A147858 -id:A147858 - cp's OEIS frontend

A147857 Differences of two positive 4th powers.

0, 15, 65, 80, 175, 240, 255, 369, 544, 609, 624, 671, 1040, 1105, 1215, 1280, 1295, 1695, 1776, 2145, 2320, 2385, 2400, 2465, 2800, 3439, 3471, 3840, 4015, 4080, 4095, 4160, 4641, 5265, 5904, 5936, 6095, 6305, 6480, 6545, 6560, 7599, 7825, 8080, 8704

Offset: 1

Max Alekseyev, Nov 15 2008, Nov 19 2008

nonn

If n belongs to this sequence then so does n*m^4 for any positive integer m. Primitive elements (i.e., not of the form n*m^4 for m>1) are listed in A147858.

There is no square in this sequence except 0. - Altug Alkan, Apr 08 2016

Robert Israel, Table of n, a(n) for n = 1..10000

Subsequence of A152044.

Cf. A147854, A147856, A147858

N:= 10^4: # to get all terms <= N
Res:= {0}:
for a from 1 to floor(sqrt(N-2)) do
  if a^4 > N then bmin:= ceil((a^4-N)^(1/4)) else bmin:= 1 fi;
  Res:= Res union {seq(a^4-b^4, b=bmin..a-1)}
od:
sort(convert(Res,list)); # Robert Israel, Sep 28 2018

nn = 50; Take[Union @@ Map[Differences, Union@ Map[Sort@ # &, Tuples[Range[Ceiling[nn/3]], {2}]]^4], nn] (* Michael De Vlieger, Apr 09 2016 *)

Offset changed by Robert Israel, Sep 28 2018

A147854 Positive integers n such that n^2 = (x^4 - y^4)*(z^4 - t^4) where the pairs of integers (x,y) and (z,t) are not proportional.

Original entry on oeis.org

520, 975, 2040, 2080, 3567, 3900, 4680, 7215, 7800, 8160, 8320, 8775, 9840, 13000, 13920, 14268, 15600, 18360, 18720, 19680, 24375, 25480, 28860, 30160, 31200, 32103, 32640, 33280, 35100, 39360, 40545, 42120, 47775, 51000, 52000, 53040

Offset: 1

Max Alekseyev, Nov 17 2008, Nov 19 2008

nonn

Positive integers n such that n^2 = s^4*A147858(m)*A147858(k) for positive integers s and kA147856.
Euler proved that if n^2 = (x^4 - y^4)*(z^4 - t^4) then a,b,c (if n is even) or 4a,4b,4c (if n is odd) form a triple of integers with all pairwise sums and differences being squares, where a=(x^4+y^4)*(z^4+t^4)/2, b=(n^2+(2xyzt)^2)/2 and c=(n^2-(2xyzt)^2)/2. Note that a,b,c are pairwise distinct if and only if (x,y) and (z,t) are not proportional.
4*A196289(n) = 4*(n^9 - n) belong to this sequence since (4*(n^9 - n))^2 = ((n^4+2*n^2-1)^4 - (n^4-2*n^2-1)^4) * (n^4 - 1).

Cf. A147856, A147857, A147858.

A147856 Positive integers n such that n^2 = (x^4 - y^4)*(z^4 - t^4) where x>y and z>t are distinct pairs of integers with gcd(x,y)=gcd(z,t)=1.

Original entry on oeis.org

520, 975, 2040, 3567, 7215, 7800, 9840, 13920, 19680, 30160, 40545, 53040, 57720, 62985, 95120, 108225, 138040, 151320, 180960, 230880, 247520, 286200, 289952, 352495, 473280, 535353, 546975, 720945, 769600, 1048560, 1141920, 1210560

Offset: 1

Max Alekseyev, Nov 18 2008

nonn

Positive integers n such that n^2 = A147858(m)*A147858(k) for positive integers kA147854: any element n of A147854 is of the form a(k)*s^2 for some positive integer s.

4*A196289(2*k) and A196289(2*k+1) belong to this sequence.

cp's OEIS Frontend

A147857 Differences of two positive 4th powers.

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A147854 Positive integers n such that n^2 = (x^4 - y^4)*(z^4 - t^4) where the pairs of integers (x,y) and (z,t) are not proportional.

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A147856 Positive integers n such that n^2 = (x^4 - y^4)*(z^4 - t^4) where x>y and z>t are distinct pairs of integers with gcd(x,y)=gcd(z,t)=1.

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