A017936 -id:A017936 - cp's OEIS frontend

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

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

A049416 Largest number whose square has n digits.

Original entry on oeis.org

3, 9, 31, 99, 316, 999, 3162, 9999, 31622, 99999, 316227, 999999, 3162277, 9999999, 31622776, 99999999, 316227766, 999999999, 3162277660, 9999999999, 31622776601, 99999999999, 316227766016, 999999999999, 3162277660168

Offset: 1

Ulrich Schimke (ulrschimke(AT)aol.com)

a(n) + A180416(n) + A180425(n) + A167615(n) = A002283(n).

			31^2 = 961, but 32^2 = 1024, hence a(3) = 31.
a(4) = 99: 99^2 = 9801 has 4 digits, while 100^2 = 10000 has 5 digits.

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

Cf. A061433, A049415. Equals A017936 - 1.

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

Ceiling[Sqrt[10^Range[40]]-1] (* Harvey P. Dale, Sep 30 2011 *)

a(n) = ceiling(sqrt(10^n)) - 1.

More terms from Larry Reeves (larryr(AT)acm.org), May 16 2001

A035076 a(n) is root of square starting with digit 9: first term of runs.

Original entry on oeis.org

30, 95, 300, 949, 3000, 9487, 30000, 94869, 300000, 948684, 3000000, 9486833, 30000000, 94868330, 300000000, 948683299, 3000000000, 9486832981, 30000000000, 94868329806, 300000000000, 948683298051, 3000000000000

Offset: 2

Patrick De Geest, Nov 15 1998

Cf. A067479 (squares), A017936.

Cf. 2: A035069, 3: A035070, 4: A035071, 5: A035072, 6: A035073, 7: A035074, 8: A035075.

Ceiling[Sqrt[9*10^Range[2,30]]] (* Harvey P. Dale, Aug 07 2013 *)

a(n) = ceiling(sqrt(9*10^n)), n > 1.

A018074 Powers of fourth root of 10 rounded up.

Original entry on oeis.org

1, 2, 4, 6, 10, 18, 32, 57, 100, 178, 317, 563, 1000, 1779, 3163, 5624, 10000, 17783, 31623, 56235, 100000, 177828, 316228, 562342, 1000000, 1778280, 3162278, 5623414, 10000000, 17782795, 31622777, 56234133, 100000000, 177827942, 316227767, 562341326, 1000000000

Offset: 0

N. J. A. Sloane

nonn

Cf. A011007, A017936 (bisection), A308467.

Ceiling[(Power[10, (4 )^-1])^Range[0,50]]  (* Harvey P. Dale, Mar 18 2011 *)

a(n) = sqrtnint(10^n-1, 4) + 1; \\ Michel Marcus, Jun 26 2024

More terms from Michel Marcus, Aug 21 2019

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

Amarnath Murthy, Jul 07 2001

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

			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.

Harry J. Smith, Table of n, a(n) for n = 1..200

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

Column k=2 of A216653.

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

je=[4]; for(n=2, 45, je=concat(je, ceil(sqrt(10^n))-ceil(sqrt(10^(n-1))))); je

{ 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

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

Corrected and extended by Dean Hickerson and Jason Earls, Jul 10 2001

Edited by R. J. Mathar, Aug 07 2008

A130080 Smallest number whose sixth power has n digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 10, 15, 22, 32, 47, 69, 100, 147, 216, 317, 465, 682, 1000, 1468, 2155, 3163, 4642, 6813, 10000, 14678, 21545, 31623, 46416, 68130, 100000, 146780, 215444, 316228, 464159, 681293, 1000000, 1467800, 2154435, 3162278, 4641589

Offset: 1

Klaus Brockhaus, May 07 2007

Powers of sixth root of 10 rounded up.

			6^6 = 46656 has five digits, 7^6 = 117649 has six digits, hence a(6) = 7.

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

Cf. A011557 (powers of 10), A017936 (smallest number whose square has n digits), A018005 (smallest number whose cube has n digits), A018074 (smallest number whose fourth power has n digits), A018143 (smallest number whose fifth power has n digits), A130081 to A130084 (smallest number whose seventh ... tenth power has n digits).

[ Ceiling(Root(10^(n-1),6)): n in [1..41] ];

Table[(Ceiling[10^((n - 1)/6)]), {n, 1, 100}] (* Vincenzo Librandi, Sep 20 2013 *)

from sympy import integer_nthroot
def A130080(n):
    a, b = integer_nthroot(10**(n-1),6)
    return a+(not b) # Chai Wah Wu, Jun 19 2024

a(n) = ceiling(10^((n-1)/6)).

A130084 Smallest number whose tenth power has at least n digits.

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 4, 6, 7, 8, 10, 13, 16, 20, 26, 32, 40, 51, 64, 80, 100, 126, 159, 200, 252, 317, 399, 502, 631, 795, 1000, 1259, 1585, 1996, 2512, 3163, 3982, 5012, 6310, 7944, 10000, 12590, 15849, 19953, 25119, 31623, 39811, 50119, 63096, 79433, 100000

Offset: 1

Klaus Brockhaus, May 07 2007

Powers of tenth root of 10 rounded up.

			3^10 = 59049 has five digits, 4^10 = 1048576 has seven digits, hence a(6) = a(7) = 4.

