A000030 -id:A000030 - cp's OEIS frontend

A060956 Leading digit of 3^n.

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

Offset: 0

Ahmed Fares (ahmedfares(AT)my-deja.com), May 08 2001

He, Xinwei; Hildebrand, A J; Li, Yuchen; Zhang, Yunyi, Complexity of Leading Digit Sequences, Discrete Mathematics and Theoretical Computer Science; 22 (2020), #14.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Dmitry Kamenetsky, First digit of 3^2020, Puzzling StackExchange.

Cf. A000030, A000244.

First[IntegerDigits[#]]&/@(3^Range[0,110]) (* Harvey P. Dale, May 16 2016 *)

a(n) = { 3^n \ 10^logint(3^n,10) } \\ Harry J. Smith, Jul 15 2009

a(n) = floor(3^n / 10^floor(log_10(3^n))) = floor( 3^n / 10^floor(n*log_10(3)) ).

a(n) = A000030(A000244(n)). - Michel Marcus, Jul 03 2018

More terms from Larry Reeves (larryr(AT)acm.org), May 11 2001

A082794 Smallest multiple of n beginning with 4.

Original entry on oeis.org

4, 4, 42, 4, 40, 42, 42, 40, 45, 40, 44, 48, 403, 42, 45, 48, 408, 414, 418, 40, 42, 44, 46, 48, 400, 416, 405, 420, 406, 420, 403, 416, 429, 408, 420, 432, 407, 418, 429, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 400, 408, 416, 424, 432, 440, 448, 456, 406, 413

Offset: 1

Amarnath Murthy, Apr 20 2003

a(n) is in {n, 2n, 3n, 4n, 5n, 6n, 7n, 8n, 11n, 12n, 13n, 14n, 15n, 16n, 21n, 22n, 23n, 24n, 31n, 32n}. - Charles R Greathouse IV, Mar 06 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A082795, A082796, A082797, A082798.

Cf. A000030.

a082794 n = until ((== 4) . a000030) (+ n) n
-- Reinhard Zumkeller, Mar 27 2012

a:= proc(n) option remember; local m; m:= 0;
      do m:=m+n; if (""||m)[1]="4" then break fi od; m
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Apr 04 2021

f[n_] := Block[{m = n}, While[ First@ IntegerDigits@ m != 4, m += n]; m]; Array[f, 56] (* Robert G. Wilson v *)

a(n)=forstep(k=n, 32*n, n, if(Vec(Str(k))[1]=="4", return(k))) \\ Charles R Greathouse IV, Mar 06 2011

def a(n):
  kn = n
  while str(kn)[0] != '4': kn += n
  return kn
print([a(n) for n in range(1, 57)]) # Michael S. Branicky, Apr 04 2021

More terms from Sean A. Irvine, Mar 06 2011

A082795 Smallest multiple of n beginning with 5.

Original entry on oeis.org

5, 50, 51, 52, 5, 54, 56, 56, 54, 50, 55, 504, 52, 56, 510, 512, 51, 54, 57, 500, 504, 506, 506, 504, 50, 52, 54, 56, 58, 510, 527, 512, 528, 510, 525, 504, 518, 532, 507, 520, 533, 504, 516, 528, 540, 506, 517, 528, 539, 50, 51, 52, 53, 54

Offset: 1

Amarnath Murthy, Apr 20 2003

a(n) is in {n, 2n, 3n, 4n, 5n, 6n, 7n, 8n, 9n, 11n, 12n, 13n, 14n, 15n, 16n, 17n, 21n, 22n, 23n, 24n, 25n, 31n, 32n, 33n, 34n, 41n, 42n}. [Charles R Greathouse IV, Mar 06 2011]

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

Cf. A082794, A082796, A082797, A082798.

Cf. A000030.

a082795 n = until ((== 5) . a000030) (+ n) n
-- Reinhard Zumkeller, Mar 27 2012

f[n_] := Block[{m = n}, While[ First@ IntegerDigits@ m != 5, m += n]; m]; Array[f, 54] (* Robert G. Wilson v *)

a(n)=forstep(k=n, 42*n, n, if(Vec(Str(k))[1]=="5", return(k))) \\ Charles R Greathouse IV, Mar 06 2011

Corrected and extended by Sean A. Irvine, Mar 06 2011

A082796 Smallest multiple of n beginning with 6.

