A068527 -id:A068527 - cp's OEIS frontend

A232395 (ceiling(sqrt(n^3 + n^2 + n + 1)))^2 - (n^3 + n^2 + n + 1).

0, 0, 1, 9, 15, 13, 30, 0, 40, 21, 45, 57, 51, 21, 70, 105, 120, 109, 66, 156, 43, 77, 81, 49, 216, 108, 217, 9, 36, 21, 293, 192, 31, 189, 309, 385, 411, 381, 289, 129, 408, 112, 281, 396, 451, 440, 357, 196, 624, 309, 613, 120, 276, 360, 366, 288, 120, 725

Offset: 0

Vladimir Shevelev, Nov 23 2013

nonn

a(n)=0, iff 1 + n + n^2 + n^3 is a perfect square. For example, a(7)=0 and we have 1 + 7 + 7^2 + 7^3 = 20^2.

a(n) = Difference between smallest square >= (n^3 + n^2 + n + 1) and (n^3 + n^2 + n + 1) - Antti Karttunen, Nov 27 2013

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

Cf. A053698, A068527.

a(n) = ceil(sqrt(n^3+n^2+n+1))^2 - (n^3+n^2+n+1); \\ Michel Marcus, Nov 23 2013

Contribution from Antti Karttunen, Nov 27 2013: (Start)
a(n) = A000290(⌈sqrt(A053698(n))⌉) - A053698(n). Where ⌈x⌉ stands for ceiling(x). This further reduces as:
a(n) = A000290(A135034(A053698(n))) - A053698(n).
a(n) = A048761(A053698(n)) - A053698(n).
a(n) = A068527(A053698(n)).
(End)

More terms from Peter J. C. Moses

A229766 Odd numbers whose square's binary reversal is also a square.

Original entry on oeis.org

1, 3, 4523, 11991, 18197, 66075, 72225, 141683, 1092489, 3168099, 6001209, 6226335, 6435309, 12489657, 17906499, 50429883, 51928701, 68301841, 295742437, 390117873, 542959199, 554456167, 566494057

Offset: 1

Alex Ratushnyak, Dec 19 2013

A003166 is a subsequence, except A003166(1)=0.

All odd numbers n such that A068527(A030101(n^2)) = 0. - Antti Karttunen, Dec 20 2013

Cf. A000290, A003166, A030101, A068527, A229687.

a(n) = sqrt(A229687(n)).

A232397 a(n) = ceiling(sqrt(n^4 + n^3 + n^2 + n + 1))^2 - (n^4 + n^3 + n^2 + n + 1).

Original entry on oeis.org

0, 4, 5, 0, 20, 3, 45, 8, 80, 15, 125, 24, 180, 35, 245, 48, 320, 63, 405, 80, 500, 99, 605, 120, 720, 143, 845, 168, 980, 195, 1125, 224, 1280, 255, 1445, 288, 1620, 323, 1805, 360, 2000, 399, 2205, 440, 2420, 483, 2645, 528, 2880, 575, 3125, 624, 3380, 675

Offset: 0

Vladimir Shevelev, Nov 23 2013

a(n) = 0 if and only if n^4 + n^3 + n^2 + n + 1 is a perfect square.
Using formula below, we immediately prove that a(n)=0 iff n=0 or n=3. This means that all nonnegative solutions of the Diophantine equation n^4 + n^3 + n^2 + n + 1 = m^2 are n=0, m=1 and n=3, m=11.
For m >=0, if we also consider negative values of n, we obtain only one more solution: n=-1, m=1.
Indeed, if one considers sequence b(n) = ceiling(sqrt(n^4 - n^3 + n^2 - n + 1))^2 - (n^4 - n^3 + n^2 - n +1 ), then, for even n, a(n) = b(n), while for odd n>=3, a(n) = b(n-2).

Robert Israel, Table of n, a(n) for n = 0..10001
Max Alekseyev, Re: oddness of iterations of sigma(), SeqFan Mailing List.

Cf. A053699, A068527, A232395, A232423.

[Ceiling(Sqrt(n^4+n^3+n^2+n+1))^2-(n^4+n^3+n^2+n+1): n in [0..60]]; // Vincenzo Librandi, Jan 31 2016

0, 4, seq(op([5*k^2, k^2-1]),k=1..100); # Robert Israel, Feb 02 2016

Table[Ceiling[Sqrt[n^4 + n^3 + n^2 + n + 1]]^2 - (n^4 + n^3 + n^2 + n + 1), {n, 0, 60}] (* Vincenzo Librandi, Jan 31 2016 *)

from math import isqrt
def A232397(n): return (1+isqrt(m:=n*(n*(n*(n+1)+1)+1)))**2-m-1 # Chai Wah Wu, Jul 29 2022

a(1) = 4, for other odd n, a(n) = ((n-1)/2)^2 - 1; for even n>=0, a(n) = 5/4 * n^2.
a(n) = A068527(A053699(n)). [Straight from the description: Difference between smallest square >= (n^4 + n^3 + n^2 + n + 1) and (n^4 + n^3 + n^2 + n + 1)]. - Antti Karttunen, Nov 28 2013
a(n) = (6*n^2-2*n-3+(4*n^2+2*n+3)*(-1)^n+20*(1-(-1)^(2^abs(n-1))))/8. - Luce ETIENNE, Jan 30 2016
G.f.: 4*x+x^2*(x^5-3*x^3-5*x^2-5)/(x^2-1)^3. - Robert Israel, Feb 02 2016

More terms from Peter J. C. Moses

A232847 Numbers k such that sum of divisors of k is a square and a triangular number (A000217). That is, numbers k such that A000203(k) is in A001110.

Original entry on oeis.org

1, 22, 17310, 20802, 23110, 24262, 25995, 26542, 29427, 31735, 33835, 38137, 39287, 39859, 40967, 13595040, 14285160, 15129504, 15378336, 15834528, 15912936, 16327008, 16555752, 16897896, 16908264, 17054388, 17145432, 17749044, 18013428, 20239146, 20713482, 21265578

Offset: 1

Alex Ratushnyak, Dec 01 2013

nonn

			Sigma(22) = 36. Because 36 is both a square and a triangular number, 22 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..11083 (terms 1..136 with sigma values <= A001110(8) from Donovan Johnson)

Cf. A000203, A000217, A000290, A001110.

A249142 Let k be the difference between the smallest square >= n and n. Sequence gives difference between the smallest square >= k and k.

Original entry on oeis.org

0, 2, 0, 0, 0, 1, 2, 0, 0, 3, 4, 0, 1, 2, 0, 0, 1, 2, 3, 4, 0, 1, 2, 0, 0, 6, 0, 1, 2, 3, 4, 0, 1, 2, 0, 0, 4, 5, 6, 0, 1, 2, 3, 4, 0, 1, 2, 0, 0, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 0, 1, 2, 0, 0, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 0, 1, 2, 0, 0, 7, 8, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 0, 1, 2, 0, 0

Offset: 1

Valtteri Raiko, Oct 22 2014

Equals A068527 applied to itself.

			For n = 13 the next biggest square is 16, thus k = 16 - 13 = 3 and for 3 the next biggest square is 4, thus a(14) = 3 - 2 = 1.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A068527.

[n - Ceiling(Sqrt(n))^2 + Ceiling(Sqrt(-n+Ceiling(Sqrt(n))^2))^2: n in [1..100]]; // Vincenzo Librandi, Oct 23 2014

A068527:= n -> ceil(sqrt(n))^2 - n:
map(A068527@@2, [$1..100]); # Robert Israel, Nov 02 2017

Table[n - Ceiling[Sqrt[n]]^2 + Ceiling[Sqrt[-n + Ceiling[Sqrt[n]]^2]]^2, {n, 1, 100}]

A068527(n)=if(issquare(n), 0, (sqrtint(n)+1)^2-n)
a(n)=A068527(A068527(n)) \\ Charles R Greathouse IV, Oct 22 2014

a(n) = A068527(A068527(n)).
a(n) = n - ceiling(sqrt(n))^2 + ceiling(sqrt(-n+ceiling(sqrt(n))^2))^2.
a(n) < (64n)^(1/4). - Charles R Greathouse IV, Oct 22 2014

Edited, old crossrefs entry moved to Comments, and first two formula lines interchanged by Wolfdieter Lang, Nov 10 2014

A300001 Side length of the smallest equilateral triangle that can be dissected into n equilateral triangles with integer sides, or 0 if no such triangle exists.

