A049415 -id:A049415 - cp's OEIS frontend

A102831 Number of n-digit 4th powers.

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

James R. Buddenhagen, Feb 27 2005

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.

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

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Biquadratic Number.

Cf. A062941, A049415.

Column k=4 of A216653.

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

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 *)

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

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

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

A371728 a(n) is the largest number that is the digit sum of an n-digit square number.

Original entry on oeis.org

9, 13, 19, 31, 40, 46, 54, 63, 70, 81, 88, 97, 106, 112, 121, 130, 136, 148, 154, 162, 171, 180, 187, 193, 205, 211, 220, 229, 235, 244, 253, 262, 271, 277, 286, 297, 301, 310, 319, 331, 334, 343, 355, 360, 367, 378, 388, 396

Offset: 1

Zhining Yang, Apr 04 2024

a(n) appears to be approximately equal to (33*n-11)/4.

			a(6) = 46 because 46 is the largest digital sum encountered among all 6-digit squares (698896, 779689, 877969).

Shouen Wang, Chinese BBS: How many of these A's are there?

Cf. A004159, A049415, A069665, A069666.

Cf. A056991, A348300, A379298.

Array[Max@Map[Total@IntegerDigits[#^2] &, Range[Floor@Sqrt[10^(#)]], Floor@Sqrt[10^(# + 1) - 1]] &, 15]

a(22)-a(48) from Zhao Hui Du, Apr 05 2024
a(49)-a(62) from Zhining Yang, May 08 2024
a(63)-a(64) from Zhining Yang, May 23 2024
Incorrect a(61) and unverified a(49) onward deleted by Zhining Yang, Mar 03 2025

A180347 The number of n-digit numbers requiring 4 nonzero squares in their representation as sum of squares.

Original entry on oeis.org

1, 14, 150, 1500, 14999, 150000, 1499999, 15000000, 149999998, 1500000001, 14999999999, 149999999999, 1500000000001, 14999999999999, 149999999999999, 1500000000000001, 14999999999999998, 149999999999999999, 1500000000000000003, 14999999999999999996

Offset: 1

Martin Renner, Jan 18 2011

A049415(n) + A180426(n) + A180429(n) + a(n) = A052268(n).

Eric W. Weisstein: MathWorld : Lagrange's Four-Square Theorem.
Eric W. Weisstein: MathWorld: Sum of Squares Function.

Cf. A004215, A167615.

Cf. A049415, A052268, A180426, A180429.

a(n) = A167615(n)-A167615(n-1).

A180426 The number of n-digit numbers requiring 2 nonzero squares in their representation as sum of squares.

Original entry on oeis.org

3, 30, 265, 2351, 21062, 191630, 1766955, 16465551, 154749588, 1464326721, 13932672360, 133165432342, 1277568139729, 12295904124627, 118665023703067, 1147922359531155

Offset: 1

Martin Renner, Jan 19 2011

A049415(n) + a(n) + A180429(n) + A180347(n) = A052268(n)

Eric W. Weisstein: MathWorld -- Lagrange's Four-Square Theorem.
Eric W. Weisstein: MathWorld -- Sum of Squares Function.

Cf. A000404, A180416.

a(n) = A180416(n)-A180416(n-1) for n>1.

a(5)-a(8) from Alois P. Heinz, Jan 20 2011
a(9)-a(10) from Donovan Johnson, Jul 01 2011
a(10) corrected and a(11)-a(16) added by Hiroaki Yamanouchi, Aug 30 2014

A180429 The number of n-digit numbers requiring 3 nonzero squares in their representation as sum of squares.

Original entry on oeis.org

2, 40, 463, 5081, 53722, 557687, 5730883, 58527612, 595228791, 6035604901, 61067111413, 616833883887, 6222429697992, 62704089037652, 631334954674157, 6352077572091621

Offset: 1

Martin Renner, Jan 19 2011

A049415(n) + A180426(n) + a(n) + A180347(n) = A052268(n)

Eric Weisstein's World of Mathematics, Lagrange's Four-Square Theorem, Sum of Squares Function.

