A000093 -id:A000093 - cp's OEIS frontend

A185543 Numbers not of the form floor(k^(3/2)); complement of A000093.

3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 26, 28, 29, 30, 32, 33, 34, 35, 37, 38, 39, 40, 42, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 83, 84, 85, 86, 87, 88, 90

Offset: 1

Clark Kimberling, Jan 30 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000
Chai Wah Wu, Algorithms for complementary sequences, arXiv:2409.05844 [math.NT], 2024.

Cf. A000093.

f[n_]=Floor[n^(3/2)]
t1=Table[f[n],{n,1,80}];t1  (* A000093 *)
t2=Complement[Range[200], Table[f[n],{n,1,80}]];t2  (* A185543 *)

def A185543(n):
    def f(x): return n-1+(a:=integer_nthroot((x+1)**2,3))[0]+(a[1]^1)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Sep 10 2024

A216155 Numbers n such that floor(sqrt(n + n^3)) = 1 + floor(sqrt(n^3)) = 1 + A000093(n).

Original entry on oeis.org

2, 13, 40, 43, 46, 52, 109, 152, 190, 243, 336, 351, 356, 366, 422, 584, 592, 741, 937, 978, 1011, 1040, 1137, 1330, 1355, 1362, 1376, 1398, 1434, 2063, 2320, 2520, 2553, 2660, 2665, 2928, 2940, 2993, 3067, 3075, 3092, 3296, 3532, 3631, 3703, 3712, 3730

Offset: 1

Zak Seidov, Sep 02 2012

nonn

The sequence is infinite. For values of n not in the sequence we have floor(sqrt(n+n^3)) = floor(sqrt(n^3)) = A000093(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000093 (floor(n^(3/2))).

Cf. A000196, A000578, A034262, A247628 (subsequence).

a216155 n = a216155_list !! (n-1)
a216155_list = filter
   (\x -> a000196 (a034262 x) == a000196 (a000578 x) + 1) [1..]
-- Reinhard Zumkeller, Sep 26 2014

Select[Range[10000], Floor[Sqrt[# + #^3]] - Floor[Sqrt[#^3]] == 1 &]

A000161 Number of partitions of n into 2 squares.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 1, 0, 0, 1, 0, 1, 0

Offset: 0

N. J. A. Sloane

Number of ways of writing n as a sum of 2 (possibly zero) squares when order does not matter.
Number of similar sublattices of square lattice with index n.
Let Pk = the number of partitions of n into k nonzero squares. Then we have A000161 = P0 + P1 + P2, A002635 = P0 + P1 + P2 + P3 + P4, A010052 = P1, A025426 = P2, A025427 = P3, A025428 = P4. - Charles R Greathouse IV, Mar 08 2010, amended by M. F. Hasler, Jan 25 2013
a(A022544(n))=0; a(A001481(n))>0; a(A125022(n))=1; a(A118882(n))>1. - Reinhard Zumkeller, Aug 16 2011

			25 = 3^2+4^2 = 5^2, so a(25) = 2.

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 339

T. D. Noe, Table of n, a(n) for n = 0..10000
B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216. [Annotated scanned copy]
Henry Bottomley, Illustration of initial terms
R. T. Bumby, Sums of four squares, in Number theory (New York, 1991-1995), 1-8, Springer, New York, 1996.
J. H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 1300-1306 (Abstract, pdf, ps).
Michael Gilleland, Some Self-Similar Integer Sequences
E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, NY, 1985, p. 84.
M. D. Hirschhorn, Some formulas for partitions into squares, Discrete Math, 211 (2000), pp. 225-228. [From _Ant King_, Oct 05 2010]
Index entries for sequences related to sublattices
Index entries for sequences related to sums of squares
Index entries for "core" sequences

Cf. A002654, A001481, A025427, A025428, A063725, A025426, A000290.
Cf. A000925, A247367.
Equivalent sequences for other numbers of squares: A010052 (1), A000164 (3), A002635 (4), A000174 (5).

