A181128 -id:A181128 - 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]]]

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]]]

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

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A181123 Numbers that are the differences of two positive cubes.

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A181124 Difference of two positive 5th powers.

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A181125 Difference of two positive 6th powers.

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A181126 Difference of two positive 7th powers.

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A181127 Difference of two positive 8th powers.

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

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