A000030 -id:A000030 - cp's OEIS frontend

A034837 Numbers that are divisible by the first, i.e., the leftmost, digit.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 24, 26, 28, 30, 33, 36, 39, 40, 44, 48, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122

Offset: 1

Erich Friedman

A 10-automatic sequence. - Charles R Greathouse IV, Jun 13 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Cf. A034709 (divisible by last digit).

import Data.Char (digitToInt)
a034837 n = a034837_list !! (n-1)
a034837_list = filter (\i -> i `mod` (a000030 i) == 0) [1..]
-- Reinhard Zumkeller, Jun 19 2011

Select[Range[150],Divisible[#,IntegerDigits[#][[1]]]&] (* Harvey P. Dale, Jul 11 2017 *)

for(n=1,1000,n%(Vecsmall(Str(n))[1]-48) || print1(n",")) \\ M. F. Hasler, Jun 19 2011

a(n)=for(k=1,1e9,k%(Vecsmall(Str(k))[1]-48) || n-- || return(k)) \\  M. F. Hasler, Jun 19 2011

def ok(n): return n and n%int(str(n)[0]) == 0
print([k for k in range(123) if ok(k)]) # Michael S. Branicky, Jan 15 2023

from itertools import count, islice
def agen(): # generator of terms
    yield from (i for e in count(0) for f in range(1, 10) for i in range(f*10**e, (f+1)*10**e, f))
print(list(islice(agen(), 64))) # Michael S. Branicky, Jan 15 2023

a(n) mod A000030(a(n)) = 0. - Reinhard Zumkeller, Sep 20 2003

Definition clarified by Harvey P. Dale, May 01 2023

A045524 Numbers k such that k! has initial digit '5'.

Original entry on oeis.org

7, 21, 38, 46, 61, 66, 81, 119, 137, 144, 150, 165, 189, 196, 206, 209, 221, 224, 235, 243, 248, 253, 258, 279, 292, 340, 342, 353, 362, 383, 413, 429, 440, 488, 508, 529, 540, 584, 597, 611, 630, 651, 662, 679, 685, 704, 711, 718, 725, 732, 764, 782, 812

Offset: 1

Jeff Burch

n such that A000030(A000142(n)) = 5. - Robert Israel, Feb 07 2017

The asymptotic density of this sequence is log_10(6/5) = 0.079181... (Kunoff, 1987). - Amiram Eldar, Jul 17 2020

			7 is a term since 7! = 5040 has the initial digit 5.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Sharon Kunoff, N! has the first digit property, The Fibonacci Quarterly, Vol. 25, No. 4 (1987), pp. 365-367.
Index entries for sequences related to factorial numbers

For factorials with initial digit d (1 <= d <= 9) see A045509, A045510, A045511, A045516, A045517, A045518, A282021, A045519; A045520, A045521, A045522, A045523, A045525, A045526, A045527, A045528, A045529.

Cf. A000030, A000142, A008905.

filter:= proc(t) local tf;
tf:= t!;
floor(tf/10^ilog10(tf)) = 5
end proc:
select(filter, [$1..1000]); # Robert Israel, Feb 07 2017

Select[ Range[ 850 ], IntegerDigits[ #! ] [[1]] == 5 & ]

isok(n) = digits(n!)[1] == 5; \\ Michel Marcus, Feb 08 2017

A008905(a(n)) = 5. - Amiram Eldar, Jul 17 2020

A241299 Initial digit of the decimal expansion of n^(n^n) or n^^3 (in Don Knuth's up-arrow notation).

Original entry on oeis.org

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

Offset: 0

Robert Munafo and Robert G. Wilson v, Apr 18 2014

0^^3 = 0 since 0^^k = 1 for even k, 0 for odd k, k >= 0.
Conjecture: the distribution of the initial digits obey Zipf's law.
The distribution of the first 1000 terms beginning with 1: 302, 196, 124, 91, 72, 46, 71, 53, 45.

			a(0) = 0, a(1) = 1, a(2) = 1 because 2^(2^2) = 16, a(3) = 7 because 3^(3^3) = 7625597484987 and its initial digit is 7, etc.

Robert P. Munafo and Robert G. Wilson v, Table of n, a(n) for n = 0..1000
Cut the Knot.org, Benford's Law and Zipf's Law, A. Bogomolny, Zipf's Law, Benford's Law from Interactive Mathematics Miscellany and Puzzles.
Hans Havermann, Next 5 terms.
M. E. J. Newman, Power laws, Pareto distributions and Zipf's law.
Eric Weisstein's World of Mathematics, Joyce Sequence.
Wikipedia, Knuth's up-arrow notation.
Wikipedia, Zipf's law.
Index entries for sequences related to Benford's law.

