cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 11 results. Next

A102831 Number of n-digit 4th powers.

Original entry on oeis.org

2, 2, 2, 4, 8, 14, 25, 43, 78, 139, 246, 437, 779, 1384, 2461, 4376, 7783, 13840, 24612, 43765, 77828, 138400, 246114, 437658, 778280, 1383998, 2461136, 4376586, 7782795, 13839982, 24611356, 43765867, 77827942, 138399825, 246113559
Offset: 1

Views

Author

James R. Buddenhagen, Feb 27 2005

Keywords

Comments

The number 0 is considered a 1-digit 4th power. This is consistent with A062941 which considers 0 a 1-digit cube, but is inconsistent with A049415 which does not consider 0 a 1-digit square.

Examples

			a(1)=2 because there are 2 1-digit 4th powers, 0 and 1.

Links

Crossrefs

Cf. A062941, A049415.
Column k=4 of A216653.

Programs

  • Maple
    r:= proc(n, k) local b; b:= iroot(n, k); b+`if`(b^k r(10^n, 4) -r(10^(n-1), 4) +`if`(n=1, 1, 0):
seq(a(n), n=1..50);  # Alois P. Heinz, Sep 12 2012
  • Mathematica
    f[n_] := If[n == 1, 2, Ceiling[ Sqrt[ Sqrt[10^n]]] - Ceiling[ Sqrt[ Sqrt[10^(n - 1)]]]]; Table[ f[n], {n, 34}] (* Robert G. Wilson v, Mar 03 2005 *)

Extensions

More terms from Robert G. Wilson v, Mar 03 2005

A062940 Number of squares (including 0) with n digits.

Original entry on oeis.org

4, 6, 22, 68, 217, 683, 2163, 6837, 21623, 68377, 216228, 683772, 2162278, 6837722, 21622777, 68377223, 216227767, 683772233, 2162277661, 6837722339, 21622776602, 68377223398, 216227766017, 683772233983, 2162277660169
Offset: 1

Views

Author

Amarnath Murthy, Jul 07 2001

Keywords

Comments

Sum of first 2n terms = 10^n. - Zak Seidov, Aug 05 2006
a(n)/a(n-1) ~ 10^(1/2). For the sequence giving the number of members of the sequence a(k)=k^r with n digits we have a(n)/a(n-1) ~ 10^(1/r). - Ctibor O. Zizka, Mar 09 2008

Examples

			a(1)=4 because there are 4 one-digit squares: 0,1,4,9. - _Zak Seidov_, Aug 05 2006
a(2)=6 because there are 6 two-digit squares: 16,25,36,49,64,81. - _Zak Seidov_, Aug 05 2006
22 squares (100=10^2, 121=11^2, ..., 961=31^2) have 3 digits, hence a(3)=22.

Links

Crossrefs

A variant of A049415. A049415(n) = A017936(n+1) - A017936(n) = A049416(n+1) - A049416(n). Cf. A000290, A062941.
Column k=2 of A216653.

Programs

  • Maple
    r:= proc(n, k) local b; b:= iroot(n, k); b+`if`(b^k r(10^n, 2) -r(10^(n-1), 2) +`if`(n=1, 1, 0):
seq(a(n), n=1..40);  # Alois P. Heinz, Sep 12 2012
  • PARI
    je=[4]; for(n=2, 45, je=concat(je, ceil(sqrt(10^n))-ceil(sqrt(10^(n-1))))); je
  • PARI
    { default(realprecision, 200); for (n=1, 200, b=ceil(10^(n/2)); if (n>1, a=b - c, a=4); c=b; write("b062940.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 14 2009

Formula

a(n) = ceiling(sqrt(10^n)) - ceiling(sqrt(10^(n-1))), n > 1.
a(n) = A017934(n) - A017934(n-1) - (-1)^n, n >= 2. - R. J. Mathar, Mar 17 2008

Extensions

Corrected and extended by Dean Hickerson and Jason Earls, Jul 10 2001
Edited by R. J. Mathar, Aug 07 2008

A062941 Number of n-digit cubes (0 is included as a single-digit number).

Original entry on oeis.org

3, 2, 5, 12, 25, 53, 116, 249, 535, 1155, 2487, 5358, 11545, 24871, 53584, 115444, 248715, 535841, 1154435, 2487154, 5358411, 11544347, 24871542, 53584111, 115443470, 248715414, 535841116, 1154434691, 2487154143, 5358411166
Offset: 1

Views

Author

Amarnath Murthy, Jul 07 2001

Keywords

Comments

Sum of first 3n terms = 10^n.

Examples

			a(3) = 5 as there are 5 three-digit cubes: 125, 216, 343, 512 and 729.

Links

Crossrefs

Cf. A062940.
Column k=3 of A216653.

Programs

  • Maple
    r:= proc(n, k) local b; b:= iroot(n, k); b+`if`(b^k r(10^n, 3) -r(10^(n-1), 3) +`if`(n=1, 1, 0):
seq(a(n), n=1..40);  # Alois P. Heinz, Sep 12 2012
  • PARI
    { default(realprecision, 300); p=1; b=-1; for (n=1, 300, p*=10; c=b; b=floor((p - 1)^(1/3)); write("b062941.txt", n, " ", b - c) ) } \\ Harry J. Smith, Aug 14 2009

Extensions

Corrected and extended by Dean Hickerson, Jul 10 2001
Offset changed from 0 to 1 by Harry J. Smith, Aug 14 2009

A063945 Number of nonnegative integers with n digits.

Original entry on oeis.org

10, 90, 900, 9000, 90000, 900000, 9000000, 90000000, 900000000, 9000000000, 90000000000, 900000000000, 9000000000000, 90000000000000, 900000000000000, 9000000000000000, 90000000000000000, 900000000000000000, 9000000000000000000, 90000000000000000000
Offset: 1

Views

Author

Shyam Sunder Gupta, Sep 01 2001

Keywords

Comments

Also, first differences of A000533. - Omar E. Pol, Feb 24 2011

Links

Crossrefs

Column k=1 of A216653.

Programs

  • Maple
    a:= n-> `if`(n=1, 10, 9*10^(n-1)):
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 12 2012
  • Mathematica
    Join[{10},NestList[10#&,90,20]] (* Harvey P. Dale, Dec 31 2022 *)

Formula

a(1) = 10, a(2) = 90, a(n) = a(n-1)*10 for n>2.
a(n) = A052268(n), n>1. - R. J. Mathar, Oct 02 2008
From Stefano Spezia, Dec 01 2024: (Start)
G.f.: 10*x*(1 - x)/(1 - 10*x).
E.g.f.: (9*exp(10*x) - 9 + 10*x)/10. (End)

A102690 Number of n-expodigital numbers (i.e., numbers m such that m^n has exactly n decimal digits).

Original entry on oeis.org

10, 6, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Lekraj Beedassy, Jan 21 2005

Keywords

Comments

a(n) = 10 - A102691(n).

Examples

			a(6) = 3 because only the 3 numbers 7, 8 and 9 are 6-expodigital: 7^6 = 117649, 8^6 = 262144, 9^6 = 531441.

Crossrefs

Cf. A102691.
Main diagonal of A216653.

Extensions

Edited by Charles R Greathouse IV, Aug 03 2010

A216654 Number of n-digit 10th powers.

Original entry on oeis.org

2, 0, 0, 1, 1, 0, 2, 1, 1, 2, 3, 3, 4, 6, 6, 8, 11, 13, 16, 20, 26, 33, 41, 52, 65, 82, 103, 129, 164, 205, 259, 326, 411, 516, 651, 819, 1030, 1298, 1634, 2056, 2590, 3259, 4104, 5166, 6504, 8188, 10308, 12977, 16337, 20567, 25893, 32597, 41037, 51662, 65039
Offset: 1

Views

Author

Alois P. Heinz, Sep 12 2012

Keywords

Examples

			a(1) = 2: 0, 1.
a(4) = 1: 1024.
a(5) = 1: 59049.
a(7) = 2: 1048576, 9765625.
a(10) = 2: 1073741824, 3486784401.
a(11) = 3: 10000000000, 25937424601, 61917364224.

Links

Crossrefs

Column k=10 of A216653.

Programs

  • Maple
    r:= proc(n, k) local b; b:= iroot(n, k); b+`if`(b^k r(10^n, 10) -r(10^(n-1), 10) +`if`(n=1, 1, 0):
seq(a(n), n=1..60);

A216655 Number of n-digit 5th powers.

Original entry on oeis.org

2, 1, 1, 3, 3, 6, 10, 14, 24, 36, 59, 93, 147, 232, 369, 585, 927, 1470, 2328, 3690, 5849, 9270, 14692, 23285, 36904, 58490, 92699, 146919, 232850, 369042, 584894, 926993, 1469185, 2328502, 3690426, 5848932, 9269933, 14691853, 23285017, 36904265, 58489320
Offset: 1

Views

Author

Alois P. Heinz, Sep 12 2012

Keywords

Examples

			a(1) = 2: 0, 1.
a(2) = 1: 32.
a(3) = 1: 243.
a(4) = 3: 1024, 3125, 7776.
a(5) = 3: 16807, 32768, 59049.
a(6) = 6: 100000, 161051, 248832, 371293, 537824, 759375.
a(7) = 10: 1048576, 1419857, 1889568, 2476099, 3200000, 4084101, 5153632, 6436343, 7962624, 9765625.

Links

Crossrefs

Column k=5 of A216653.

Programs

  • Maple
    r:= proc(n, k) local b; b:= iroot(n, k); b+`if`(b^k r(10^n, 5) -r(10^(n-1), 5) +`if`(n=1, 1, 0):
seq(a(n), n=1..50);

A216656 Number of n-digit 6th powers.

Original entry on oeis.org

2, 1, 1, 1, 2, 3, 5, 7, 10, 15, 22, 31, 47, 69, 101, 148, 217, 318, 468, 687, 1008, 1479, 2171, 3187, 4678, 6867, 10078, 14793, 21714, 31870, 46780, 68664, 100784, 147931, 217134, 318707, 467800, 686635, 1007843, 1479311, 2171332, 3187079, 4677993, 6866354
Offset: 1

Views

Author

Alois P. Heinz, Sep 12 2012

Keywords

Comments

a(6*k+1) + a(6*k+2) + a(6*k+3) + a(6*k+4) + a(6*k+5) + a(6*k+6) = 9*10^k for k >= 1. - Robert Israel, Jul 22 2018

Examples

			a(1) = 2: 0, 1.
a(2) = 1: 64.
a(3) = 1: 729.
a(4) = 1: 4096.
a(5) = 2: 15625, 46656.
a(6) = 3: 117649, 262144, 531441.
a(7) = 5: 1000000, 1771561, 2985984, 4826809, 7529536.
a(8) = 7: 11390625, 16777216, 24137569, 34012224, 47045881, 64000000, 85766121.

Links

Crossrefs

Column k=6 of A216653.

Programs

  • Maple
    r:= proc(n, k) local b; b:= iroot(n, k); b+`if`(b^k r(10^n, 6) -r(10^(n-1), 6) +`if`(n=1, 1, 0):
seq(a(n), n=1..50);

A216657 Number of n-digit 7th powers.

Original entry on oeis.org

2, 0, 1, 1, 2, 2, 2, 4, 6, 7, 11, 14, 20, 28, 39, 55, 75, 104, 145, 202, 280, 390, 541, 752, 1045, 1452, 2017, 2803, 3895, 5412, 7520, 10449, 14519, 20174, 28031, 38950, 54120, 75200, 104490, 145188, 201738, 280314, 389496, 541202, 751998, 1044898, 1451881
Offset: 1

Views

Author

Alois P. Heinz, Sep 12 2012

Keywords

Examples

			a(1) = 2: 0, 1.
a(3) = 1: 128.
a(4) = 1: 2187.
a(5) = 2: 16384, 78125.
a(6) = 2: 279936, 823543.
a(7) = 2: 2097152, 4782969.
a(8) = 4: 10000000, 19487171, 35831808, 62748517.
a(9) = 6: 105413504, 170859375, 268435456, 410338673, 612220032, 893871739.

Links

Crossrefs

Column k=7 of A216653.

Programs

  • Maple
    r:= proc(n, k) local b; b:= iroot(n, k); b+`if`(b^k r(10^n, 7) -r(10^(n-1), 7) +`if`(n=1, 1, 0):
seq(a(n), n=1..50);

A216658 Number of n-digit 8th powers.

Original entry on oeis.org

2, 0, 1, 1, 1, 1, 2, 2, 4, 4, 6, 8, 11, 14, 18, 25, 34, 44, 60, 79, 105, 141, 187, 250, 334, 445, 593, 791, 1054, 1407, 1875, 2501, 3336, 4447, 5931, 7909, 10547, 14065, 18755, 25010, 33353, 44475, 59310, 79090, 105469, 140645, 187553, 250105, 333522, 444758
Offset: 1

Views

Author

Alois P. Heinz, Sep 12 2012

Keywords

Examples

			a(1) = 2: 0, 1.
a(3) = 1: 256.
a(4) = 1: 6561.
a(9) = 4: 100000000, 214358881, 429981696, 815730721.
a(10) = 4: 1475789056, 2562890625, 4294967296, 6975757441.
a(11) = 6: 11019960576, 16983563041, 25600000000, 37822859361, 54875873536, 78310985281.

Links

Crossrefs

Column k=8 of A216653.

Programs

  • Maple
    r:= proc(n, k) local b; b:= iroot(n, k); b+`if`(b^k r(10^n, 8) -r(10^(n-1), 8) +`if`(n=1, 1, 0):
seq(a(n), n=1..60);
  • Mathematica
    Join[{2,0},Tally[IntegerLength[Range[2,18*10^5]^8]][[;;,2]]] (* Harvey P. Dale, Sep 15 2024 *)
Showing 1-10 of 11 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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