A048760 -id:A048760 - cp's OEIS frontend

A065730 Largest square <= n-th prime.

1, 1, 4, 4, 9, 9, 16, 16, 16, 25, 25, 36, 36, 36, 36, 49, 49, 49, 64, 64, 64, 64, 81, 81, 81, 100, 100, 100, 100, 100, 121, 121, 121, 121, 144, 144, 144, 144, 144, 169, 169, 169, 169, 169, 196, 196, 196, 196, 225, 225, 225, 225, 225, 225, 256, 256, 256, 256, 256

Offset: 1

Labos Elemer, Nov 15 2001

For n > 2: a(n) = A257053(n,0). - Reinhard Zumkeller, Apr 15 2015

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

Cf. A048760, A000040, A065731-A065741.

Cf. A257053.

a065730 = a048760 . a000040  -- Reinhard Zumkeller, Apr 15 2015

[Floor(Sqrt(NthPrime(n)))^2: n in [1..70]]; // Vincenzo Librandi, Jan 05 2018

Table[Floor[Sqrt@ Prime@ n]^2, {n, 60}] (* Michael De Vlieger, Jul 03 2016 *)

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

a(n) = A048760(A000040(n)).

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)).

A260740 a(n) = n minus the number of positive squares needed to sum to n using the greedy algorithm: a(n) = n - A053610(n).

Original entry on oeis.org

0, 0, 0, 0, 3, 3, 3, 3, 6, 8, 8, 8, 8, 11, 11, 11, 15, 15, 15, 15, 18, 18, 18, 18, 21, 24, 24, 24, 24, 27, 27, 27, 27, 30, 32, 32, 35, 35, 35, 35, 38, 38, 38, 38, 41, 43, 43, 43, 43, 48, 48, 48, 48, 51, 51, 51, 51, 54, 56, 56, 56, 56, 59, 59, 63, 63, 63, 63, 66, 66, 66, 66, 69, 71, 71, 71, 71, 74, 74, 74, 78, 80

Offset: 0

Antti Karttunen, Aug 12 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A048760, A053610, A062535, A255131, A261221, A261222, A261223, A261224.

Cf. also A261225.

a(n) = n - A053610(n).
As a recurrence:
a(0) = 0; for n >= 1, a(n) = -1 + A048760(n) + a(n-A048760(n)). [Where A048760(n) gives the largest square <= n.]
Other identities. For all n >= 1:
a(n) = A255131(n) - A062535(n).

A060984 a(1) = 1; a(n+1) = a(n) + (largest square <= a(n)).

Original entry on oeis.org

1, 2, 3, 4, 8, 12, 21, 37, 73, 137, 258, 514, 998, 1959, 3895, 7739, 15308, 30437, 60713, 121229, 242333, 484397, 967422, 1933711, 3865811, 7730967, 15459367, 30912128, 61814609, 123625653, 247235577, 494448306, 988888002, 1977738918, 3955408759, 7910812423

Offset: 1

R. K. Guy, May 11 2001

Arises in analyzing "put-or-take" games (see Winning Ways, 484-486, 501-503), the prototype being Epstein's Put-or-Take-a-Square game.

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982.
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section E26.

Henry J. Smith and Harvey P. Dale, Table of n, a(n) for n = 1..1000 (first 500 terms from Henry J. Smith)

Cf. A060985, A237888.

a060984 n = a060984_list !! (n-1)
a060984_list = iterate (\x -> x + a048760 x) 1
-- Reinhard Zumkeller, Dec 24 2013

a[1] = 1; a[n_] := a[n] = a[n - 1] + Floor[ Sqrt[ a[n - 1] ] ]^2; Table[ a[n], {n, 1, 40} ]
RecurrenceTable[{a[1]==1,a[n]==a[n-1]+Floor[Sqrt[a[n-1]]]^2},a,{n,40}] (* Harvey P. Dale, Nov 19 2011 *)
NestList[#+Floor[Sqrt[#]]^2&,1,40] (* Harvey P. Dale, Jan 22 2013 *)

{ default(realprecision, 100); for (n=1, 500, if (n==1, a=1, a+=floor(sqrt(a))^2); write("b060984.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 15 2009

from sympy import integer_nthroot
A060984_list = [1]
for i in range(10**3): A060984_list.append(A060984_list[-1]+integer_nthroot(A060984_list[-1],2)[0]**2) # Chai Wah Wu, Apr 02 2021

from math import isqrt
from itertools import accumulate
def f(an, _): return an + isqrt(an)**2
print(list(accumulate([1]*36, f))) # Michael S. Branicky, Apr 02 2021

a(n+1) = a(n)+[sqrt(a(n))]^2 = a(n)+A061886(n) = a(n)+A048760(a(n)) = A061887(a(n)). - Henry Bottomley, May 12 2001

a(n) ~ c * 2^n, where c = 0.11511532187216693... (see A237888). - Vaclav Kotesovec, Feb 15 2014

More terms from David W. Wilson, Henry Bottomley and Robert G. Wilson v, May 12 2001

A154333 Difference between n^3 and the next smaller square.