Original entry on oeis.org

6, 6, 6, 60, 60, 6, 63, 64, 63, 60, 66, 60, 65, 602, 60, 64, 68, 612, 608, 60, 63, 66, 69, 600, 600, 624, 621, 616, 609, 60, 62, 64, 66, 68, 630, 612, 629, 608, 624, 600, 615, 630, 602, 616, 630, 644, 611, 624, 637, 600, 612, 624, 636, 648, 605, 616, 627, 638

Offset: 1

Amarnath Murthy, Apr 20 2003

a(n) is in {n, 2n, 3n, 4n, 5n, 6n, 7n, 8n, 9n, 11n, 12n, 13n, 14n, 15n, 16n, 17n, 18n, 21n, 22n, 23n, 24n, 25n, 26n, 31n, 32n, 33n, 34n, 35n, 41n, 42n, 43n, 51n, 52n}. - Charles R Greathouse IV, Mar 06 2011

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

Cf. A082794, A082795, A082797, A082798.

Cf. A000030.

a082796 n = until ((== 6) . a000030) (+ n) n
-- Reinhard Zumkeller, Mar 27 2012

f[n_] := Block[{m = n}, While[ First@ IntegerDigits@ m != 6, m += n]; m]; Array[f, 55] (* Robert G. Wilson v *)

a(n)=forstep(k=n, 52*n, n, if(Vec(Str(k))[1]=="6", return(k))) \\ Charles R Greathouse IV, Mar 06 2011

def a(n):
    m = n
    while str(m)[0] != '6': m += n
    return m
print([a(n) for n in range(1, 59)]) # Michael S. Branicky, Jun 06 2021

More terms from Sean A. Irvine, Mar 06 2011

A208259 Numbers starting and ending with digit 1.

Original entry on oeis.org

1, 11, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 1001, 1011, 1021, 1031, 1041, 1051, 1061, 1071, 1081, 1091, 1101, 1111, 1121, 1131, 1141, 1151, 1161, 1171, 1181, 1191, 1201, 1211, 1221, 1231, 1241, 1251, 1261, 1271, 1281, 1291, 1301, 1311, 1321, 1331

Offset: 1

Jaroslav Krizek, Feb 24 2012

A000030(a(n)) = a(n) mod 10 = 1. - Reinhard Zumkeller, Jul 16 2014

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

Intersection of A017281 and A131835. Union of A062332 and A208260.
Supersequence of A208262 (numbers with all divisors starting and ending with digit 1).
Cf. A062332 (primes starting and ending with a digit 1), A208260 (nonprime numbers starting and ending with a digit 1).
Cf. A017281, A131835, A000030.

a208259 n = a208259_list !! (n-1)
a208259_list = 1 : map ((+ 1) . (* 10)) a131835_list
-- Reinhard Zumkeller, Jul 16 2014

Select[Range[2000], First[IntegerDigits[#]] == 1 && Last[IntegerDigits[#]] == 1 &] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2012 *)

A254338 Initial digits of A254143 in decimal representation.

Original entry on oeis.org

1, 4, 7, 1, 2, 3, 3, 4, 6, 1, 1, 2, 2, 2, 3, 3, 3, 4, 6, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 6, 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, 4, 4, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Reinhard Zumkeller, Feb 27 2015

a(n) = A000030(A254143(n));
also initial digits of A254323: a(n) = A000030(A254323(n)).
all terms are of the form u*v mod 10, where u <= v and belonging to {1,3,4,6,7}, the distinct elements of A254397:
length of k-th run of consecutive 1s = A005993(k-2), k > 1;
length of k-th run of consecutive 2s = k*(k+1)/2 = A000217(k), k >= 1;
length of k-th run of consecutive 3s = k+1, k >= 1;
length of k-th run of consecutive 4s = A065033(k-1);
n with a(n) = 4: A237424(n) = (10^a+10^b+1)/3 with b = 0, see also A093137, A133384;
n with a(n) = 6: A237424(n) = (10^a+10^b+1)/3 with a = b; A005994(a(n)) = 6 for n > 1; see also A199682;

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

Cf. A254143, A254323, A000030, A254339, A005993, A000217, A065033.

Haskell
```
a254338 = a000030 . a254143
```

listA237424(lim)=my(v=List(),a,t); while(1, for(b=0,a, t=(10^a+10^b+1)/3; if(t>lim, return(Set(v))); listput(v, t)); a++)
do(lim)=my(v=List(),u=listA237424(lim),t); for(i=1,#u, for(j=1,i, t=u[i]*u[j]; if(t>lim,break); listput(v,t))); apply(n->digits(n)[1], Set(v)) \\ Charles R Greathouse IV, May 13 2015

A082797 Smallest multiple of n beginning with 7.

