A048761 -id:A048761 - cp's OEIS frontend

A165775 n + (least square >= n), i.e., n + A048761(n).

0, 2, 6, 7, 8, 14, 15, 16, 17, 18, 26, 27, 28, 29, 30, 31, 32, 42, 43, 44, 45, 46, 47, 48, 49, 50, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 146

Offset: 0

M. F. Hasler, Oct 05 2009

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Table[n+Ceiling[Sqrt[n]]^2,{n,0,70}] (* Harvey P. Dale, Feb 04 2019 *)

PARI
```
A165775(n)=n+(sqrtint(n-1)+1)^2
```

A165775 = A001477 + A048761

A072690 a(n) = (n - A048760(n)) * (A048761(n) - n).

Original entry on oeis.org

0, 2, 2, 0, 4, 6, 6, 4, 0, 6, 10, 12, 12, 10, 6, 0, 8, 14, 18, 20, 20, 18, 14, 8, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12, 0, 14, 26, 36, 44, 50, 54, 56, 56, 54, 50, 44, 36, 26, 14, 0, 16, 30, 42, 52, 60, 66, 70, 72, 72, 70, 66

Offset: 1

Reinhard Zumkeller, Jul 02 2002

nonn

a(n)=0 iff n is a square.

a(n) = A053186(n) * A068527(n).

a(n) = A053186(n) * (A072689(n) - A053186(n)).

A165776 n + (least square > n), i.e., n + A048761(n+1).

Original entry on oeis.org

1, 5, 6, 7, 13, 14, 15, 16, 17, 25, 26, 27, 28, 29, 30, 31, 41, 42, 43, 44, 45, 46, 47, 48, 49, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 145, 146

Offset: 0

M. F. Hasler, Oct 05 2009

			a(4)= 13 = 4+9 because 9 is the least square > 4.

Cf. A165775.

PARI
```
A165776(n)=n+(sqrtint(n)+1)^2
```

a(n) = A001477(n) + A048761(n+1)

A048760 Largest square <= n.

Original entry on oeis.org

0, 1, 1, 1, 4, 4, 4, 4, 4, 9, 9, 9, 9, 9, 9, 9, 16, 16, 16, 16, 16, 16, 16, 16, 16, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64

Offset: 0

Charles T. Le (charlestle(AT)yahoo.com)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Krassimir T. Atanassov, On Some of Smarandache's Problems, 1999.
Henry Bottomley, Illustration of A000196, A048760, A053186.
Jose Castillo, Other Smarandache Type Functions: Inferior/Superior Smarandache f-part of x, Smarandache Notions Journal, Vol. 10, No. 1-2-3 (1999), pp. 202-204.
Valentina V. Radeva and Krassimir T. Atanassov, On the 40-th and 41-st Smarandache's problems, Notes on Number Theory and Discrete Mathematics, Vol. 4, No. 3 (1998), pp. 101-104.
Florentin Smarandache, Only Problems, Not Solutions!, 1993.
Michael Somos, Sequences used for indexing triangular or square arrays.

Cf. A000196, A000290, A048761, A345202.

a048760 = (^ 2) . a000196  -- Reinhard Zumkeller, Feb 12 2012

A048760 := proc(n)
    floor(sqrt(n)) ;
    %^2 ;
end proc: # R. J. Mathar, May 19 2016

Array[Floor[Sqrt[#]]^2&,80,0] (* Harvey P. Dale, Mar 30 2012 *)
Table[PadRight[{},2n+1,n^2],{n,0,10}]//Flatten (* Harvey P. Dale, Feb 28 2025 *)

a(n) = sqrtint(n)^2; \\ Michel Marcus, Jun 06 2015

a(n) = floor(n^(1/2))^2 = A000290(A000196(n)). - Reinhard Zumkeller, Feb 12 2012, Sep 03 2002
n^2 repeated (2n+1) times, n=0,1,... - Zak Seidov, Oct 25 2008
Sum_{n>=1} (1/a(n) - 1/n) = gamma + zeta(2) (= A345202). - Amiram Eldar, Jun 12 2021
Sum_{n>=1} 1/a(n)^2 = 2*zeta(3) + Pi^4/90. - Amiram Eldar, Aug 15 2022

A068527 Difference between smallest square >= n and n.

