A065730 -id:A065730 - cp's OEIS frontend

A065733 Largest square <= n^3.

0, 1, 4, 25, 64, 121, 196, 324, 484, 729, 961, 1296, 1681, 2116, 2704, 3364, 4096, 4900, 5776, 6724, 7921, 9216, 10609, 12100, 13689, 15625, 17424, 19600, 21904, 24336, 26896, 29584, 32761, 35721, 39204, 42849, 46656, 50625, 54756, 59049, 63504

Offset: 0

Labos Elemer, Nov 15 2001

			a(10) = 961, as 961 = 31^2 is the largest square <= 1000 = 10^3.

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. A000093, A048760, A065730-A065741, A087285.

Cf. A077115, A077116, A077118, A077121.

a065733 n = head [x | x <- reverse [0.. n^3], a010052 x == 1] -- Reinhard Zumkeller, Oct 10 2013

Table[Floor[Sqrt[w^3]//N]^2, {w, 1, 50}]

A065733(n)=sqrtint(n^3)^2  \\ M. F. Hasler, Oct 05 2013

a(n) + A077116(n) = n^3.
a(n) = A048760(n^3).
n^3 - 2*n^(3/2) <= a(n) <= n^3. - Charles R Greathouse IV, Dec 05 2022
a(n) = A000093(n)^2. - Amiram Eldar, Jul 14 2024

A065741 Largest square <= sum of squares of divisors of n.

Original entry on oeis.org

1, 4, 9, 16, 25, 49, 49, 81, 81, 121, 121, 196, 169, 225, 256, 324, 289, 441, 361, 529, 484, 576, 529, 841, 625, 841, 784, 1024, 841, 1296, 961, 1296, 1156, 1444, 1296, 1849, 1369, 1764, 1681, 2209, 1681, 2500, 1849, 2500, 2304, 2601, 2209, 3364, 2401

Offset: 1

Labos Elemer, Nov 15 2001

Harry J. Smith, Table of n, a(n) for n=1..1000

Cf. A001157, A048760, A065730-A065741.

Table[Floor[Sqrt[DivisorSigma[2, w]]//N]^2, {w, 1, 100}]

a(n) = { sqrtint(sigma(n,2))^2 } \\ Harry J. Smith, Oct 29 2009

a(n) = A048760(A001157(n)).

A257053 Primes in enhanced squares representation, cf. A256913.

Original entry on oeis.org

2, 3, 4, 1, 4, 3, 9, 2, 9, 4, 16, 1, 16, 3, 16, 4, 3, 25, 4, 25, 4, 2, 36, 1, 36, 4, 1, 36, 4, 3, 36, 9, 2, 49, 4, 49, 9, 1, 49, 9, 3, 64, 3, 64, 4, 3, 64, 9, 64, 9, 4, 2, 81, 2, 81, 4, 3, 1, 81, 16, 100, 1, 100, 3, 100, 4, 3, 100, 9, 100, 9, 4, 121, 4, 2

Offset: 1

Reinhard Zumkeller, Apr 15 2015

A257070(n) = length of n-th row;
T(n,k) = A256913(A000040(n),k), k = 0..A257070(n)-1;
T(n,0) = A065730(n) for n > 2;
T(n,A257071(n)-1) = A257070(n).

			.   n | prime(n) |  ESR, row sum = prime(n)
.  ---+----------+-------------------------
.   1 |        2 |  [2]
.   2 |        3 |  [3]
.   3 |        5 |  [4, 1]
.   4 |        7 |  [4, 3]
.   5 |       11 |  [9, 2]
.   6 |       13 |  [9, 4]
.   7 |       17 |  [16, 1]
.   8 |       19 |  [16, 3]
.   9 |       23 |  [16, 4, 3]
.  10 |       29 |  [25, 4]
.  11 |       31 |  [25, 4, 2]
.  12 |       37 |  [36, 1]
.  13 |       41 |  [36, 4, 1]
.  14 |       43 |  [36, 4, 3]
.  15 |       47 |  [36, 9, 2]
.  16 |       53 |  [49, 4]
.  17 |       59 |  [49, 9, 1]
.  18 |       61 |  [49, 9, 3]
.  19 |       67 |  [64, 3]
.  20 |       71 |  [64, 4, 3]
.  21 |       73 |  [64, 9]
.  22 |       79 |  [64, 9, 4, 2]
.  23 |       83 |  [81, 2]
.  24 |       89 |  [81, 4, 3, 1]
.  25 |       97 |  [81, 16]

