A083482 -id:A083482 - cp's OEIS frontend

A083481 Squarefree part of the n-th oblong number.

2, 6, 3, 5, 30, 42, 14, 2, 10, 110, 33, 39, 182, 210, 15, 17, 34, 38, 95, 105, 462, 506, 138, 6, 26, 78, 21, 203, 870, 930, 62, 66, 1122, 1190, 35, 37, 1406, 1482, 390, 410, 1722, 1806, 473, 55, 230, 2162, 141, 3, 2, 102, 663, 689, 318, 330, 770, 798, 3306, 3422, 885

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 03 2003

Smallest k such that n*(n+1)*k is a perfect square.

k = p*q*r... is a squarefree number where p,q,r are those prime factors of n(n+1) which have odd index.

Harvey P. Dale, Table of n, a(n) for n = 1..500
John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011.

Cf. A002378, A007913, A083482.

A083481 := proc(n)
    A007913(n)*A007913(n+1) ;
end proc:
seq( A083481(n),n=1..40) ; # R. J. Mathar, Mar 15 2023

sk[n_]:=Module[{k=1},While[!IntegerQ[Sqrt[n*k]],k++];k]; sk/@Table[n(n+1),{n,60}] (* Harvey P. Dale, Mar 28 2013 *)

PARI
```
a(n)=core(n*(n+1))
```

from sympy.ntheory.factor_ import core
def A083481(n): return core(n*(n+1)) # Chai Wah Wu, Mar 20 2023

a(n) = A007913(n*(n+1)). - R. J. Mathar, Nov 02 2011

a(n) = A007913(n) * A007913(n+1). - Amiram Eldar, Jul 10 2024

More terms from Benoit Cloitre, May 04 2003

A243813 Table read by antidiagonals: T(n,k) is the curvature (truncated to integer) of a circle in a variation of nested Pappus chains (see Comments for details).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 2, 9, 1, 1, 1, 1, 3, 13, 1, 1, 1, 1, 2, 5, 19, 1, 1, 1, 1, 1, 3, 7, 25, 1, 1, 1, 1, 1, 2, 4, 9, 33, 1, 1, 1, 1, 1, 1, 2, 5, 11, 41, 1, 1, 1, 1, 1, 1, 2, 3, 6, 14, 51, 1, 1, 1, 1, 1, 1, 1, 2, 4, 7, 17, 61, 1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 9, 21

Offset: 0

Kival Ngaokrajang, Jun 11 2014

Refer to the construction rule used in A243618. For this case, the curvature is defined by (-1/k, 1/(k-1), 1), the circle radius will diverge to infinity (zero curvature). The integral curvatures appearing as periodic, i.e., 2, 6, 6, 10, 30, 42, 28, 12, ..., = A083482(k-1). The integral curvatures seem to align as some sequence, e.g., 3, 7, 13, 21, 31, 43, ..., = A002061(k) and 9, 25, 49, ..., = A016754(k-1). See illustration.

			Table begins:
  n/k  2   3   4   5   6   7  ...
   0   1   1   1   1   1   1  ...
   1   1   1   1   1   1   1  ...
   2   3   1   1   1   1   1  ...
   3   5   2   1   1   1   1  ...
   4   9   3   2   1   1   1  ...
   5  13   5   3   2   1   1  ...
   6  19   7   4   2   2   1  ...
   7  25   9   5   3   2   2  ...
   8  33  11   6   4   3   2  ...
   9  41  14   7   5   3   2  ...
  10  51  17   9   6   4   3  ...
  11  61  21  11   7   5   3  ...
  12  73  25  13   8   5   4  ...
  ...

Kival Ngaokrajang, Illustration for k = 2..7

Cf. A243618, A083482, A002061, A016754.
Cf. Column 1 = A080827(n), column 2 = A056827(n) + 1.
Cf. Integral curvature in column 1..6: [A058331, A227776, A056107, A212656, A158558, A158604].

cp's OEIS Frontend

A083481 Squarefree part of the n-th oblong number.

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