Original entry on oeis.org

0, 0, 2, 1, 0, 4, 3, 2, 1, 0, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9

Offset: 0

Vladeta Jovovic, Mar 21 2002

The greedy inverse (sequence of the smallest k such that a(k)=n) starts 0, 3, 2, 6, 5, 11, 10, 18, 17, 27, 26, 38, 37, 51, 50, ... and appears to be given by A010000 and A002522, interleaved. - R. J. Mathar, Nov 17 2014

Ivan Panchenko, Table of n, a(n) for n = 0..1000

Cf. A053186, A068869, A066857.

Bisections: A348596, A350962.

[ Ceiling(Sqrt(n))^2-n : n in [0..50] ]; // Wesley Ivan Hurt, Jun 11 2014

A068527:=n->ceil(sqrt(n))^2-n; seq(A068527(n), n=0..100); # Wesley Ivan Hurt, Jun 11 2014

Table[Ceiling[Sqrt[n]]^2-n,{n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)

a(n)=if(issquare(n), 0, (sqrtint(n)+1)^2-n) \\ Charles R Greathouse IV, Oct 22 2014

from math import isqrt
def A068527(n): return 0 if n == 0 else (isqrt(n-1)+1)**2-n # Chai Wah Wu, Feb 22 2022

a(n) = A048761(n) - n = ceiling(sqrt(n))^2 - n.

G.f.: (-x^2 + (x-x^2)*Sum_{m>=1} (1+2*m)*x^(m^2))/(1-x)^2. This sum is related to Jacobi Theta functions. - Robert Israel, Nov 17 2014

A068869 Smallest number k such that n! + k is a square.

Original entry on oeis.org

0, 2, 3, 1, 1, 9, 1, 81, 729, 225, 324, 39169, 82944, 176400, 215296, 3444736, 26167684, 114349225, 255004929, 1158920361, 11638526761, 42128246889, 191052974116, 97216010329, 2430400258225, 1553580508516, 4666092737476, 565986718738441, 2137864362693921

Offset: 1

Amarnath Murthy, Mar 13 2002

nonn

Observation: for n < 2000, only for n = 1, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16 is a(n) a square (see A360210).
According to my conjecture that n! + n^2 != m^2 for n >= 1, m >= 0 (see A004664), for all terms: a(n) != n^2. - Alexander R. Povolotsky, Oct 06 2008
There are two cases: a(n) > sqrt(n!) in A182203 and a(n) < sqrt(n!) in A182204. - Artur Jasinski, Apr 13 2012

			a(6) = 9 as 6! + 9 = 729 is a square.

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

Cf. A038202, A048761, A055228, A066857, A360210.

Cf. A182203, A182204.

Table[ Ceiling[ Sqrt[n! ]]^2 - n!, {n, 1, 28}]

A068869(n)=(sqrtint(n!-1)+1)^2-n!  \\ M. F. Hasler, Apr 01 2012

from math import factorial, isqrt
def a(n): return (isqrt((f:=factorial(n))-1)+1)**2 - f
print([a(n) for n in range(1, 30)]) # Michael S. Branicky, Jan 30 2023

a(n) = A055228(n)^2 - n! = ceiling(sqrt(n!))^2 - n! = A048761(n!) - n!.

a(n) <= A038202(n)^2, with equality for the n listed in the first comment. - M. F. Hasler, Apr 01 2012

More terms from Vladeta Jovovic, Mar 21 2002

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

A077115 Least integer square >= n^3.

