A181124 -id:A181124 - 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

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]]]

A258807 a(n) = n^5 - 1.

Original entry on oeis.org

0, 31, 242, 1023, 3124, 7775, 16806, 32767, 59048, 99999, 161050, 248831, 371292, 537823, 759374, 1048575, 1419856, 1889567, 2476098, 3199999, 4084100, 5153631, 6436342, 7962623, 9765624, 11881375, 14348906, 17210367, 20511148, 24299999, 28629150, 33554431

Offset: 1

Vincenzo Librandi, Jun 11 2015

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Subsequence of A181124.
Cf. A024003, A024049, A300656.
Sequences of the type n^k-1: A132411 (k=2), A068601 (k=3), A123865 (k=4), this sequence (k=5), A123866 (k=6), A258808 (k=7), A258809 (k=8), A258810 (k=9), A123867 (k=10), A258812 (k=11), A123868 (k=12).

List([1..35],n->n^5-1); # Muniru A Asiru, Oct 28 2018

Magma
```
[n^5-1: n in [1..50]];
```

I:=[0,31,242,1023, 3124,7775]; [n le 6 select I[n] else 6*Self(n-1)-15*Self(n-2)+20*Self(n-3)-15*Self(n-4)+ 6*Self(n-5)-Self(n-6): n in [1..50]];

seq(n^5-1,n=1..35); # Muniru A Asiru, Oct 28 2018

Table[n^5 - 1, {n, 1, 50}] (* or *) LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 31, 242, 1023, 3124, 7775}, 50]

a(n)=n^5-1 \\ Charles R Greathouse IV, Jun 11 2015

for n in range(1, 50): print(n**5 - 1, end=', ') # Stefano Spezia, Oct 28 2018

[n^5-1 for n in (1..50)] # Bruno Berselli, Jun 11 2015

G.f.: x^2*(31 + 56*x + 36*x^2 - 4*x^3 + x^4)/(1 - x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
a(n) = -A024003(n). - Bruno Berselli, Jun 11 2015
Sum_{n>=2} 1/a(n) = Sum_{n>=1} (zeta(5*n) - 1) = 0.0379539032... - Amiram Eldar, Nov 06 2020

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]]]

A247099 Numbers which are the sum or difference of two fifth powers.

Original entry on oeis.org

0, 1, 2, 31, 32, 33, 64, 211, 242, 243, 244, 275, 486, 781, 992, 1023, 1024, 1025, 1056, 1267, 2048, 2101, 2882, 3093, 3124, 3125, 3126, 3157, 3368, 4149, 4651, 6250, 6752, 7533, 7744, 7775, 7776, 7777, 7808, 8019, 8800, 9031, 10901, 13682, 15552, 15783, 15961

Offset: 1

Charles R Greathouse IV, Nov 29 2014

nonn

			31 = 2^5 - 1^5, 32 = 2^5 + 0^5, 33 = 2^5 + 1^5.

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

Cf. A000584, A004842, A004831, A181124.

Union[Flatten[{Total[#],Abs[#[[2]]-#[[1]]]}&/@Tuples[Range[0,10]^5,2]]] (* Harvey P. Dale, Mar 30 2015 *)

T=thueinit('z^5+1);
is(n)=n==0 || #thue(T, n)>0

a(n) << n^(5/2). Can this be improved?

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

A181128 Difference of two positive 9th powers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A258807 a(n) = n^5 - 1.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Magma

Maple

Mathematica

PARI

Python

Sage

Formula

A181125 Difference of two positive 6th powers.

Views

Author

Keywords

Comments

Links

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

Links

Crossrefs

Programs

Mathematica

A247099 Numbers which are the sum or difference of two fifth powers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments