cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A038598 First differences between numbers that are a difference between 2 positive cubes.

Original entry on oeis.org

7, 12, 7, 11, 19, 5, 2, 28, 7, 19, 7, 3, 25, 17, 20, 19, 7, 2, 1, 53, 8, 17, 20, 15, 4, 7, 44, 1, 10, 51, 21, 16, 3, 16, 7, 2, 34, 55, 2, 27, 26, 8, 37, 19, 7, 56, 33, 2, 47, 9, 44, 17, 37, 15, 4, 7, 17, 11, 88, 26, 37, 19, 9, 10, 45, 6, 37, 19, 7, 22, 33, 2, 26, 55, 44, 7, 19, 65, 44, 10, 7
Offset: 1

Views

Author

Jeff Burch

Keywords

Crossrefs

Cf. A038593-A038597, A093330.
First differences of A181123.

Extensions

Extended by Ray Chandler, Nov 29 2008
Offset corrected and a(1)=7 inserted by Sean A. Irvine, Jan 23 2021

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

Views

Author

T. D. Noe, Oct 06 2010

Keywords

Comments

No term is a prime number.

Links

Crossrefs

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

Programs

  • Mathematica
    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]]]

A225908 Numbers that are both a sum and a difference of two positive cubes.

Original entry on oeis.org

91, 152, 189, 217, 513, 728, 1027, 1216, 1512, 1736, 2457, 3087, 4104, 4706, 4921, 4977, 5103, 5256, 5824, 5859, 6832, 7657, 8216, 8587, 9728, 10712, 11375, 12096, 12691, 13851, 13888, 14911, 15093, 15561, 16120, 16263, 19000, 19656, 21014, 23058, 23625, 24696
Offset: 1

Views

Author

Jonathan Sondow, Jun 21 2013

Keywords

Comments

Solutions x to the equations x = a^3 + b^3 = c^3 - d^3 in positive integers.
The intersection of A003325 and A181123. See those sequences for additional comments, references, links and cross-refs.
Suggested by Shiraishi's solutions to Gokai Ampon's equation u^3 + v^3 + w^3 = n^3 (transpose a term from the left side to the right side). See A023042 and A226903.
An infinite subsequence is (A226904(n)+1)^3 - A226904(n)^3.

Examples

			3^3 + 4^3 + 5^3 = 6^3, so 3^3 + 4^3 = 91 and 3^3 + 5^3 = 152 and 4^3 + 5^3 = 189 are members.

References

  • Shiraishi Chochu (aka Shiraishi Nagatada), Shamei Sampu (Sacred Mathematics), 1826.

Links

Crossrefs

Cf. A003325, A023042, A181123, A226903, A226904.

Programs

  • Mathematica
    nn = 3*10^4; t1 = Union[Flatten[Table[x^3 + y^3, {x, nn^(1/3)}, {y, x, (nn - x^3)^(1/3)}]]]; p = 3; t2 = Union[Reap[Do[n = i^p - j^p; If[n <= nn, Sow[n]], {i, Ceiling[(nn/p)^(1/(p - 1))]}, {j, i}]][[2, 1]]]; Intersection[t1, t2] (* T. D. Noe, Jun 21 2013 *)

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

Views

Author

T. D. Noe, Oct 06 2010

Keywords

Comments

No term is a prime number.

Links

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

T. D. Noe, Oct 06 2010

Keywords

Comments

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?

Links

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

T. D. Noe, Oct 06 2010

Keywords

Comments

No term is a prime number.

Links

Crossrefs

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

Programs

  • Mathematica
    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]]]

A225909 Numbers that are both a sum of two positive cubes and a difference of two consecutive cubes.

Original entry on oeis.org

91, 217, 1027, 4921, 8587, 14911, 31519, 39331, 106597, 117019, 136747, 185257, 195841, 265519, 281827, 616987, 636181, 684019, 712969, 724717, 736561, 955981, 1200169, 1352737, 1405621, 1771777, 2481571, 2756167, 2937331, 4251871, 4996171, 5262901
Offset: 1

Views

Author

Jonathan Sondow, Jun 21 2013

Keywords

Comments

Solutions x to the equations x = a^3 + b^3 = (c+1)^3 - c^3 in positive integers. The values of c are A226902.
The intersection of A003325 and A003215.
Subsequence of A225908 = numbers that are both a sum and a difference of two positive cubes.
Shiraishi's solution to Gokai Ampon's equation u^3 + v^3 + w^3 = n^3 (see A023042 and A226903) shows that the sequence is infinite.

Examples

			3^3 + 4^3 = 6^3 - 5^3 = 91, so 91 is a member.

References

  • Shiraishi Chochu (aka Shiraishi Nagatada), Shamei Sampu (Sacred Mathematics), 1826.

Links

Crossrefs

Cf. A003215, A003325, A023042, A181123, A225908, A226902, A226903.

Programs

Formula

a(n) = (A226902(n)+1)^3 - A226902(n)^3.

A098110 Smallest number that is the difference between two positive cubes in n ways.

Original entry on oeis.org

7, 721, 3367, 4118877, 1412774811, 424910390480793
Offset: 1

Views

Author

Jeff Burch, Sep 23 2004

Keywords

Comments

a(7) <= 15490327057569000, a(8) <= 123922616460552000. - Giovanni Resta, Mar 19 2020

Examples

			Pairs (x, y) such that x^3 - y^3 = a(1), ..., a(6):
7 = (2, 1);
721 = (16, 15), (9, 2);
3367 = (34, 33), (16, 9), (15, 2)l
4118877 = (162, 51), (165, 72), (178, 115), (678, 675);
1412774811 = (1134, 357), (1155, 504), (1246, 805), (2115, 2004), (4746, 4725);
424910390480793 = (596001, 595602), (317982, 316575), (141705, 134268), (83482, 53935), (77385, 33768), (75978, 23919).

Crossrefs

Cf. A014439, A014440, A014441, A181123, A034179, A265625, A333376, A333377.
Cf. A011541,A047696.

Extensions

a(6) from Giovanni Resta, Mar 19 2020

A246589 A maximal expulsion set containing the cubes as a subset.

Original entry on oeis.org

1, 3, 8, 10, 12, 14, 21, 23, 25, 27, 34, 36, 38, 40, 45, 47, 49, 51, 58, 60, 62, 64, 69, 71, 73, 75, 82, 84, 86, 88, 93, 95, 97, 99, 106, 112, 125, 136, 149, 160, 173, 184, 197, 210, 214, 216, 221, 223, 225, 227, 232, 234, 236, 238, 245, 247, 249, 251, 258
Offset: 1

Views

Author

N. J. A. Sloane, Sep 03 2014

Keywords

Comments

Brown's article indicates 176 terms up to 1000, however it appears that there are only 173 terms up to 1000. - Rémy Sigrist, Feb 17 2019

Links

Crossrefs

Cf. A000578, A181123.

Programs

  • PARI
    See Links section.

Extensions

More terms from Rémy Sigrist, Feb 17 2019

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

Views

Author

Adam Kertesz, Apr 29 2018

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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).

Programs

  • Maple
    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
Previous Showing 11-20 of 28 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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