A002993 -id:A002993 - 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

A367360 Comma transform of squares.

Original entry on oeis.org

1, 14, 49, 91, 62, 53, 64, 96, 48, 11, 1, 11, 41, 91, 62, 52, 62, 93, 43, 14, 4, 14, 45, 95, 66, 56, 67, 97, 48, 19, 9, 11, 41, 91, 61, 51, 61, 91, 41, 11, 1, 11, 41, 91, 62, 52, 62, 92, 42, 12, 2, 12, 42, 92, 63, 53, 63, 93, 43, 13, 3, 13, 43, 94, 64, 54, 64, 94, 44, 14, 5, 15, 45, 95, 65, 55, 65, 96, 46, 16, 6, 16, 46, 97

Offset: 0

N. J. A. Sloane, Nov 22 2023

To compute the comma transform of a sequence [b,c,d,e,f,...], concatenate the last digit of each term with the first digit of the following term. In other words, these are the numbers formed by the pairs of digits that surround the commas that separate the terms of the original sequence.
The comma transform CT(S) of a sequence S of positive numbers maps S into the set F consisting of finite or infinite sequences of positive numbers each with one or two digits. The inverse comma transform CTi maps an element of F to an element of F.
Inspired by Eric Angelini's A121805.

			The squares are 0, 1, 4, 9, 16, 25, ..., so the comma transform is [0]1, 14, 49, 91, 62, ...

Michael S. Branicky, Table of n, a(n) for n = 0..10000

Cf. A121805, A008959, A002993.

A166499 is the comma transform of the primes, A367361 of the powers of 2, A367362 of the nonnegative integers. See also A368362.

Maple code for comma transform (CT(a)) of a sequence a:
# leading digit, from A000030
Ldigit:=proc(n) local v; v:=convert(n, base, 10); v[-1]; end;
CT:=proc(a) local b,i; b:=[];
for i from 1 to nops(a)-1 do
b := [op(b), 10*(a[i] mod 10) + Ldigit(a[i+1])]; od: b; end;
# Inverse comma transform of sequence A calculated in base "bas": - N. J. A. Sloane, Jan 03 2024
bas := 10;
Ldigit:=proc(n) local v; v:=convert(n, base, bas); v[-1]; end;
CTi := proc(A) local B,i,L,R;
for i from 1 to nops(A) do
   if A[i]>=bas^2 then error("all terms must have 1 or 2 digits"); fi; od:
B:=Array(1..nops(A),-1);
if A[1] >= bas then B[1]:= Ldigit(A[1]); L:=(A[1] mod bas);
else B[1]:=10; L:=A[1];
fi;
for i from 2 to nops(A) do
  if A[i] >= bas then R := Ldigit(A[i]) else R:=0; fi;
  B[i] := L*bas + R;
  L := (A[i] mod bas);
od;
B;
end;
# second Maple program:
a:= n-> parse(cat(""||(n^2)[-1],""||((n+1)^2)[1])):
seq(a(n), n=0..99);  # Alois P. Heinz, Nov 22 2023

a[n_]:=FromDigits[{Last[IntegerDigits[n^2]],First[IntegerDigits[(n+1)^2]]}];
a/@Range[0,83] (* Ivan N. Ianakiev, Nov 24 2023 *)

from itertools import count, islice, pairwise
def S(): yield from (str(i**2) for i in count(0))
def agen(): yield from (int(t[-1]+u[0]) for t, u in pairwise(S()))
print(list(islice(agen(), 84))) # Michael S. Branicky, Nov 22 2023

def A367360(n): return (0, 10, 40, 90, 60, 50, 60, 90, 40, 10)[n%10]+int(str((n+1)**2)[0]) # Chai Wah Wu, Dec 22 2023

a(n) = 10 * A008959(n) + A002993(n+1). - Alois P. Heinz, Nov 22 2023

A002994 Initial digit of cubes.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

a(n) = A000030(A000578(n)). - Reinhard Zumkeller, Aug 17 2008

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, A000578, A144582, A002993.

a002994 = a000030 . a000578  -- Reinhard Zumkeller, Apr 01 2015

a:= n-> parse(""||(n^3)[1]):
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 06 2021

a[n_] := First[IntegerDigits[n^3]]; Table[a[n], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Dec 21 2008 *)

a(n) = if (n, digits(n^3)[1], 0); \\ Michel Marcus, Dec 13 2017

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

A089951 Numbers having the same leading decimal digits as their squares.

Original entry on oeis.org

0, 1, 10, 11, 12, 13, 14, 95, 96, 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, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 895

Offset: 1

Reinhard Zumkeller, Jan 12 2004

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

			895*895 = 801025, therefore 895 is a term: a(55)=895.

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

Cf. A018834.
Cf. A144582. - Reinhard Zumkeller, Aug 17 2008
Cf. A000030, A002993, A000290, A256523 (subsequence).

a089951 n = a089951_list !! (n-1)
a089951_list = [x | x <- [0..], a000030 x == a000030 (x ^ 2)]
-- Reinhard Zumkeller, Apr 01 2015

F:= proc(d) $10^d .. floor(sqrt(2)*10^d), $ ceil(sqrt(80)*10^d) .. 9*10^d - 1, $ ceil(sqrt(90)*10^d) .. 10^(d+1)-1 end proc:
0, F(0), F(1), F(2), F(3); # Robert Israel, Mar 18 2015

d[n_] := IntegerDigits[n]; Select[Range[895],
First[d[#]] == First[d[#^2]] &] (* Jayanta Basu, Jun 03 2013 *)

a(n)={my(v = [1, sqrt(80), sqrt(90)], w=[(k)->10^k * ((sqrt(2) - 1))\1 + 1, (k)->9 * 10^k - ceil(sqrt(80) * 10^k), (k)->10 * 10^k - ceil(sqrt(90) * 10^k)],i = 1,k = 0); if(n==1, 0, n--; while(n>w[i](k), n-=w[i](k); i++; if(i == 4, i = 1; k++)); ceil(v[i]*10^k)+n-1)} \\ David A. Corneth, Feb 26 2015

isok(n) = (n == 0) || (digits(n)[1] == digits(n^2)[1]); \\ Michel Marcus, Mar 18 2015

A number n is in the sequence iff n = 0 or n/10^floor(log_10(n)) lies in one of the half-open intervals [1, sqrt(2)), [sqrt(80), 9) or [sqrt(90), 10). - David W. Wilson, May 29 2008

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

cp's OEIS Frontend

A000030 Initial digit of n.

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A367360 Comma transform of squares.

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A002994 Initial digit of cubes.

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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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A089951 Numbers having the same leading decimal digits as their squares.

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