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Feb 20 2018

nonn

No solutions exist for n = 2, 3 and 5.
a(n) = A290820(n) for n <= 8. It is conjectured that a(n) < A290820(n) for all n > 12.
The seven numbers mentioned by Peter Munn in the Formula section [1, 2, 4, 5, 7, 10, 13] coincide with the seven terms of A123120. - M. F. Hasler and Omar E. Pol, Feb 23 2018

			            a(9)=3               a(10)=4                a(11)=5
              *                     *                      *
             / \                   / \                    / \
            *---*                 *---*                  +   +
           / \ / \               / \ / \                /     \
          *---*---*             *---*---*              +       +
         / \ / \ / \           / \ / \ / \            /         \
        *---*---*---*         +   *---*   +          *---+---+---*
                             /     \ /     \        / \ / \     / \
                            *---+---*---+---*      *---*---*   +   +
                                                  / \ / \ / \ /     \
                                                 *---*---*---*---+---*
.
           a(12)=6                a(13)=4                a(14)=5
              *                      *                      *
             / \                    / \                    / \
            *---*                  *---*                  +   +
           / \ / \                / \ / \                /     \
          *---*---*              *---*---*              +       +
         / \ / \ / \            / \ / \ / \            /         \
        *---*---*---*          *---*   *---*          *---+---+---*
       / \         / \        / \ /     \ / \        / \ / \ / \ / \
      *   +       +   +      *---*---*---*---*      *---*---*---*   +
     /     \     /     \                           / \ / \ / \ /     \
    +       +   +       +                         *---*---*---*---+---*
   /         \ /         \
  *---+---+---*---+---+---*
.
           a(15)=6                 a(16)=4                a(17)=5
              *                       *                      *
             / \                     / \                    / \
            +   +                   *---*                  +   +
           /     \                 / \ / \                /     \
          +       +               *---*---*              +       +
         /         \             / \ / \ / \            /         \
        +           +           *---*---*---*          *---*---*---*
       /             \         / \ / \ / \ / \        / \ / \ / \ / \
      *---*---*---*---*       *---*---*---*---*      *---*---*---*---*
     / \     / \     / \                            / \ / \ / \ / \ / \
    *---*   *---*   *---*                          *---*---*---*---*---*
   / \ / \ / \ / \ / \ / \
  *---*---*---*---*---*---*
.
           a(18)=6                 a(19)=5                 a(20)=6
              *                       *                       *
             / \                     / \                     / \
            +   +                   +   +                   *---*
           /     \                 /     \                 / \ / \
          +       +               *---*---*               *---*---*
         /         \             / \     / \             / \ / \ / \
        +           +           *---*   *---*           *---*---*---*
       /             \         / \ / \ / \ / \         / \ / \ / \ / \
      *---*---*---*---*       *---*---*---*---*       +   *---*---*   +
     / \ / \ / \ / \ / \     / \ / \ / \ / \ / \     /     \ / \ /     \
    *---*---*   *---*---*   *---*---*---*---*---*   +       *---*       +
   / \ / \ /     \ / \ / \                         /         \ /         \
  *---*---*---+---*---*---*                       *---+---+---*---+---+---*

Ales Drapal, Carlo Hamalainen, An enumeration of equilateral triangle dissections, arXiv:0910.5199 [math.CO], 2009-2010.

Cf. A068527, A123120, A290820, A299705.

a(n^2) = n for all n>=1, a(n^2-3) = n for all n>=3. - Corrected by Peter Munn, Feb 24 2018

For n > 23, if A068527(n) = 1, 2, 4, 5, 7, 10 or 13 then a(n) = ceiling(sqrt(n)) + 1 else a(n) = ceiling(sqrt(n)). - Peter Munn, Feb 23 2018

a(21)-a(100) from Peter Munn, Feb 24 2018

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)).

A099135 a(n) = smallest number m, not occurring earlier, such that a(k)+m is a square for some k

Original entry on oeis.org

1, 3, 6, 8, 10, 13, 12, 4, 5, 11, 14, 2, 7, 9, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72

Offset: 1

Reinhard Zumkeller, Sep 29 2004

Permutation of the natural numbers with inverse A099136;

a(n) <> n iff 1 < n <= 14.

Cf. A048761, A068527.

a(n)=if(n>14,n,[1, 3, 6, 8, 10, 13, 12, 4, 5, 11, 14, 2, 7, 9][n]) \\ Charles R Greathouse IV, Sep 02 2011

A249298 Smallest positive integer k, such that s-kn is a square where s is the smallest square >= kn.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 3, 1, 1, 4, 5, 1, 5, 6, 1, 1, 9, 2, 9, 2, 1, 12, 13, 1, 1, 14, 1, 3, 17, 2, 19, 1, 3, 20, 1, 1, 23, 24, 3, 1, 25, 2, 27, 6, 1, 30, 31, 1, 1, 2, 3, 7, 35, 4, 1, 2, 3, 40, 41, 1, 41, 42, 1, 1, 1, 4, 47, 10, 5, 2, 51, 1, 51, 52, 3, 12, 1, 6, 57, 1, 1, 60, 61, 1, 3, 62, 7, 3, 65, 2

Offset: 1

Valtteri Raiko, Oct 24 2014

nonn

For any n>=3, there exists at least one positive integer k, 1 <= k <= n-1 such that the difference between the smallest square >= k*n and k*n is a square. To prove this, consider the multiplier k = n-2. Then (n-2)*n = (n-1)^2-1, thus the difference from the next square is 1, which is a square. If n = 1, k = 1 and if n = 2, k = 2.