a000161 n =
   sum $ map (a010052 . (n -)) $ takeWhile (<= n `div` 2) a000290_list
a000161_list = map a000161 [0..]
-- Reinhard Zumkeller, Aug 16 2011

A000161 := proc(n) local i,j,ans; ans := 0; for i from 0 to n do for j from i to n do if i^2+j^2=n then ans := ans+1 fi od od; RETURN(ans); end; [ seq(A000161(i), i=0..50) ];
A000161 := n -> nops( numtheory[sum2sqr](n) ); # M. F. Hasler, Nov 23 2007

Length[PowersRepresentations[ #,2,2]] &/@Range[0,150] (* Ant King, Oct 05 2010 *)

a(n)=sum(i=0,n,sum(j=0,i,if(i^2+j^2-n,0,1))) \\ for illustrative purpose

A000161(n)=sum(k=sqrtint((n-1)\2)+1,sqrtint(n),issquare(n-k^2)) \\ Charles R Greathouse IV, Mar 21 2014, improves earlier code by M. F. Hasler, Nov 23 2007

A000161(n)=#sum2sqr(n) \\ See A133388 for sum2sqr(). - M. F. Hasler, May 13 2018

from math import prod
from sympy import factorint
def A000161(n):
    f = factorint(n)
    return int(not any(e&1 for e in f.values())) + (((m:=prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in f.items()))+((((~n & n-1).bit_length()&1)<<1)-1 if m&1 else 0))>>1) if n else 1 # Chai Wah Wu, Sep 08 2022

a(n) = card { { a,b } c N | a^2+b^2 = n }. - M. F. Hasler, Nov 23 2007

Let f(n)= the number of divisors of n that are congruent to 1 modulo 4 minus the number of its divisors that are congruent to 3 modulo 4, and define delta(n) to be 1 if n is a perfect square and 0 otherwise. Then a(n)=1/2 (f(n)+delta(n)+delta(1/2 n)). - Ant King, Oct 05 2010

A094683 Juggler sequence: if n mod 2 = 0 then floor(sqrt(n)) else floor(n^(3/2)).

Original entry on oeis.org

0, 1, 1, 5, 2, 11, 2, 18, 2, 27, 3, 36, 3, 46, 3, 58, 4, 70, 4, 82, 4, 96, 4, 110, 4, 125, 5, 140, 5, 156, 5, 172, 5, 189, 5, 207, 6, 225, 6, 243, 6, 262, 6, 281, 6, 301, 6, 322, 6, 343, 7, 364, 7, 385, 7, 407, 7, 430, 7, 453, 7, 476, 7, 500, 8, 524, 8, 548, 8, 573, 8, 598, 8, 623, 8, 649

Offset: 0

N. J. A. Sloane, Jun 09 2004

nonn

Interspersion of A000093 and A000196. - Michel Marcus, Nov 11 2013

C. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, p. 233.

Karl V. Boddy, Table of n, a(n) for n = 0..10000
Vikram Prasad and M. A. Prasad, Estimates of the maximum excursion constant and stopping constant of juggler-like sequences, ResearchGate, 2025.
Harry J. Smith, Juggler Sequence
Eric Weisstein's World of Mathematics, Juggler Sequence
Wikipedia, Juggler sequence

Cf. A007320, A007321, A094684, A094685, A094716.

Cf. A000093, A000196.

A094683 :=proc(n) if n mod 2 = 0 then RETURN(floor(sqrt(n))) else RETURN(floor(n^(3/2))); end if; end proc;

Table[If[EvenQ[n], Floor[Sqrt[n]], Floor[n^(3/2)]], {n, 0, 100}] (* Indranil Ghosh, Apr 07 2017 *)

a(n) = if(n%2,sqrtint(n^3), sqrtint(n)); \\ Indranil Ghosh, Apr 08 2017

import math
from sympy import sqrt
def a(n): return int(math.floor(sqrt(n))) if n%2 == 0 else int(math.floor(n**(3/2)))
print([a(n) for n in range(51)]) # Indranil Ghosh, Apr 08 2017

from math import isqrt
def A094683(n): return isqrt(n**3 if n % 2 else n) # Chai Wah Wu, Feb 18 2022

A065733 Largest square <= n^3.

Original entry on oeis.org

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

A077121 Number of integer squares <= n^3.

Original entry on oeis.org

1, 2, 3, 6, 9, 12, 15, 19, 23, 28, 32, 37, 42, 47, 53, 59, 65, 71, 77, 83, 90, 97, 104, 111, 118, 126, 133, 141, 149, 157, 165, 173, 182, 190, 199, 208, 217, 226, 235, 244, 253, 263, 273, 282, 292, 302, 312, 323, 333, 344, 354, 365, 375, 386, 397, 408, 420

Offset: 0

Reinhard Zumkeller, Oct 29 2002

a(n) = number of terms in n-th row of A167222. - Reinhard Zumkeller, Oct 31 2009

			Squares <= 3^3 = 27: 0, 1, 4, 9, 16 and 25, hence a(3) = 6.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000093, A065733, A077113, A167222, A167223.

a[n_] := Floor[Sqrt[n^3]] + 1; Array[a, 100, 0] (* Amiram Eldar, Apr 06 2025 *)

from math import isqrt
def A077121(n): return isqrt(n**3)+1 # Chai Wah Wu, Sep 08 2024

a(n) = floor(n^(3/2))+1 = A000093(n) + 1.

A155013 Integer part of square root of n^5 = A000584(n).

Original entry on oeis.org

1, 5, 15, 32, 55, 88, 129, 181, 243, 316, 401, 498, 609, 733, 871, 1024, 1191, 1374, 1573, 1788, 2020, 2270, 2536, 2821, 3125, 3446, 3787, 4148, 4528, 4929, 5350, 5792, 6255, 6740, 7247, 7776, 8327, 8901, 9498, 10119, 10763, 11432, 12124, 12841, 13584, 14351

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 19 2009

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000093.

Integer part of square root of n^k: A000196 (k=1), A000093 (k=3), this sequence (k=5), A155014 (k=7), A155015 (k=11), A155016 (k=13), A155018 (k=15), A155019 (k=17),

[Floor(Sqrt(n^5)): n in [1..30]]; // G. C. Greubel, Dec 30 2017

a={};Do[AppendTo[a,IntegerPart[(n^5)^(1/2)]],{n,5!}];a
IntegerPart[Sqrt[Range[50]^5]] (* Harvey P. Dale, May 14 2012 *)
Table[Floor[Sqrt[n^5]], {n,1,30}] (* G. C. Greubel, Dec 30 2017 *)

for(n=1,30, print1(sqrtint(n^5), ", ")) \\ G. C. Greubel, Dec 30 2017

from math import isqrt
def A155013(n): return isqrt(n**5) # Chai Wah Wu, Aug 08 2025

a(n) = floor(n^2 * sqrt(n)). - Davide Rotondo, Dec 01 2024

A002821 a(n) = nearest integer to n^(3/2).

Original entry on oeis.org

0, 1, 3, 5, 8, 11, 15, 19, 23, 27, 32, 36, 42, 47, 52, 58, 64, 70, 76, 83, 89, 96, 103, 110, 118, 125, 133, 140, 148, 156, 164, 173, 181, 190, 198, 207, 216, 225, 234, 244, 253, 263, 272, 282, 292, 302, 312, 322, 333, 343, 354, 364, 375, 386, 397

Offset: 0

N. J. A. Sloane

M. Boll, Tables Numériques Universelles. Dunod, Paris, 1947, p. 46.
M. Hall, Jr., The Diophantine equation x^3-y^2=k, pp. 173-198 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
A. V. Lebedev and R. M. Fedorova, A Guide to Mathematical Tables. Pergamon, Oxford, 1960, p. 17.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A000093, A077118.

a002821 = round . sqrt . fromIntegral . (^ 3)
-- Reinhard Zumkeller, Jul 11 2014

A002821 := proc(n)
    round(n^(3/2)) ;
end proc:
seq(A002821(n),n=0..100) ;

t[n_]:=Module[{flt=Floor[n],cet=Ceiling[n]},If[n-fltHarvey P. Dale, May 12 2011 *)

from math import isqrt
def A002821(n): return (m:=isqrt(k:=n**3))+int((k-m*(m+1)<<2)>=1) # Chai Wah Wu, Jul 30 2022

a(n) = floor(n^(3/2) + 1/2). - Ridouane Oudra, Nov 13 2019

a(n) = sqrt(A077118(n)). - Chai Wah Wu, Jul 30 2022

A155016 Integer part of square root of A010801.

Original entry on oeis.org

0, 1, 90, 1262, 8192, 34938, 114283, 311269, 741455, 1594323, 3162277, 5875603, 10343751, 17403307, 28172943, 44115700, 67108864, 99521746, 144301645, 205068240, 286216701, 393029741, 531798888, 709955183, 936209559, 1220703125

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jan 19 2009

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 100 terms from Karl V. Keller, Jr.)

Integer part of square root of n^k: A000196 (k=1), A000093 (k=3), A155013 (k=5), A155014 (k=7), A155015 (k=11), this sequence (k=13), A155018 (k=15), A155019 (k=17).

[Floor(Sqrt(n^13)): n in [1..30]]; // G. C. Greubel, Dec 30 2017

a={};Do[AppendTo[a,IntegerPart[(n^13)^(1/2)]],{n,0,5!}];a
Table[Floor[Sqrt[n^13]], {n,1,30}] (* G. C. Greubel, Dec 30 2017 *)

for(n=1,30, print1(floor(sqrt(n^13)), ", ")) \\ G. C. Greubel, Dec 30 2017

Offset corrected by Karl V. Keller, Jr., Sep 27 2014

A155018 Integer part of square root of n^15 = A010803(n).

Original entry on oeis.org

0, 1, 181, 3787, 32768, 174692, 685700, 2178889, 5931641, 14348907, 31622776, 64631634, 124125023, 226242995, 394421215, 661735513, 1073741824, 1691869691, 2597429617, 3896296578, 5724334022, 8253624572, 11699575548, 16328969210

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jan 19 2009

nonn

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

Integer part of square root of n^k: A000196 (k=1), A000093 (k=3), A155013 (k=5), A155014 (k=7), A155015 (k=11), A155016 (k=13), this sequence (k=15), A155019 (k=17).

[Floor(Sqrt(n^15)): n in [1..30]]; // G. C. Greubel, Dec 30 2017

a={};Do[AppendTo[a,IntegerPart[(n^15)^(1/2)]],{n,0,5!}];a
Table[Floor[Sqrt[n^15]], {n,1,30}] (* G. C. Greubel, Dec 30 2017 *)

for(n=1,30, print1(floor(sqrt(n^15)), ", ")) \\ G. C. Greubel, Dec 30 2017

Offset corrected by Alois P. Heinz, Sep 27 2014

cp's OEIS Frontend

A185543 Numbers not of the form floor(k^(3/2)); complement of A000093.

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A216155 Numbers n such that floor(sqrt(n + n^3)) = 1 + floor(sqrt(n^3)) = 1 + A000093(n).

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A000161 Number of partitions of n into 2 squares.

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A094683 Juggler sequence: if n mod 2 = 0 then floor(sqrt(n)) else floor(n^(3/2)).

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A065733 Largest square <= n^3.

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A077121 Number of integer squares <= n^3.

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A155013 Integer part of square root of n^5 = A000584(n).

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