Initial decimal digit of n^5.

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A097413 Initial decimal digit of n^9.

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A097410 Initial decimal digit of n^6.

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A097411 Initial decimal digit of n^7.

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