A147857 -id:A147857 - cp's OEIS frontend

A181123 Numbers that are the differences of two positive cubes.

0, 7, 19, 26, 37, 56, 61, 63, 91, 98, 117, 124, 127, 152, 169, 189, 208, 215, 217, 218, 271, 279, 296, 316, 331, 335, 342, 386, 387, 397, 448, 469, 485, 488, 504, 511, 513, 547, 602, 604, 631, 657, 665, 702, 721, 728, 784, 817, 819, 866, 875, 919, 936, 973

Offset: 1

T. D. Noe, Oct 06 2010

Because x^3-y^3 = (x-y)(x^2+xy+y^2), the difference of two cubes is a prime number only if x=y+1, in which case all the primes are cuban, see A002407.
The difference can be a square (see A038597), but Fermat's Last Theorem prevents the difference from ever being a cube. Beal's Conjecture implies that there are no higher odd powers in this sequence.
If n is in the sequence, it must be x^3-y^3 where 0 < y <= x < n^(1/2). - Robert Israel, Dec 24 2017

T. D. Noe and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 from Noe)

Cf. A014439, A014440, A014441, A333376, A333377, A038593 (subsequence), A086120.

Cf. A024352 (squares), A147857 (4th powers), A181124-A181128 (5th to 9th powers).

N:= 10^4: # to get all terms <= N
sort(convert(select(`<=`, {0, seq(seq(x^3-y^3, y=1..x-1),x=1..floor(sqrt(N)))}, N),list)); # Robert Israel, Dec 24 2017

nn=10^5; p=3; Union[Reap[Do[n=i^p-j^p; If[n<=nn, Sow[n]], {i,Ceiling[(nn/p)^(1/(p-1))]}, {j,i}]][[2,1]]]
With[{nn=60},Take[Union[Abs[Flatten[Differences/@Tuples[ Range[ nn]^3,2]]]], nn]] (* Harvey P. Dale, May 11 2014 *)

list(lim)=my(v=List([0]),a3); for(a=2,sqrtint(lim\3), a3=a^3; for(b=if(a3>lim,sqrtnint(a3-lim-1,3)+1,1), a-1, listput(v,a3-b^3))); Set(v) \\ Charles R Greathouse IV, Jan 25 2018

A181124 Difference of two positive 5th powers.

Original entry on oeis.org

0, 31, 211, 242, 781, 992, 1023, 2101, 2882, 3093, 3124, 4651, 6752, 7533, 7744, 7775, 9031, 13682, 15783, 15961, 16564, 16775, 16806, 24992, 26281, 29643, 31744, 32525, 32736, 32767, 40951, 42242, 51273, 55924, 58025, 58806, 59017, 59048, 61051

Offset: 1

T. D. Noe, Oct 06 2010

nonn

Because x^5-y^5 = (x-y)(x^4+x^3*y+x^2*y^2+x*y^3+y^4), the difference of two 5th powers is a prime number only if x=y+1, in which case all the primes are in A121616. The number 7744 is the first of an infinite number of squares in this sequence.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A024352 (squares), A181123 (cubes), A147857 (4th powers), A181125-A181128 (6th to 9th powers)

nn=10^9; p=5; Union[Reap[Do[n=i^p-j^p; If[n<=nn, Sow[n]], {i,Ceiling[(nn/p)^(1/(p-1))]}, {j,i}]][[2,1]]]

A181128 Difference of two positive 9th powers.

Original entry on oeis.org

0, 511, 19171, 19682, 242461, 261632, 262143, 1690981, 1933442, 1952613, 1953124, 8124571, 9815552, 10058013, 10077184, 10077695, 30275911, 38400482, 40091463, 40333924, 40353095, 40353606, 93864121, 124140032, 132264603, 133955584

Offset: 1

T. D. Noe, Oct 06 2010

nonn

No term is a prime number.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A024352 (squares), A181123 (cubes), A147857 (4th powers), A181124-A181127 (5th to 8th powers)

nn=10^15; p=9; Union[Reap[Do[n=i^p-j^p; If[n<=nn, Sow[n]], {i,Ceiling[(nn/p)^(1/(p-1))]}, {j,i}]][[2,1]]]

A152044 Numbers expressible as the difference of two nonnegative fourth powers.

Original entry on oeis.org

0, 1, 15, 16, 65, 80, 81, 175, 240, 255, 256, 369, 544, 609, 624, 625, 671, 1040, 1105, 1215, 1280, 1295, 1296, 1695, 1776, 2145, 2320, 2385, 2400, 2401, 2465, 2800, 3439, 3471, 3840, 4015, 4080, 4095, 4096, 4160, 4641, 5265, 5904, 5936, 6095, 6305, 6480

Offset: 1

Mark Taggart (mt2612f(AT)aol.com), Nov 21 2008

nonn

This sequence seems to grow quadratically. Does a(n) ~ k*n^2 for some k? - Charles R Greathouse IV, Jan 16 2025

			E.g. 15=2^4-1^4, 175=4^4-3^4

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Contains A000583 and A147857 as subsequences. - Chandler

Cf. A042965, A152043, A152045.

Select[Abs[Differences/@Tuples[Range[0,12]^4,2]]//Flatten//Union,#<= 6500&] (* Harvey P. Dale, Sep 14 2020 *)

is(n)=if(n<1,return(!n)); for(m=sqrtnint(n-1,4)+1, sqrtnint(n\4,3)+1, if(ispower(m^4-n,4),return(1))); 0 \\ Charles R Greathouse IV, Sep 04 2013

lst(lim)=my(v=List([0]),t); lim\=1; for(n=1,sqrtnint(lim\4,3)+1, for(m=sqrtnint(max(n^4-lim,0),4), n-1, t=n^4-m^4; if(t<=lim, listput(v,t)))); vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Sep 04 2013

Extended by Ray Chandler, Dec 04 2008

Definition corrected by Harvey P. Dale, Jan 19 2018

A181125 Difference of two positive 6th powers.

Original entry on oeis.org

0, 63, 665, 728, 3367, 4032, 4095, 11529, 14896, 15561, 15624, 31031, 42560, 45927, 46592, 46655, 70993, 102024, 113553, 116920, 117585, 117648, 144495, 215488, 246519, 258048, 261415, 262080, 262143, 269297, 413792, 468559, 484785, 515816

Offset: 1

T. D. Noe, Oct 06 2010

nonn

No term is a prime number.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A024352 (squares), A181123 (cubes), A147857 (4th powers), A181124-A181128 (5th to 9th powers).

Cf. A022522 (a subsequence, except its first term). - Mathew Englander, Jun 01 2014

nn=10^10; p=6; Union[Reap[Do[n=i^p-j^p; If[n<=nn, Sow[n]], {i,Ceiling[(nn/p)^(1/(p-1))]}, {j,i}]][[2,1]]]

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 *)

A181126 Difference of two positive 7th powers.

Original entry on oeis.org

0, 127, 2059, 2186, 14197, 16256, 16383, 61741, 75938, 77997, 78124, 201811, 263552, 277749, 279808, 279935, 543607, 745418, 807159, 821356, 823415, 823542, 1273609, 1817216, 2019027, 2080768, 2094965, 2097024, 2097151, 2685817

Offset: 1

T. D. Noe, Oct 06 2010

nonn

Because x^7-y^7 = (x-y)(x^6+x^5*y+x^4*y^2+x^3*y^3+x^2*y^4+x*y^5+y^6), the difference of two 7th powers is a prime number only if x=y+1, in which case all the primes are in A121618.

The number 67675234241018881 = 127^8 is the first of an infinite number of squares of the form (b^(7k)-1)^8 in this sequence. Are any other squares possible?

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A024352 (squares), A181123 (cubes), A147857 (4th powers), A181124-A181128 (5th to 9th powers)

nn=10^12; p=7; Union[Reap[Do[n=i^p-j^p; If[n<=nn, Sow[n]], {i,Ceiling[(nn/p)^(1/(p-1))]}, {j,i}]][[2,1]]]
Join[{0},#[[2]]-#[[1]]&/@Subsets[Range[10]^7,{2}]//Union] (* Harvey P. Dale, Oct 23 2024 *)

A181127 Difference of two positive 8th powers.

Original entry on oeis.org

0, 255, 6305, 6560, 58975, 65280, 65535, 325089, 384064, 390369, 390624, 1288991, 1614080, 1673055, 1679360, 1679615, 4085185, 5374176, 5699265, 5758240, 5764545, 5764800, 11012415, 15097600, 16386591, 16711680, 16770655, 16776960

Offset: 1

T. D. Noe, Oct 06 2010

nonn

No term is a prime number.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A024352 (squares), A181123 (cubes), A147857 (4th powers), A181124-A181128 (5th to 9th powers)

nn=10^14; p=8; Union[Reap[Do[n=i^p-j^p; If[n<=nn, Sow[n]], {i,Ceiling[(nn/p)^(1/(p-1))]}, {j,i}]][[2,1]]]

A303744 Numbers that are not a difference between same powers (greater than 1) of positive numbers.

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 18, 22, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 222, 226, 230, 234, 238, 246, 250, 254, 258, 262, 266, 270, 274, 278, 282, 286, 290

Offset: 1

Adam Kertesz, Apr 29 2018

nonn

Apart from 1 and 4, all terms == 2 (mod 4). - Robert Israel, Jun 25 2018

			Odd numbers greater than 1 are differences of squares, so they are not here.
8 is not a term, 9 - 1: difference of two squares;
26 is not a term, 27 - 1: difference of two cubes.

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

Sequences of numbers that are difference of powers: A024352 (squares), A181123 (cubes).

And of further n-th powers: A147857 (4th), A181124 (5th), A181125 (6th), A181126 (7th), A181127 (8th), A181128 (9th).

N:= 1000: # to get all terms <= N
S:= {1,2,4,seq(i,i=6..N,4)}:
for p from 3 to ilog2(N+1) do
  for n from 1 while n^p - (n-1)^p <= N do
    if n^p > N then m0:= ceil((n^p - N)^(1/p)) else m0:= 1 fi;
    for m from m0 to n-1 do
      v:= n^p-m^p;
      S:= S minus {v};
    od
od od:
sort(convert(S,list)); # Robert Israel, Jun 25 2018

cp's OEIS Frontend

A181123 Numbers that are the differences of two positive cubes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A181124 Difference of two positive 5th powers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A181128 Difference of two positive 9th powers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A152044 Numbers expressible as the difference of two nonnegative fourth powers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A181125 Difference of two positive 6th powers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Crossrefs

A147858 Differences of two coprime 4th powers.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A181126 Difference of two positive 7th powers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A181127 Difference of two positive 8th powers.

Views

Author

Keywords

Comments