Cf. A000419, A180425.

a(n) = A180425(n)-A180425(n-1) for n>1.

a(6) from Lars Blomberg, Jun 29 2011
a(7)-a(10) from Donovan Johnson, Jul 01 2011
a(10) corrected and a(11)-a(16) added by Hiroaki Yamanouchi, Aug 30 2014

A177952 a(n) = number of n-digit squares in base 10 such that there is at least one permutation that is also a square in base 10. Initial zeros are not allowed for any square.

Original entry on oeis.org

0, 0, 7, 13, 86, 293, 1212, 4699, 17380, 60623, 203799, 664953, 2135649, 6800449, 21572602, 68311990, 216144075, 683666674

Offset: 1

Roger Bilisoly (bilisolyr(AT)ccsu.edu), May 15 2010

The ratio of a(n) to the n-th entry of sequence A049415 goes to 1. Bilisoly (2008), listed below, has a proof of this. Squares of this type are called "anasquares" in this reference (short for "anagram of squares").

			For instance, a(3) = 7 because (1) 144, 441 are both squares and permutations of each other as is 256, 625 and 169, 196, 961 and (2) there are no other 3 digit squares that can be permuted to another square (because initial zeros are forbidden, 100 and 001, etc., do not count).

R. Bilisoly, 92.45 Anasquares: Square anagrams of squares, The Mathematical Gazette, 92(2008), 58-63.

a(n) converges to A049415 in the sense that the ratio of the two sequences goes to 1 as n goes to infinity.

nAnasquares[ndigits_] := Module[{nsquares = 0, nkeys = 0, nanapat = 0, upper, lower, square, key, dictionary}, lower = Sqrt[10^(ndigits - 1)] // Ceiling; upper = Sqrt[10^ndigits - 1] // Floor; Do[ ++nsquares; square = i^2; key = ToString[FromDigits[Sort[IntegerDigits[square]]]]; If[StringQ[ dictionary[ key]] && (Length[StringPosition[dictionary[key], ","]] == 0), ++nanapat, Null] If[StringQ[dictionary[key]], dictionary[key] = dictionary[key] <> "," <> ToString[square], dictionary[key] = ToString[square]; ++nkeys], {i, lower, upper}]; Return[nsquares - nkeys + nanapat] ] Table[nAnasquares[n], {n, 1, 10}]

a(16)-a(18) from Donovan Johnson, Jun 10 2010

A366940 a(n) is the number of positive squares with n digits, all distinct.

Original entry on oeis.org

3, 6, 13, 36, 66, 96, 123, 97, 83, 87, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Tanya Khovanova, Oct 29 2023

a(n) = 0, for n > 10.

			a(1)=3 because all three 1-digit squares, 1, 4, and 9, have trivially distinct digits.
a(2)=6 because all six 2-digit squares, 16, 25, 36, 49, 64, and 81, have distinct digits.
158407396 = 12586^2: has 9 distinct digits. Thus, this number contributes to a(9). On the other hand, 158382225 = 12585^2 has repeated digits. Thus, it doesn't contribute.

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000290, A045540, A049415, A073532, A078255.

Table[Length[Select[Range[100000], Length[Union[IntegerDigits[#^2]]] == k &&  Length[IntegerDigits[#^2]] == k &]], {k, 10}]

from math import isqrt
from itertools import permutations
def sqr(n): return isqrt(n)**2 == n
def a(n):
    if n > 10: return 0
    return sum(1 for p in permutations("0123456789", n) if p[0] != '0' and sqr(int("".join(p))))
print([a(n) for n in range(1, 31)]) # Michael S. Branicky, Oct 29 2023

cp's OEIS Frontend

A102831 Number of n-digit 4th powers.

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A049416 Largest number whose square has n digits.

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

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A371728 a(n) is the largest number that is the digit sum of an n-digit square number.

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A180347 The number of n-digit numbers requiring 4 nonzero squares in their representation as sum of squares.

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A180426 The number of n-digit numbers requiring 2 nonzero squares in their representation as sum of squares.

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A180429 The number of n-digit numbers requiring 3 nonzero squares in their representation as sum of squares.

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A177952 a(n) = number of n-digit squares in base 10 such that there is at least one permutation that is also a square in base 10. Initial zeros are not allowed for any square.

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