Smallest positive integer k such that ceiling(sqrt(k*n))^2-k*n is a square.

			a(10) = 4, for ceiling(sqrt(10))^2-10 = 6, ceiling(sqrt(2*10))^2-2*10 = 5, ceiling(sqrt(3*10))^2-3*10 = 6 and ceiling(sqrt(4*10))^2-4*10 = 9 = 3^2.

Cf. A000290, A145236 (equals a(A000040)), A068527 (difference for k=1).

Cf. A145016, A145022, A145023, A145047, A145048, A145049, A145050, A145215.

dif[n_] := Ceiling[Sqrt[n]]^2 - n;a[k_] := Module[{n = 1}, While[dif[dif[n*k]] != 0, n++]; Return[n]];Table[a[k], {k, 1, 90}]

a(n) = {k=1; while(!issquare(ceil(sqrt(k*n))^2-k*n), k++); k;} \\ Michel Marcus, Oct 24 2014

A256173 Numbers k such that ceiling(sqrt(k))^2 - k is a square.

Original entry on oeis.org

0, 1, 3, 4, 5, 8, 9, 12, 15, 16, 21, 24, 25, 27, 32, 35, 36, 40, 45, 48, 49, 55, 60, 63, 64, 65, 72, 77, 80, 81, 84, 91, 96, 99, 100, 105, 112, 117, 120, 121, 128, 135, 140, 143, 144, 153, 160, 165, 168, 169, 171, 180, 187, 192, 195, 196, 200, 209, 216, 221, 224, 225, 231, 240, 247, 252, 255, 256, 264, 273, 280, 285, 288, 289, 299

Offset: 1

Valtteri Raiko, Mar 17 2015

nonn

Numbers k such that A068527(k) is a square. k is in the sequence if and only if k - ceiling(sqrt(k))^2 + ceiling(sqrt(ceiling(sqrt(k))^2 - k))^2 = 0.
A000290 is a subsequence since for a square k, ceiling(sqrt(k))^2 - k = 0, a square too.
Also, numbers k such that A249298(k) is 1.
Also, numbers k such that A249142(k) is 0.
The only prime numbers in the sequence are 3 and 5.
No number from A016825 appears in the sequence.
If p and q are terms of A065091 and if q satisfies the inequality p - 2*sqrt(2p) + 2 < q < p + 2*sqrt(2p) + 2, then p*q is in the sequence. Thus infinitely many numbers from A046315 appear in the sequence.

			Ceiling(sqrt(27))^2 - 27 = 9 = 3^2, so 27 is in the sequence.

Cf. A000290, A068527, A145236, A249142, A249298, A046315.

[n: n in [0..200] | IsSquare(Ceiling(Sqrt(n))^2-n)]; // Vincenzo Librandi, Mar 18 2015

Flatten[Position[Table[n - Ceiling[Sqrt[n]]^2 + Ceiling[Sqrt[-n + Ceiling[Sqrt[n]]^2]]^2, {n, 0, 300}], 0]] - 1
Select[Range[0,300],IntegerQ[Sqrt[Ceiling[Sqrt[#]]^2-#]]&] (* Harvey P. Dale, Sep 06 2023 *)

isok(n) = issquare(ceil(sqrt(n))^2-n); \\ Michel Marcus, Mar 18 2015

cp's OEIS Frontend

A232395 (ceiling(sqrt(n^3 + n^2 + n + 1)))^2 - (n^3 + n^2 + n + 1).

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A229766 Odd numbers whose square's binary reversal is also a square.

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A232397 a(n) = ceiling(sqrt(n^4 + n^3 + n^2 + n + 1))^2 - (n^4 + n^3 + n^2 + n + 1).

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A232847 Numbers k such that sum of divisors of k is a square and a triangular number (A000217). That is, numbers k such that A000203(k) is in A001110.

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A249142 Let k be the difference between the smallest square >= n and n. Sequence gives difference between the smallest square >= k and k.

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A300001 Side length of the smallest equilateral triangle that can be dissected into n equilateral triangles with integer sides, or 0 if no such triangle exists.

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

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A099135 a(n) = smallest number m, not occurring earlier, such that a(k)+m is a square for some k

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A249298 Smallest positive integer k, such that s-kn is a square where s is the smallest square >= kn.