Original entry on oeis.org

1, 4, 2, 15, 4, 20, 19, 28, 53, 39, 35, 47, 81, 40, 11, 127, 13, 56, 135, 79, 45, 39, 67, 135, 249, 152, 83, 48, 53, 104, 207, 7, 216, 100, 26, 431, 28, 116, 270, 496, 277, 104, 546, 503, 524, 615, 139, 368, 685, 391, 155, 732, 652, 648, 726, 55, 293, 631, 170, 704, 405

Offset: 1

M. F. Hasler, Jan 07 2009

nonn

The sequence A077116(n) = n^3-[sqrt(n^3)]^2 satisfies A077116(n)=0 <=> n^3 is a square <=> n is a square. It differs from the present sequence (which is always positive) only in these indices, where a(k^2)=2k^3-1.

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

Cf. A087285 (range of this sequence, excluding the initial term 1).

A154333 := proc(n)
    A071797(n^3) ;
end proc: # R. J. Mathar, May 29 2016

nss[n_]:=Module[{n3=n^3,s},s=Floor[Sqrt[n3]]^2;If[s==n3,s=(Sqrt[s]- 1)^2, s]]; Table[n^3-nss[n],{n,70}] (* Harvey P. Dale, Jan 19 2017 *)

A154333(n) = n^3-sqrtint(n^3-1)^2
a154333 = vector(90,n,n^3-sqrtint(n^3-1)^2)

a(n) = n^3 - [sqrt(n^3 - 1)]^2 = A000578(n) - A048760(n^3-1). a(k^2) = 2 k^3 - 1.

a(n) = A071797(n^3). - R. J. Mathar, May 29 2016

A202304 a(n) = floor(sqrt(3*n)).

Original entry on oeis.org

0, 1, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14

Offset: 0

Vincenzo Librandi, Jan 17 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Cf. A060018, A172471.

Magma
```
[Isqrt(3*n) : n in [0..80]];
```
Mathematica
```
Table[Floor[Sqrt[3n]],{n,0,80}]
```

a(n)=sqrtint(3*n) \\ Charles R Greathouse IV, Jan 17 2012

a(n) = A000196(3n), a(n)^2 = A048760(3n). - Bruno Berselli, Jan 18 2012

A005240 P-positions in Epstein's Put or Take a Square game.

Original entry on oeis.org

0, 5, 20, 29, 45, 80, 101, 116, 135, 145, 165, 173, 236, 257, 397, 404, 445, 477, 540, 565, 580, 585, 629, 666, 836, 845, 885, 909, 944, 949, 954, 975, 1125, 1177

Offset: 1

N. J. A. Sloane

nonn

The game is played with two players alternatingly removing or adding chips on a heap. If C denotes the number of chips on the heap, a player must either put or take the largest possible square number of chips in his move, C -> C +- A048760(C). The player capable of taking all chips wins. The P positions are numbers of chips where the player to draw first will lose (assuming the opponent has a full analysis of the game). - R. J. Mathar, May 06 2016

			5 is a term because either putting 4 or taking 4 leads to squares (9 or 1) and the opponent wins by taking.
20 is a term because either putting 16 or taking 16 leads to squares (36 or 4) and the opponent wins by taking.

R. K. Guy, Unsolved Problems in Number Theory, E26.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

E. R. Berlekamp, J. H. Conway, R. K. Guy, Gewinnen (Strategien fur mathematische Spiele), Vieweg, (1986) p. 58.
R. K. Guy, Letter to N. J. A. Sloane, Aug 1975.

Cf. A005241, A048760.

A056893 Smallest prime with square excess of n.

Original entry on oeis.org

2, 3, 7, 13, 41, 31, 23, 89, 73, 59, 47, 61, 113, 239, 79, 97, 593, 139, 163, 461, 277, 191, 167, 193, 281, 251, 223, 317, 353, 991, 431, 761, 433, 563, 359, 397, 521, 479, 439, 569, 617, 571, 619, 773, 829, 887, 947, 673, 1493, 1571, 727, 1013, 953, 1279

Offset: 1

Henry Bottomley, Jul 05 2000

nonn

			a(4)=13 since 13=3^2+4, while 2, 3, 5, 7 and 11 have square excesses of 1, 2, 1, 3 and 3 respectively.

Cf. A053186, A000196, A002496, A048760, A056892-A056898.

A056893 := proc(n)
    local p ;
    p :=2 ;
    while A053186(p) <> n do
        p := nextprime(p) ;
    end do:
    return p ;
end proc: # R. J. Mathar, Jul 28 2013

A056893(n)={
    local(p=2) ;
    while( A053186(p)!=n,
        p=nextprime(p+1)
    ) ;
    return(p)
} /* R. J. Mathar, Jul 28 2013 */

a(n) =n+A056894(n).

a(n) = min{p in A000040: A053186(p) = n}. - R. J. Mathar, Jul 28 2013

A066857 Smallest number k such that n! - k is a square.

Original entry on oeis.org

0, 1, 2, 8, 20, 44, 140, 320, 476, 3584, 12311, 4604, 74879, 414119, 2071775, 5703551, 11551671, 45680444, 442548224, 1960632176, 2657058876, 24923993276, 130518272975, 1478154932316, 5446454455004, 38610655379975

Offset: 1

Reinhard Zumkeller, Jan 21 2002

nonn

Sequence is not monotonic: a(n) < a(n-1) for n = 12, 71, 90, 143, 145, 151, 172, 218, 257. - Zak Seidov, Jun 25 2013

			a(10) = 3628800 - 1904 * 1904 = 3628800 - 3625216 = 3584.

T. D. Noe, Table of n, a(n) for n = 1..100

Cf. A068869.

Table[n! - Floor[Sqrt[n! ]]^2, {n, 1, 27}]

a(n)=my(N=n!); N-sqrtint(N)^2 \\ Charles R Greathouse IV, Jun 25 2013

a(n) = A053186(n!) = n!-A048760(n!) = n!-floor(sqrt(n!))^2 = n!-A055226(n)^2.

More terms from Vladeta Jovovic, Mar 21 2002

Edited by Robert G. Wilson v and N. J. A. Sloane, Mar 22 2002

A174060 a(n) = Sum_{k=0..n} floor(sqrt(k))^2.

Original entry on oeis.org

0, 1, 2, 3, 7, 11, 15, 19, 23, 32, 41, 50, 59, 68, 77, 86, 102, 118, 134, 150, 166, 182, 198, 214, 230, 255, 280, 305, 330, 355, 380, 405, 430, 455, 480, 505, 541, 577, 613, 649, 685, 721, 757, 793, 829, 865, 901, 937, 973, 1022, 1071, 1120, 1169, 1218, 1267, 1316

Offset: 0

Vladimir Joseph Stephan Orlovsky, Mar 06 2010

Partial sums of A048760. - R. J. Mathar, Mar 31 2010

Karl-Heinz Hofmann, Table of n, a(n) for n = 0..10000

Cf. A022554 (1st), this sequence (2nd), A363497 (3rd).

Cf. A363498 (4th), A363499 (5th), A048760.

Accumulate[Table[Floor[Sqrt[k]]^2, {k, 0, 59}]] (* Harvey P. Dale, Jul 13 2013 *)

a(n)=my(m=sqrtint(n+1));(n+1)*m^2-m*(m+1)*(3*m^2+m-1)/6 \\ Charles R Greathouse IV, Jul 04 2013

a(n) = sum(k=0, n, sqrtint(k)^2); \\ Karl-Heinz Hofmann, Jun 15 2023

from math import isqrt
A174060 = [0]
for n in range(1,56): A174060.append(A174060[-1] + isqrt(n)**2)
print(A174060) # Karl-Heinz Hofmann, Jun 15 2023

from math import isqrt
def A174060(n): return ((m:=isqrt(n+1))*(6*m*(n+1) - (m+1)*(3*m**2+m-1)))//6
# Karl-Heinz Hofmann, Jun 15 2023

a(n) = (1/6)*m*(6*m*n - (m+1)*(3*m^2+m-1)) with m = floor(sqrt(n)). - Yalcin Aktar, Jan 30 2012

cp's OEIS Frontend

A065730 Largest square <= n-th prime.

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

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A260740 a(n) = n minus the number of positive squares needed to sum to n using the greedy algorithm: a(n) = n - A053610(n).

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A060984 a(1) = 1; a(n+1) = a(n) + (largest square <= a(n)).

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A154333 Difference between n^3 and the next smaller square.

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A202304 a(n) = floor(sqrt(3*n)).

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A005240 P-positions in Epstein's Put or Take a Square game.

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A056893 Smallest prime with square excess of n.