Original entry on oeis.org

7, 70, 72, 72, 70, 72, 7, 72, 72, 70, 77, 72, 78, 70, 75, 704, 714, 72, 76, 700, 714, 704, 713, 72, 75, 78, 702, 700, 725, 720, 713, 704, 726, 714, 70, 72, 74, 76, 78, 720, 738, 714, 731, 704, 720, 736, 705, 720, 735, 700, 714, 728, 742, 702, 715

Offset: 1

Amarnath Murthy, Apr 20 2003

a(n) is in {n, 2n, 3n, 4n, 5n, 6n, 7n, 8n, 9n, 11n, 12n, 13n, 14n, 15n, 16n, 17n, 18n, 21n, 22n, 23n, 24n, 25n, 26n, 27n, 31n, 32n, 33n, 34n, 35n, 41n, 42n, 43n, 44n, 51n, 52n, 53n, 61n, 62n}. [Charles R Greathouse IV, Mar 06 2011]

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

Cf. A082794, A082795, A082796, A082798.

Cf. A000030.

a082797 n = until ((== 7) . a000030) (+ n) n
-- Reinhard Zumkeller, Mar 27 2012

f[n_] := Block[{m = n}, While[ First@ IntegerDigits@ m != 7, m += n]; m]; Array[f, 55] (* Robert G. Wilson v *)

a(n)=forstep(k=n, 62*n, n, if(Vec(Str(k))[1]=="7", return(k))) \\ Charles R Greathouse IV, Mar 06 2011

Corrected and extended by Sean A. Irvine, Mar 06 2011

A082798 Smallest multiple of n beginning with 8.

Original entry on oeis.org

8, 8, 81, 8, 80, 84, 84, 8, 81, 80, 88, 84, 806, 84, 810, 80, 85, 810, 817, 80, 84, 88, 805, 816, 800, 806, 81, 84, 87, 810, 806, 800, 825, 816, 805, 828, 814, 836, 819, 80, 82, 84, 86, 88, 810, 828, 846, 816, 833, 800, 816, 832, 848, 810, 825, 840, 855, 812

Offset: 1

Amarnath Murthy, Apr 20 2003

a(n) is in {n, 2n, 3n, 4n, 5n, 6n, 7n, 8n, 9n, 11n, 12n, 13n, 14n, 15n, 16n, 17n, 18n, 21n, 22n, 23n, 24n, 25n, 26n, 27n, 31n, 32n, 33n, 34n, 35n, 36n, 41n, 42n, 43n, 44n, 45n, 51n, 52n, 53n, 54n, 61n, 62n, 63n, 71n, 72n}. [Charles R Greathouse IV, Mar 06 2011]

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

Cf. A082794, A082795, A082796, A082797.

Cf. A000030.

a082798 n = until ((== 8) . a000030) (+ n) n
-- Reinhard Zumkeller, Mar 27 2012

f[n_] := Block[{m = n}, While[ First@ IntegerDigits@ m != 8, m += n]; m]; Array[f, 55] (* Robert G. Wilson v *)

a(n)=forstep(k=n, 72*n, n, if(Vec(Str(k))[1]=="8", return(k))) \\ Charles R Greathouse IV, Mar 06 2011

def a(n):
    m = n
    while str(m)[0] != '8': m += n
    return m
print([a(n) for n in range(1, 59)]) # Michael S. Branicky, Aug 08 2021

Corrected and extended by Sean A. Irvine, Mar 06 2011

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

A098174 a(n) is the smallest e > 0 such that the initial digit of n^e = 1 in decimal representation.

Original entry on oeis.org

1, 4, 9, 2, 3, 4, 5, 8, 16, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 3, 3, 3, 3, 3, 3, 5, 7, 9, 25, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 11, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 10, 10, 11, 12, 13, 14, 16, 18, 20, 23, 27, 32, 40, 53

Offset: 1

Reinhard Zumkeller, Aug 30 2004

A000030(n^a(n)) = 1; A098175(n) = n^a(n).
From Rémy Sigrist, Jun 25 2018: (Start)
We can extend this sequence to every Gaussian integers as follows:
- for any Gaussian integer z, let f(z) be the least k > 0 such that the initial decimal digit of the real part of z^k equals 1, or -1 if no such k exists,
- naturally f(n) = a(n) for any n > 0,
- apparently f(z) = -1 iff z = 0,
- see Links section for the color plot of f.
(End)

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Color plot of f(x + i*y) for x = -200..1000 and y = -200..500

Cf. A000030, A098175, A131835.

a(n, base=10) = my (nk=n); for (k=1, oo, my (z); logint(nk, base, &z); if (nk\z==1, return (k), nk*=n)) \\ Rémy Sigrist, Jun 21 2018

cp's OEIS Frontend

A060956 Leading digit of 3^n.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A082794 Smallest multiple of n beginning with 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Extensions

A082795 Smallest multiple of n beginning with 5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Extensions

A082796 Smallest multiple of n beginning with 6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Extensions

A208259 Numbers starting and ending with digit 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

A254338 Initial digits of A254143 in decimal representation.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

PARI

A082797 Smallest multiple of n beginning with 7.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs