A062940 -id:A062940 - cp's OEIS frontend

A216653 Number A(n,k) of n-digit k-th powers; square array A(n,k), n>=1, k>=1, read by antidiagonals.

10, 4, 90, 3, 6, 900, 2, 2, 22, 9000, 2, 2, 5, 68, 90000, 2, 1, 2, 12, 217, 900000, 2, 1, 1, 4, 25, 683, 9000000, 2, 0, 1, 3, 8, 53, 2163, 90000000, 2, 0, 1, 1, 3, 14, 116, 6837, 900000000, 2, 0, 1, 1, 2, 6, 25, 249, 21623, 9000000000

Offset: 1

Alois P. Heinz, Sep 12 2012

			A(1,1) = 10: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
A(1,2) = 4: 0, 1, 4, 9.
A(2,2) = 6: 16, 25, 36, 49, 64, 81.
A(3,3) = 5: 125, 216, 343, 512, 729.
A(4,4) = 4: 1296, 2401, 4096, 6561.
A(5,5) = 3: 16807, 32768, 59049.
A(6,6) = 3: 117649, 262144, 531441.
Square array A(n,k) begins:
:n\k|        1:     2:    3:   4:   5:  6:  7:  8
+---+--------------------------------------------
: 1 |       10,     4,    3,   2,   2,  2,  2,  2
: 2 |       90,     6,    2,   2,   1,  1,  0,  0
: 3 |      900,    22,    5,   2,   1,  1,  1,  1
: 4 |     9000,    68,   12,   4,   3,  1,  1,  1
: 5 |    90000,   217,   25,   8,   3,  2,  2,  1
: 6 |   900000,   683,   53,  14,   6,  3,  2,  1
: 7 |  9000000,  2163,  116,  25,  10,  5,  2,  2
: 8 | 90000000,  6837,  249,  43,  14,  7,  4,  2

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Columns k=1-10 give: A063945, A062940, A062941, A102831, A216655, A216656, A216657, A216658, A216659, A216654.

Main diagonal gives: A102690.

r:= proc(n, k) local b; b:= iroot(n, k); b+`if`(b^k r(10^n, k) -r(10^(n-1), k) +`if`(n=1, 1, 0):
seq(seq(A(n, 1+d-n), n=1..d), d=1..10);

A049415 Number of squares (of positive integers) with n digits.

Original entry on oeis.org

3, 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

Ulrich Schimke (ulrschimke(AT)aol.com)

a(n) + A180426(n) + A180429(n) + A180347(n) = A052268(n).
Lim_{n->infinity} a(2n)/10^n = 1 - 1/sqrt(10);
lim_{n->infinity} a(2n-1)/10^n = 1/sqrt(10) - 1/10. - Robert G. Wilson v, Aug 29 2012

			22 squares (100=10^2, 121=11^2, ...., 961=31^2) have 3 digits, hence a(3)=22.

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

A049415(n) = A017936(n+1) - A017936(n) = A049416(n+1) - A049416(n).

Cf. A062940.

[Ceiling(Sqrt(10^n))-Ceiling(Sqrt(10^(n-1))) : n in [1..30]]; // Vincenzo Librandi, Oct 01 2011

f[n_] := Ceiling[Sqrt[10^n - 1]] - Ceiling[Sqrt[10^(n - 1)]]; f[1] = 3; Array[f, 24] (* Robert G. Wilson v, Aug 29 2012 *)

a(n) = ceiling(sqrt(10^n)) - ceiling(sqrt(10^(n-1))).
From Jon E. Schoenfield, Nov 30 2019: (Start)
a(2n) = floor(10^n * (1 - 1/sqrt(10))), so each even-indexed term a(2n) is given by the first n digits (after the decimal point) of 1 - 1/sqrt(10) = 0.68377223398316...;
a(2n-1) = ceiling(10^n * (1/sqrt(10) - 1/10)), so each odd-indexed term a(2n-1) is given by the first n digits (after the decimal point) of 1/sqrt(10) - 1/10 = 0.21622776601683..., plus 1. (End)

More terms from Dean Hickerson, Jul 10 2001

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

Amarnath Murthy, Jul 07 2001

Sum of first 3n terms = 10^n.

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

Harry J. Smith, Table of n, a(n) for n = 1..300

Cf. A062940.

Column k=3 of A216653.

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

{ 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

Corrected and extended by Dean Hickerson, Jul 10 2001

Offset changed from 0 to 1 by Harry J. Smith, Aug 14 2009

A126726 Number of squares (of nonnegative integers) that require n binary (base-2) digits.

Original entry on oeis.org

0, 2, 0, 1, 1, 2, 2, 4, 4, 7, 9, 14, 18, 27, 37, 54, 74, 107, 149, 213, 299, 425, 599, 849, 1199, 1697, 2399, 3394, 4798, 6787, 9597, 13573, 19195, 27146, 38390, 54292, 76780, 108584, 153560, 217168, 307120, 434335, 614241, 868669, 1228483, 1737338, 2456966

Offset: 0

Andrew G. West (westa(AT)wlu.edu), Mar 13 2007

Binary equivalent to A062940, which uses decimal representation.

			21^2 = 441 = 110111001_2, requiring 9 digits.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A062940.

r:= proc(n) local b; b:= isqrt(n); b+`if`(b^2 `if`(n=0, 0, r(2^n) -r(2^(n-1)) +`if`(n=1, 1, 0)):
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 13 2012

Conjecture: a(n) = A190568(n)-(-1)^n for n>0. - R. J. Mathar, May 21 2025

A173906 Total number of digits of the squares of all n-digit numbers.

Original entry on oeis.org

16, 338, 5183, 69837, 878377, 10583772, 123837722, 1418377223, 15983772233, 177837722339, 1958377223398, 21383772233983, 231837722339831, 2498377223398316, 26783772233983162, 285837722339831620, 3038377223398316206, 32183772233983162066, 339837722339831620668, 3578377223398316206680

Offset: 1

Robert G. Wilson v, Nov 26 2010

Sequence inspired by Claudio Meller's email to [seqfan] 07:01 a.m., Nov 26 2010, subject: About a problem of Bernardo Recamán Santos found in http://www.mathpuzzle.com/

Cf. A000290, A062940, A113119.

f[n_] := Block[{br = 1+ Floor[10^(n-1)*Sqrt@10]},(2n-1)(br-10^(n-1)) + 2n(10^n-br)]; Array[f, 20]

Limit_{n->oo} a(n)/(n*10^n) = 9/5. [corrected by Jason Yuen, Feb 07 2025]
From Jason Yuen, Feb 07 2025: (Start)
a(n) = (2*n-1)*A062940(2*n-1) + (2*n)*A062940(2*n).
a(n) = (9/5)*n*10^n - ceiling((sqrt(10)-1)*10^(n-1)) for n > 1. (End)

cp's OEIS Frontend

A216653 Number A(n,k) of n-digit k-th powers; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A049415 Number of squares (of positive integers) with n digits.

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

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A126726 Number of squares (of nonnegative integers) that require n binary (base-2) digits.

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A173906 Total number of digits of the squares of all n-digit numbers.

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