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened

%Cf. A256913, A000040, A065730, A257070 (traces), A257071 (row lengths).

a257053 n k = a257053_tabf !! (n-1) !! k
a257053_row n = a257053_tabf !! (n-1)
a257053_tabf = map (a256913_row . fromIntegral) a000040_list

A065731 Largest square <= n!.

Original entry on oeis.org

1, 1, 1, 4, 16, 100, 676, 4900, 40000, 362404, 3625216, 39904489, 478996996, 6226945921, 87177877081, 1307672296225, 20922784184449, 355687416544329, 6402373660047556, 121645099966283776, 2432902006216007824, 51090942169052381124, 1124000727752683686724

Offset: 0

Labos Elemer, Nov 15 2001

Harry J. Smith, Table of n, a(n) for n = 0..100

Cf. A048760, A000142, A065730-A065741.

[Floor(Sqrt(Factorial(n)))^2: n in [0..25]]; // Vincenzo Librandi, Dec 07 2018

Array[Floor@ Sqrt[#!]^2 &, 21, 0] (* Michael De Vlieger, Dec 06 2018 *)

a(n) = { sqrtint(n!)^2 } \\ Harry J. Smith, Oct 28 2009

a(n) = floor(sqrt(n!))^2.

a(n) = A048760(A000142(n)).

Offset changed from 1 to 0 by Harry J. Smith, Oct 28 2009

A065732 Largest square <= 2^n.

Original entry on oeis.org

1, 4, 4, 16, 25, 64, 121, 256, 484, 1024, 2025, 4096, 8100, 16384, 32761, 65536, 131044, 262144, 524176, 1048576, 2096704, 4194304, 8386816, 16777216, 33547264, 67108864, 134212225, 268435456, 536848900, 1073741824, 2147395600, 4294967296, 8589767761, 17179869184

Offset: 1

Labos Elemer, Nov 15 2001

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

Cf. A048760, A000079, A065730-A065741.

Table[Floor[Sqrt[2^w]//N]^2, {w, 1, 50}]

a(n) = { sqrtint(2^n)^2 } \\ Harry J. Smith, Oct 28 2009

a(n) = A048760(A000079(n)) = A048760(2^n).

A065736 Largest square <= 10^n.

Original entry on oeis.org

9, 100, 961, 10000, 99856, 1000000, 9998244, 100000000, 999950884, 10000000000, 99999515529, 1000000000000, 9999995824729, 100000000000000, 999999961946176, 10000000000000000, 99999999989350756, 1000000000000000000, 9999999998935075600, 100000000000000000000, 999999999956753113201

Offset: 1

Labos Elemer, Nov 15 2001

Harry J. Smith, Table of n, a(n) for n = 1..100

Cf. A048760, A011557, A065730-A065741.

Table[Floor[Sqrt[10^n]]^2,{n,20}] (* Harvey P. Dale, Dec 04 2014 *)

a(n) = { sqrtint(10^n)^2 } \\ Harry J. Smith, Oct 28 2009

a(n) = A048760(A011557(n)) = A048760(10^n).

A065734 Largest square <= sigma(n).

Original entry on oeis.org

1, 1, 4, 4, 4, 9, 4, 9, 9, 16, 9, 25, 9, 16, 16, 25, 16, 36, 16, 36, 25, 36, 16, 49, 25, 36, 36, 49, 25, 64, 25, 49, 36, 49, 36, 81, 36, 49, 49, 81, 36, 81, 36, 81, 64, 64, 36, 121, 49, 81, 64, 81, 49, 100, 64, 100, 64, 81, 49, 144, 49, 81, 100, 121, 81, 144, 64, 121, 81, 144

Offset: 1

Labos Elemer, Nov 15 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A048760, A000203, A065730-A065741.