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

Cf. A011279, A011557 (powers of 10), A017936 (smallest number whose square has n digits), A018005 (smallest number whose cube has n digits), A018074 (smallest number whose fourth power has n digits), A018143 (smallest number whose fifth power has n digits), A130080 to A130083 (smallest number whose sixth ... ninth power has n digits).

[Ceiling(Root(10^(n-1),10)): n in [1..51]];

Table[(Ceiling[10^((n - 1)/10)]), {n, 1, 60}] (* Vincenzo Librandi, Sep 20 2013 *)

from sympy import integer_nthroot
def A130084(n): return (lambda x:x[0]+(not x[1]))(integer_nthroot(10**(n-1),10)) # Chai Wah Wu, Jun 20 2024

a(n) = ceiling(10^((n-1)/10)).

A130083 Smallest number whose ninth power has at least n digits.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 5, 6, 8, 10, 13, 17, 22, 28, 36, 47, 60, 78, 100, 130, 167, 216, 279, 360, 465, 600, 775, 1000, 1292, 1669, 2155, 2783, 3594, 4642, 5995, 7743, 10000, 12916, 16682, 21545, 27826, 35939, 46416, 59949, 77427, 100000, 129155, 166811, 215444

Offset: 1

Klaus Brockhaus, May 07 2007

Powers of ninth root of 10 rounded up.

			2^9 = 512 has three digits, 3^9 = 19683 has five digits, hence a(4) = a(5) = 3.

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

Cf. A011278, A011557 (powers of 10), A017936 (smallest number whose square has n digits), A018005 (smallest number whose cube has n digits), A018074 (smallest number whose fourth power has n digits), A018143 (smallest number whose fifth power has n digits), A130080 to A130084 (smallest number whose sixth ... tenth power has n digits).

[ Ceiling(Root(10^(n-1),9)): n in [1..49] ];

Table[(Ceiling[10^((n - 1)/9)]), {n, 1, 60}] (* Vincenzo Librandi, Sep 21 2013 *)

from sympy import integer_nthroot
def A130083(n): return (lambda x:x[0]+(not x[1]))(integer_nthroot(10**(n-1),9)) # Chai Wah Wu, Jun 20 2024

a(n) = ceiling(10^((n-1)/9)).

A130081 Smallest number whose seventh power has at least n digits.

Original entry on oeis.org

1, 2, 2, 3, 4, 6, 8, 10, 14, 20, 27, 38, 52, 72, 100, 139, 194, 269, 373, 518, 720, 1000, 1390, 1931, 2683, 3728, 5180, 7197, 10000, 13895, 19307, 26827, 37276, 51795, 71969, 100000, 138950, 193070, 268270, 372760, 517948, 719686, 1000000, 1389496

Offset: 1

Klaus Brockhaus, May 07 2007

Powers of seventh root of 10 rounded up.

			1^7 = 1 has 1 digit, 2^7 = 128 has three digits, hence a(2) = a(3) = 2.

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

Cf. A011276, A011557 (powers of 10), A017936 (smallest number whose square has n digits), A018005 (smallest number whose cube has n digits), A018074 (smallest number whose fourth power has n digits), A018143 (smallest number whose fifth power has n digits), A130080 to A130084 (smallest number whose sixth ... tenth power has n digits).

[Ceiling(Root(10^(n-1),7)): n in [1..44]];

Table[(Ceiling[10^((n - 1)/7)]), {n, 1, 60}] (* Vincenzo Librandi, Sep 20 2013 *)

from sympy import integer_nthroot
def A130081(n): return (lambda x:x[0]+(not x[1]))(integer_nthroot(10**(n-1),7)) # Chai Wah Wu, Jun 20 2024

a(n) = ceiling(10^((n-1)/7)).

A317774 a(n) = A322667(n) + 1.

Original entry on oeis.org

8, 60, 532, 5100, 50317, 501000, 5003163, 50010000, 500031623, 5000100000, 50000316228, 500001000000, 5000003162278, 50000010000000, 500000031622777, 5000000100000000, 50000000316227767, 500000001000000000, 5000000003162277661, 50000000010000000000

Offset: 1

Jianing Song, Dec 22 2018

nonn

a(n) is the smallest positive integer k such that floor(k^2/10^n) - floor((k - 1)^2/10^n) = 2. The main entries are A322666 and A322667. However, some people may search for this entry.

			floor(7^2/10) = 4, floor(8^2/10) = 6, and 8 is the smallest k such that floor(k^2/10) - floor((k - 1)^2/10) = 2, so a(1) = 8.
floor(59^2/10) = 34, floor(60^2/10) = 36, and 60 is the smallest k such that floor(k^2/100) - floor((k - 1)^2/100) = 2, so a(2) = 60.

Cf. A017936, A322666, A322667.

a(n) = if(n==1, 8, 5*10^(n-1) + ceil(10^(n/2)))

a(n) = 5*10^(n-1) + ceiling(10^(n/2)) for n >= 2.

cp's OEIS Frontend

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

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A049416 Largest number whose square has n digits.

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A035076 a(n) is root of square starting with digit 9: first term of runs.

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A018074 Powers of fourth root of 10 rounded up.

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

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Maple

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A130080 Smallest number whose sixth power has n digits.

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Magma

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A130084 Smallest number whose tenth power has at least n digits.

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