A147856 -id:A147856 - 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.

A147858 Differences of two coprime 4th powers.

Original entry on oeis.org

0, 15, 65, 80, 175, 255, 369, 544, 609, 624, 671, 1105, 1295, 1695, 1776, 2145, 2320, 2385, 2400, 2465, 3439, 3471, 4015, 4095, 4160, 4641, 5936, 6095, 6305, 6545, 6560, 7599, 7825, 8080, 9855, 9919, 9999, 10545, 12209, 12240, 13345, 13920, 14016

Offset: 0

Max Alekseyev, Nov 15 2008

nonn

Primitive elements of A147857: any element n of A147857 is of the form a(k)*m^4 for some positive integer m.

Cf. A147854, A147856, A147857.

Join[{0},Rest[#[[2]]-#[[1]]&/@Select[Subsets[Range[0,20]^4,{2}], CoprimeQ@@#&] //Union]] (* Harvey P. Dale, Dec 08 2016 *)

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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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A147858 Differences of two coprime 4th powers.

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Mathematica