Original entry on oeis.org

0, 1, 9, 36, 64, 144, 225, 361, 529, 729, 1024, 1369, 1764, 2209, 2809, 3481, 4096, 5041, 5929, 6889, 8100, 9409, 10816, 12321, 13924, 15625, 17689, 19881, 22201, 24649, 27225, 29929, 33124, 36100, 39601, 43264, 46656, 51076, 55225, 59536

Offset: 0

Reinhard Zumkeller, Oct 29 2002

			a(10) = 1024, as 1024 = 32^2 is the least square >= 1000 = 10^3.

Cf. A000290, A000578, A048761, A065733, A077107, A077118, A185549.

[Ceiling(n^(3/2))^2: n in [0..50]]; // Vincenzo Librandi, Feb 17 2015

lis[n_]:=Module[{c=Sqrt[n^3]},If[IntegerQ[c],c^2,(Floor[c]+1)^2]]; Array[lis,40,0] (* Harvey P. Dale, Jan 22 2013 *)

a(n) - A070929(n) = n^3.
a(n) = ceiling(n^(3/2))^2. - Benoit Cloitre, Nov 01 2002
a(n) = A185549(n)^2. - Amiram Eldar, May 17 2025
a(n) = A048761(n^3). - Michel Marcus, May 17 2025

A243091 Least number k > n such that n concatenated with k is a perfect square.

Original entry on oeis.org

1, 6, 5, 6, 9, 29, 25, 29, 41, 61, 24, 56, 25, 69, 44, 21, 81, 64, 49, 36, 25, 316, 201, 104, 336, 281, 244, 225, 224, 241, 276, 36, 49, 64, 81, 344, 100, 249, 44, 69, 96, 209, 436, 56, 89, 369, 225, 61, 400, 284, 176, 84, 441, 361, 76, 225, 169, 76, 564, 536, 84, 504, 500, 504, 516, 536

Offset: 0

Derek Orr, Aug 18 2014

Records occur at: 0, 1, 4, 5, 8, 9, 13, 16, 21, 24, 35, 42, 52, 58, 67, 75, 80, ..., . - Robert G. Wilson v, Nov 23 2015

			a(1) = 6 since 6>1 and 16 = 4^2.
a(2) = 5 since 5>2 and 25 = 5^2.

Robert G. Wilson v, Table of n, a(n) for n = 0..1000

Cf. A090566.

f[n_] := Block[{x = n, d = 1 + Floor@ Log10@ n}, q = (Floor@ Sqrt[(10^d + 1) x] + 1)^2; If[q < (10^d) (x + 1), Mod[q, 10^d], Mod[(Floor@ Sqrt[(10^d) (10 x + 1) - 1] + 1)^2, 10^(d + 1)]]]; Array[f, 65] (* Robert G. Wilson v, Nov 23 2015, after the algorithm of David W. Wilson in A090566 *)
lnk[n_]:=Module[{k=n+1},While[!IntegerQ[Sqrt[n 10^IntegerLength[k]+k]],k++];k]; Array[lnk,70,0] (* Harvey P. Dale, Sep 01 2023 *)

a(n)=s=Str(n); k=n+1; while(!issquare(eval(concat(s,Str(k)))), k++); return(k)
vector(100, n, a(n))

A048761 = t->(sqrtint(t-1)+1)^2
A243091(n)={my(d=#Str(n),a=A048761((1+10^d)*n)); a>=(n+1)*10^d && a=A048761((n*10+1)*10^d); a%10^(d+(a>=100^d))} \\ M. F. Hasler, Nov 24 2015

a(0)=1 added by N. J. A. Sloane, Nov 24 2015

A080817 Leading diagonal of triangle in A080818.

Original entry on oeis.org

1, 3, 6, 10, 6, 10, 15, 8, 13, 19, 26, 15, 22, 30, 16, 24, 33, 43, 25, 35, 46, 25, 36, 48, 61, 36, 49, 63, 35, 49, 64, 33, 48, 64, 81, 46, 63, 81, 43, 61, 80, 100, 58, 78, 99, 54, 75, 97, 49, 71, 94, 118, 66, 90, 115, 60, 85, 111, 138, 79, 106, 134, 72, 100, 129, 159, 93, 123

Offset: 1

Amarnath Murthy, Mar 21 2003

nonn

Cf. A080818, A080819, A080820.