Table[Floor[Sqrt[DivisorSigma[1,n]]]^2,{n,70}] (* Harvey P. Dale, Oct 05 2015 *)

A000203(n) := divsum(n) ;
A048760(n) := (floor(sqrt(n)))^2 ;
A065734(n) := A048760(A000203(n)) ;
for n from 1 thru 10 do print(A065734(n)) ; /* R. J. Mathar, Feb 13 2012 */

a(n) = { sqrtint(sigma(n))^2 } \\ Harry J. Smith, Oct 28 2009

a(n) = A048760(A000203(n)).

Previous Mathematica program replaced by Harvey P. Dale, Oct 05 2015

A065735 Largest square <= product of first n primes.

Original entry on oeis.org

1, 4, 25, 196, 2304, 29929, 509796, 9696996, 223084096, 6469628356, 200559769921, 7420737154609, 304250260158441, 13082761295250201, 614889781233144561, 32589158470911505225, 1922760350143213168900

Offset: 1

Labos Elemer, Nov 15 2001

Harry J. Smith, Table of n, a(n) for n = 1..100

Cf. A048760, A002110, A065730-A065741.

Floor[Sqrt[#]]^2&/@FoldList[Times,Prime[Range[20]]] (* Harvey P. Dale, Jan 13 2024 *)

{ m=1; for (n=1, 100, m*=prime(n); a=sqrtint(m)^2; write("b065735.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 28 2009

a(n) = A048760(A002110(n)).

A065737 Largest square <= binomial(n,2).

Original entry on oeis.org

0, 1, 1, 4, 9, 9, 16, 25, 36, 36, 49, 64, 64, 81, 100, 100, 121, 144, 169, 169, 196, 225, 225, 256, 289, 324, 324, 361, 400, 400, 441, 484, 484, 529, 576, 625, 625, 676, 729, 729, 784, 841, 900, 900, 961, 1024, 1024, 1089, 1156, 1225, 1225, 1296, 1369, 1369, 1444

Offset: 1

Labos Elemer, Nov 15 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A048760, A000217, A000290, A061288, A065730-A065741.

Floor[Sqrt[#]]^2&/@Binomial[Range[50],2] (* Harvey P. Dale, Feb 28 2020 *)

a(n) = { sqrtint(binomial(n, 2))^2 } \\ Harry J. Smith, Oct 28 2009

a(n) = A048760(binomial(n,2)).

a(n) = A061288(n-1)^2 = A000290(A061288(n-1)) for n>=2. - Alois P. Heinz, Nov 07 2024

A065738 Largest square <= binomial(2n,n).

Original entry on oeis.org

1, 4, 16, 64, 225, 900, 3364, 12769, 48400, 184041, 703921, 2702736, 10394176, 40106889, 155102116, 601034256, 2333566249, 9075039169, 35345128009, 137845868176, 538256995600, 2104098203601, 8233427666025, 32247599617809

Offset: 1

Labos Elemer, Nov 15 2001

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

Cf. A048760, A000984, A065730-A065741.

a(n) = { sqrtint(binomial(2*n, n))^2 } \\ Harry J. Smith, Oct 29 2009

a(n) = A048760(A000984(n)).

Terms corrected and extended by Harry J. Smith, Oct 29 2009

cp's OEIS Frontend

A065733 Largest square <= n^3.

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A065741 Largest square <= sum of squares of divisors of n.

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A257053 Primes in enhanced squares representation, cf. A256913.

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A065731 Largest square <= n!.

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A065732 Largest square <= 2^n.

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A065736 Largest square <= 10^n.

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A065734 Largest square <= sigma(n).

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A065735 Largest square <= product of first n primes.

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