Cf. A000030, A000312, A002488, A066022, A241291, A241292, A241293, A241294, A241295, A241296, A241297, A241298.

g[n_] := Quotient[n^p, 10^(Floor[ p*Log10@ n] - (1004 + p))]; f[n_] := Block[{p = n}, Quotient[ Nest[ g@ # &, p, p], 10^(1004 + p)]]; Array[f, 105, 0]

For n > 0, a(n) = floor(t/10^floor(log_10(t))) where t = n^(n^n).

a(n) = A000030(A002488(n)). - Omar E. Pol, Jul 04 2019

A364185 Leading digit of 11^n.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Jul 15 2023

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

Cf. A008952, A060956, A111395, A362871, A363093, A363249.

Cf. A000030, A001020, A008571.

a[n_] := IntegerDigits[11^n][[1]]; Array[a, 100, 0] (* Amiram Eldar, Jul 15 2023 *)

PARI
```
a(n) = digits(11^n)[1];
```

a(n) = A000030(A001020(n)).

A040997 Absolute value of first digit of n minus sum of other digits of n.

Original entry on oeis.org

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

Offset: 1

Felice Russo

This is different from |A055017(n)| = |(x1 + x3 + ...) - (x2 + x4 + ...)|, where x1,...,xk are the digits of n. - M. F. Hasler, Nov 09 2019

			a(371) = |3-7-1| = 5.

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

Cf. A037904, A040114, A040115, A040163, A055017.

Cf. A000030, A007953, A217657.

a040997 n = abs $ a000030 n - a007953 (a217657 n) -- Reinhard Zumkeller, Oct 10 2012

apply( A040997(n)={abs(vecsum(n=digits(n))-n[1]*2)}, [1..199]) \\ M. F. Hasler, Nov 09 2019

If decimal expansion of n is x1 x2 ... xk then a(n) = |x1-x2-x3- ... -xk|.

a(n) = abs(A000030(n) - A007953(A217657(n))). - Reinhard Zumkeller, Oct 10 2012

Name edited and incorrect formula deleted by M. F. Hasler, Nov 09 2019

A077648 Initial digits of prime numbers.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Nov 19 2002

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A000030, A000040.

[Intseq(p)[#Intseq(p)]: p in PrimesUpTo(600)]; // Bruno Berselli, Feb 14 2013

Table[First[IntegerDigits[Prime[n]]], {n, 1, 120}]

a(n) = digits(prime(n))[1]; \\ Michel Marcus, Feb 11 2017

from sympy import prime
def A077648(n): return int(str(prime(n))[0]) # Chai Wah Wu, Oct 16 2024

a(n) = A000030(A000040(n)).

A217398 Numbers starting with 5.

Original entry on oeis.org

5, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542

Offset: 1

Jeremy Gardiner, Oct 02 2012

The lower and upper asymptotic densities of this sequence are 1/45 and 5/27, respectively. - Amiram Eldar, Feb 27 2021

Also numbers such that when the leftmost digit is moved to the unit's place the result is divisible by 5. - Stefano Spezia, Jul 08 2025

Jeremy Gardiner, Table of n, a(n) for n = 1..1111
Index entries for 10-automatic sequences.

Cf. A000867, A011535.
Subsequences include: A045711, A077330, A077681, A106415, A106425.
Cf. A131835, A217394, A217395, A217397, A217399, A217400, A217401, A217402.

a217398 n = a217398_list !! (n-1)
a217398_list = filter ((== 5) . a000030) [1..]
-- Reinhard Zumkeller, Mar 13 2014

Select[Range[1000], IntegerDigits[#][[1]] == 5 &] (* T. D. Noe, Oct 02 2012 *)

def A217398(n): return n+(44*10**(len(str(9*n-8))-1))//9 # Chai Wah Wu, Dec 07 2024

A000030(a(n)) = 5; A143473(a(n)) = a(n). - Reinhard Zumkeller, Mar 13 2014

a(n) = n + (44*10^floor(log_10(9*n-8))-8)/9. - Alan Michael Gómez Calderón, May 17 2023

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

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

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

cp's OEIS Frontend

A034837 Numbers that are divisible by the first, i.e., the leftmost, digit.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Python

Python

Formula

Extensions

A045524 Numbers k such that k! has initial digit '5'.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A241299 Initial digit of the decimal expansion of n^(n^n) or n^^3 (in Don Knuth's up-arrow notation).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A364185 Leading digit of 11^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A040997 Absolute value of first digit of n minus sum of other digits of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

PARI

Formula

Extensions

A077648 Initial digits of prime numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

A217398 Numbers starting with 5.

Views

Author

Keywords