From David Wasserman, May 13 2004: (Start)
a(n) = (ceiling(sqrt(n(n+1)/2)))^2 - n(n-1)/2.
a(n) = A048761(A000217(n)) - A000217(n-1). (End)

More terms from David Wasserman, May 13 2004

A232091 Smallest square or promic (oblong) number greater than or equal to n.

Original entry on oeis.org

0, 1, 2, 4, 4, 6, 6, 9, 9, 9, 12, 12, 12, 16, 16, 16, 16, 20, 20, 20, 20, 25, 25, 25, 25, 25, 30, 30, 30, 30, 30, 36, 36, 36, 36, 36, 36, 42, 42, 42, 42, 42, 42, 49, 49, 49, 49, 49, 49, 49, 56, 56, 56, 56, 56, 56, 56, 64, 64, 64, 64, 64, 64, 64, 64, 72, 72, 72, 72, 72, 72, 72, 72, 81

Offset: 0

L. Edson Jeffery, Nov 18 2013

Result attributed to the students Daring, et al., in the links section.
a(n) appears in floor(sqrt(a(n))) = A000194(n) successive terms.
Counting successive equal terms give sequence: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ... (see A008619). - Michel Marcus, Jan 10 2014

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
David Applegate, Proof of the equality A216607(n) = A232091(n) - n.
E. Daring, I. Guadarrama, S. Sprague, and C. Winterer, WhaleConjecture.
Casey Douglas, The Next Square or Pronic, June 2012. [Wayback Machine copy]

Cf. A048761, A235382, A259225.

Cf. A000290 (squares), A002378 (promic or oblong numbers), A002620 (A000290 union A002378).

[(Ceiling(n /Ceiling(Sqrt(n)))*Ceiling(Sqrt(n))): n in [1..80]]; // Vincenzo Librandi, Jun 22 2015

Join[{0}, Table[Ceiling[n/Ceiling[Sqrt[n]]] Ceiling[Sqrt[n]], {n, 100}]] (* Alonso del Arte, Nov 18 2013 *)

a(n)=my(t=sqrtint(n-1)+1);t*((n-1)\t+1) \\ Charles R Greathouse IV, Nov 18 2013

a(n) = ceiling(n/ceiling(sqrt(n)))*ceiling(sqrt(n)).
a(n) = min(k : k >= n, k in A002620).
a(k^2) = k^2; a(k*(k+1)) = k*(k+1).
It appears that a(n) = A216607(n) + n. (Verified for all n<10^9 by Lars Blomberg, Jan 09 2014.) This conjecture now follows from a proof given by David Applegate, Jan 10 2014 (see [Applegate]).
a(n) = min(A048761(n), A259225(n)). - Michel Marcus, Jun 22 2015
Sum_{n>=1} 1/a(n)^2 = 2 - Pi^2/6 + zeta(3). - Amiram Eldar, Aug 16 2022

Extended by Charles R Greathouse IV, Nov 18 2013

a(0)=0 prepended by Michel Marcus, Jun 22 2015

cp's OEIS Frontend

A165775 n + (least square >= n), i.e., n + A048761(n).

Views

Author

Keywords

Links

Programs

Mathematica

PARI

Formula

A072690 a(n) = (n - A048760(n)) * (A048761(n) - n).

Views

Author

Keywords

Comments

Crossrefs

Formula

A165776 n + (least square > n), i.e., n + A048761(n+1).

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A048760 Largest square <= n.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A068527 Difference between smallest square >= n and n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

A068869 Smallest number k such that n! + k is a square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A077115 Least integer square >= n^3.

Views

Author

Keywords

Examples

Crossrefs

Programs

Magma

Mathematica

Formula

A243091 Least number k > n such that n concatenated with k is a perfect square.

Views

Author

Keywords