A027434 -id:A027434 - cp's OEIS frontend

A350634 Products of the parts s,t in each partition of k (= 2,3,...) into two parts, ordered by increasing k and then by increasing values of s*t (see example).

1, 2, 3, 4, 4, 6, 5, 8, 9, 6, 10, 12, 7, 12, 15, 16, 8, 14, 18, 20, 9, 16, 21, 24, 25, 10, 18, 24, 28, 30, 11, 20, 27, 32, 35, 36, 12, 22, 30, 36, 40, 42, 13, 24, 33, 40, 45, 48, 49, 14, 26, 36, 44, 50, 54, 56, 15, 28, 39, 48, 55, 60, 63, 64, 16, 30, 42, 52, 60, 66, 70, 72

Offset: 1

Wesley Ivan Hurt, Jan 09 2022

nonn

If b > 0 and c > 0 are the integer coefficients of a monic quadratic x^2 + b*x + c, it has integer roots if its discriminant d^2 = b^2 - 4c is a perfect square. This sequence is the values of c for increasing b sorted by b then c. The first pair of (b, c) = (2, 1) and has d = 0. The n-th pair of (b, c) = (A027434(n),a(n)) and has d = A082375(n-1). - Frank M Jackson, Jan 22 2024

			---------------------------------------------------------------------------
The products of the parts start: 1*1, 1*2, 1*3, 2*2, 1*4, 2*3, etc., which are precisely the values of a(n): 1, 2, 3, 4, 4, 6, ...
                                                                     [1,9]
                                                     [1,7]   [1,8]   [2,8]
                                     [1,5]   [1,6]   [2,6]   [2,7]   [3,7]
                     [1,3]   [1,4]   [2,4]   [2,5]   [3,5]   [3,6]   [4,6]
     [1,1]   [1,2]   [2,2]   [2,3]   [3,3]   [3,4]   [4,4]   [4,5]   [5,5]
  k    2       3       4       5       6       7       8       9      10
---------------------------------------------------------------------------

Cf. A027434, A082375, A122197, A199474, A339399.

Times@@@Flatten[Table[IntegerPartitions[k, {2}], {k, 2, 100}], 1] (* Frank M Jackson, Jan 22 2024 *)
lst={}; Do[If[IntegerQ[d=Sqrt[b^2-4c]], AppendTo[lst, c]], {b, 1, 100}, {c, 1, b^2/4}]; lst (* Frank M Jackson, Jan 22 2024 *)

a(n) = A122197(n) * A199474(n).
a(n) = A339399(2n-1) * A339399(2n).
a(n) = ((A027434(n))^2 - (A082375(n))^2)/4. - Frank M Jackson, Jan 22 2024

A248928 Interleave (2n+2)^2 with (2n+3)^2, both listed n+1 times.

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Oct 17 2014

Discovered via Janet's sequence A167268: the result of adding to A167268 the smallest increasing sequence (2, 7, 10, 14, 19, 23, 26, 30, 34, 39, 43, 47, ...) as to get a sequence of nondecreasing squares.
Even terms: 4, 16, 16, 36, 36, 36, ... = 4*A093995(n+1).
Odd terms: (A131507(n) + 2)^2.

			Seen as an irregular triangle:
4;
9;
16, 16;
25, 25;
36, 36, 36;
49, 49, 49;
64, 64, 64, 64;
81, 81, 81, 81;
...

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

Cf. A000267, A007395, A027434, A093995, A131507, A167268.

Module[{nn=10,a,b},a=Table[PadRight[{},n+1,(2n+2)^2],{n,0,nn}];b= Table[ PadRight[ {},n+1,(2n+3)^2],{n,0,nn}];Riffle[a,b]]//Flatten (* Harvey P. Dale, Jun 10 2022 *)

vector(60, n, (sqrtint(4*n-3)+1)^2) \\ after Charles R Greathouse IV, Michel Marcus, Oct 27 2014

a(n) = A027434(n+1)^2.

A259227 Hydropronic numbers: numbers n that can be written as a product of 2 integers whose sum is equal to ceiling(n/ceiling(sqrt(n))) + ceiling(sqrt(n)).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 28, 30, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 56, 60, 63, 64, 66, 70, 72, 77, 80, 81, 84, 88, 90, 91, 96, 99, 100, 104, 108, 110, 112, 117, 120, 121, 126, 130, 132, 135, 140, 143, 144, 150, 154, 156, 160

Offset: 1

Michel Marcus, Jun 21 2015

nonn

It appears that ceiling(n/ceiling(sqrt(n))) + ceiling(sqrt(n)) is A027434(n).

Ivan Neretin, Table of n, a(n) for n = 1..10000
Casey Douglas, The Next Square or Pronic, June 2012.
Casey Douglas, Hydropronic Sequence?, June 2012.
Casey Douglas, Another Quickie, July 2012.

Select[Range@160, IntegerQ@Sqrt[((r = Ceiling@Sqrt@#) + Ceiling[#/r])^2 - 4 #] &] (* Ivan Neretin, Oct 16 2016 *)

isok(n) = {d = divisors(n); for (k=1, #d, if ((d[k] + n/d[k]) == ceil(n/ceil(sqrt(n)))+ceil(sqrt(n)), return (1)););}

cp's OEIS Frontend

A350634 Products of the parts s,t in each partition of k (= 2,3,...) into two parts, ordered by increasing k and then by increasing values of s*t (see example).

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A248928 Interleave (2n+2)^2 with (2n+3)^2, both listed n+1 times.

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A259227 Hydropronic numbers: numbers n that can be written as a product of 2 integers whose sum is equal to ceiling(n/ceiling(sqrt(n))) + ceiling(sqrt(n)).

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cp's OEIS Frontend

A350634 Products of the parts s,t in each partition of k (= 2,3,...) into two parts, ordered by increasing k and then by increasing values of s*t (see example).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A248928 Interleave (2*n+2)^2 with (2*n+3)^2, both listed n+1 times.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A259227 Hydropronic numbers: numbers n that can be written as a product of 2 integers whose sum is equal to ceiling(n/ceiling(sqrt(n))) + ceiling(sqrt(n)).

Views

Author

Keywords

Comments

Links

Programs

Mathematica

PARI

A248928 Interleave (2n+2)^2 with (2n+3)^2, both listed n+1 times.