A024352 -id:A024352 - cp's OEIS frontend

A174337 94*n^2.

0, 94, 376, 846, 1504, 2350, 3384, 4606, 6016, 7614, 9400, 11374, 13536, 15886, 18424, 21150, 24064, 27166, 30456, 33934, 37600, 41454, 45496, 49726, 54144, 58750, 63544, 68526, 73696, 79054, 84600, 90334, 96256, 102366, 108664

Offset: 0

Vincenzo Librandi, Mar 16 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A024352.

Magma
```
[94*n^2: n in [0..50]];
```

I:=[0, 94, 376]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Aug 21 2014

Table[94 n^2, {n, 0, 40}] (* or *) CoefficientList[Series[94 x (1 + x)/(1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 21 2014 *)
LinearRecurrence[{3,-3,1},{0,94,376},40] (* Harvey P. Dale, Aug 15 2016 *)

a(n)=94*n^2 \\ Charles R Greathouse IV, Jun 17 2017

G.f. 94*x*(1+x)/(1-x)^3 . - R. J. Mathar, Oct 01 2011

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Aug 21 2014

A334078 a(n) is the smallest positive integer that can be expressed as the difference of two positive squares in at least n ways.

Original entry on oeis.org

3, 15, 45, 96, 192, 240, 480, 480, 720, 960, 1440, 1440, 2880, 2880, 2880, 3360, 5040, 5040, 6720, 6720, 10080, 10080, 10080, 10080, 20160, 20160, 20160, 20160, 20160, 20160, 30240, 30240, 40320, 40320, 40320, 40320, 60480, 60480, 60480, 60480, 80640, 80640

Offset: 1

Ilya Gutkovskiy, Apr 13 2020

nonn

Eric Weisstein's World of Mathematics, Square Number

Cf. A000290, A024352, A048610, A068314, A094191, A100073, A214771, A334077.

A375463 Numbers appearing on all three positions in ordered primitive Pythagorean triples.

Original entry on oeis.org

221, 325, 377, 425, 493, 629, 697, 725, 925, 1025, 1073, 1189, 1325, 1517, 1537, 1769, 1885, 1961, 2173, 2257, 2405, 2501, 2665, 2701, 2993, 3145, 3233, 3293, 3445, 3485, 3649, 3869, 3965, 3977, 4453, 4505, 4717, 4745, 5141, 5185, 5353, 5429, 5777, 5785, 5917

Offset: 1

Piotr Lipski, Aug 16 2024

nonn

			221 is a term since the following primitive Pythagorean triples have 221 in first, second and third position: (221, 24420, 24421), (60, 221, 229), (21, 220, 221).

Mathematics Stack Exchange, Is there an integer that serves as the short leg of a primitive Pythagorean triple, the long leg of another, and the hypotenuse of a third?
Rémy Sigrist, PARI program

Intersection of A008846, A024352 and A024354.

Cf. A263728.

PARI
```
\\ See Links section.
```

a(n) == 1 (mod 4). - Hugo Pfoertner, Aug 18 2024

More terms from Rémy Sigrist, Aug 17 2024

A300903 a(n) is the smallest k such that k^2 - n^2 is a prime power (A000961), or 0 if no such k exists.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 9, 10, 15, 12, 13, 14, 15, 16, 48, 0, 19, 0, 21, 22, 0, 24, 25, 0, 27, 54, 36, 30, 31, 33, 96, 34, 0, 36, 37, 0, 0, 40, 41, 42, 0, 0, 45, 0, 0, 0, 49, 0, 51, 52, 0, 54, 55, 66, 57, 0, 0, 0, 61, 0, 63, 64, 192, 66, 0, 0, 69, 70, 0, 0, 0, 0, 75, 76, 0, 0, 79, 0, 0, 82, 0, 84, 85, 0, 87, 0, 0, 90, 91, 0, 0, 0, 0, 96, 97

Offset: 0

Altug Alkan, Mar 14 2018

If such k exists (for n > 0), then the maximum ratio of k / n is (p + 1)/(p - 1) with p = 2 where p is prime root of corresponding prime power. So a(n) <= 3*n.
If 2*n+1 is in A000961 (in particular if n is in A005097), then a(n) = n + 1.
Numbers n such that a(n) = 0 are 17, 19, 22, 25, 34, 37, 38, 42, 43, 45, 46, ...
Initial corresponding prime powers are 1, 3, 5, 7, 9, 11, 13, 32, 17, 19, 125, 23, 25, 27, 29, 31, 2048.

			a(17) = 0 because there is no k such that k^2 - 17^2 = (k + 17)*(k - 17) is a prime power.
a(21) = 22 because 22^2 - 21^2 = 43 and 22 is the least number with this property.
a(27) = 54 because 54^2 - 27^2 = 3^7 and 54 is the only number with this property.

Cf. A000961, A005097, A024352.

f:= proc(n) local p,k,a,b,r;
  if nops(numtheory:-factorset(2*n+1))<=1 then return n+1 fi;
  k:= infinity;
  for p in numtheory:-factorset(2*n) do
    b:= padic:-ordp(2*n,p);
    r:= 2*n + p^b;
    a:= padic:-ordp(r,p);
    if r = p^a then
      k:= min(k, (p^a+p^b)/2)
    fi
  od;
  if k = infinity then 0 else k fi
end proc:
map(f, [$0..1000]); # Robert Israel, Mar 15 2018

Table[Boole[n == 0] + Block[{k = n + 1, m = 3 n}, While[Nor[PrimePowerQ[k^2 - n^2], k > m], k++]; If[k > m, 0, k]], {n, 0, 96}] (* Michael De Vlieger, Mar 16 2018 *)

a(n) = if(n==0, 1, for(k=n+1, 3*n, if(isprimepower(k^2-n^2), return(k)));0)

A363863 Numbers expressible as j^2 - k^2, 1 <= k <= j-2 ("squares with a square hole").

Original entry on oeis.org

8, 12, 15, 16, 20, 21, 24, 27, 28, 32, 33, 35, 36, 39, 40, 44, 45, 48, 51, 52, 55, 56, 57, 60, 63, 64, 65, 68, 69, 72, 75, 76, 77, 80, 81, 84, 85, 87, 88, 91, 92, 93, 95, 96, 99, 100, 104, 105, 108, 111, 112, 115, 116, 117, 119, 120, 123, 124, 125, 128, 129, 132, 133, 135, 136

Offset: 1

Thomas A. Fisher, Jun 25 2023

Inspired by my 4-year-old son, who loves Numberblocks, I decided to work out which numbers appear in the "squares with [square] holes club". These are numbers which, when configured as a square, have a square wholly removed. For example, 8 is 3 X 3 with a 1 X 1 hole in the middle. 24 is both a 5 X 5 with a 1 X 1 hole in the middle and a 7 X 7 with a 5 X 5 hole in the middle. The hole has to be "wholly contained", meaning I can't, for example, have 3^2 - 2^2 = 9 - 4 = 5, as removing a 2 X 2 square from a 3 X 3 square doesn't leave a "hole", as we are working with blocks, i.e., integers.

This sequence contains all natural numbers which factor as (j - k)*(j + k), where j - k >= 2 and k >= 1. That is, all natural numbers which have at least one factor pair of the form u*v such that u and v have the same parity, are distinct, and are both strictly greater than 1. This precisely rules out 1, primes, squares of primes, and the even numbers which are congruent to 2 modulo 4. In other words, this sequence is equal to A080257\A016825.

			8 = 3^2 - 1^2, 12 = 4^2 - 2^2, 15 = 4^2 - 1^2, ...

Subsequence of A024352.

Equals A080257\A016825.

isok(k) = ((omega(k)>1) || (isprimepower(k)>2)) && ((k % 4) != 2); \\ Michel Marcus, Jun 30 2023

cp's OEIS Frontend

A174337 94*n^2.

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A334078 a(n) is the smallest positive integer that can be expressed as the difference of two positive squares in at least n ways.

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A375463 Numbers appearing on all three positions in ordered primitive Pythagorean triples.

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PARI

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Extensions

A300903 a(n) is the smallest k such that k^2 - n^2 is a prime power (A000961), or 0 if no such k exists.

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Maple

Mathematica

PARI

A363863 Numbers expressible as j^2 - k^2, 1 <= k <= j-2 ("squares with a square hole").

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Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI