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.

Showing 1-3 of 3 results.

A106265 Numbers a > 0 such that the Diophantine equation a + b^2 = c^3 has integer solutions b and c.

Original entry on oeis.org

1, 2, 4, 7, 8, 11, 13, 15, 18, 19, 20, 23, 25, 26, 27, 28, 35, 39, 40, 44, 45, 47, 48, 49, 53, 54, 55, 56, 60, 61, 63, 64, 67, 71, 72, 74, 76, 79, 81, 83, 87, 89, 95, 100, 104, 106, 107, 109, 112, 116, 118, 121, 124, 125, 126, 127, 128, 135, 139, 143, 146, 147, 148, 150, 151, 152, 153
Offset: 1

Views

Author

Zak Seidov, Apr 28 2005

Keywords

Comments

A given a(n) can have multiple solutions with distinct (b,c), e.g., a=4 with b=2, c=2 (4 + 2^2 = 2^3) or with b=11, c=5 (4 + 11^2 = 5^3). (See also A181138.) Sequences A106266 and A106267 list the minimal values. - M. F. Hasler, Oct 04 2013
The cubes A000578 = (1, 8, 27, 64, ...) form a subsequence of this sequence, corresponding to b=0, a=c^3. If b=0 is excluded, these terms are not present, except for a few exceptions, a = 216, 343, 12167, ... (6^3 + 28^2 = 10^3, 7^3 + 13^2 = 8^3, 23^3 + 588^2 = 71^3, ...), cf. A038597 for the possible b-values. - M. F. Hasler, Oct 05 2013
This is the complement of A081121. The values do indeed correspond to solutions listed in Gebel's file. - M. F. Hasler, Oct 05 2013
B-file corrected following a remark by Alois P. Heinz, May 24 2019. A double-check would be appreciated in view of two values that were missing, for unknown reasons, in the earlier version of the b-file. - M. F. Hasler, Aug 10 2024

Examples

			a = 1,2,4,7,8,11,13,15,18,19,20,23,25,26,27,28,35,39,40,44,45,47,48,49,53, ...
b = 0,5,2,1,0, 4,70, 7, 3,18,14, 2,10, 1, 0, 6,36, 5,52, 9,96,13,4,524,26, ...
c = 1,3,2,2,2, 3,17, 4, 3, 7, 6, 3, 5, 3, 3, 4,11, 4,14, 5,21, 6, 4,65, 9, ...
Here are the values grouped together:
{{1, 0, 1}, {2, 5, 3}, {4, 2, 2}, {7, 1, 2}, {8, 0, 2}, {11, 4, 3}, {13, 70, 17}, {15, 7, 4}, {18, 3, 3}, {19, 18, 7}, {20, 14, 6}, {23, 2, 3}, {25, 10, 5}, {26, 1, 3}, {27, 0, 3}, {28, 6, 4}, {35, 36, 11}, {39, 5, 4}, {40, 52, 14}, {44, 9, 5}, {45, 96, 21}, {47, 13, 6}, {48, 4, 4}, {49, 524, 65}, {53, 26, 9}, {54, 17, 7}, {55, 3, 4}, {56, 76, 18}, {60, 2, 4}, {61, 8, 5}, {63, 1, 4}, {64, 0, 4}, {67, 110, 23}, {71, 21, 8}, ... }
a(2243) = 10000 = 25^3 - 75^2. - _M. F. Hasler_, Oct 05 2013, index corrected Aug 10 2024
a(136) = 366 = 11815^3 - 1284253^2 (has c/a(n) ~ 32.3); a(939) = 3607 = 244772^3 - 121099571^2 (has c/a(n) ~ 67.9); a(1090) = 4265 = 84521^3 - 24572364^2 (has c/a(n) ~ 19.8). - _M. F. Hasler_, Aug 10 2024

Links

Crossrefs

Cf. A106266, A106267 for respective minimal values of b and c.
Cf. A023055: (Apparent) differences between adjacent perfect powers (integers of form a^b, a >= 1, b >= 2); A076438: n which appear to have a unique representation as the difference of two perfect powers; that is, there is only one solution to Pillai's equation a^x - b^y = n, with a>0, b>0, x>1, y>1; A076440: n which appear to have a unique representation as the difference of two perfect powers and one of those powers is odd; that is, there is only one solution to Pillai's equation a^x - b^y = n, with a>0, b>0, x>1, y>1 and that solution has odd x or odd y (or both odd); A075772: Difference between n-th perfect power and the closest perfect power, etc.
Cf. A023055, A075772, A076438, A076440.
Cf. A054504, A081121, A081120; A179386 - A179388.

Programs

  • Mathematica
    f[n_] := Block[{k = Floor[n^(1/3) + 1]}, While[k < 10^6 && !IntegerQ[ Sqrt[k^3 - n]], k++ ]; If[k == 10^6, 0, k]]; Select[ Range[ 154], f[ # ] != 0 &] (* Robert G. Wilson v, Apr 28 2005 *)
  • PARI
    select( {is_A106265(a, L=99)=for(c=sqrtnint(a, 3), (a+9)*L, issquare(c^3-a, &b) && return(c))}, [1..199]) \\ The function is_A106265 returns 0 if n isn't a term, or else the c-value (A106267) which can't be zero if n is a term. The L-value can be used to increase the search limit but so far no instance is known that requires L>68. - M. F. Hasler, Aug 10 2024

Formula

a(n) = A106267(n)^3 - A106266(n)^2.

Extensions

More terms from Robert G. Wilson v, Apr 28 2005
Definition corrected, solutions with b=0 added by M. F. Hasler, Sep 30 2013

A076438 Numbers k which appear to have a unique representation as the difference of two perfect powers; that is, there is only one solution to Pillai's equation a^x - b^y = k, with a > 0, b > 0, x > 1, y > 1.

Original entry on oeis.org

1, 2, 10, 29, 30, 38, 43, 46, 52, 59, 122, 126, 138, 142, 146, 150, 154, 166, 170, 173, 181, 190, 194, 214, 222, 234, 263, 270, 282, 283, 298, 317, 318, 332, 338, 342, 347, 349, 354, 361, 370, 379, 382, 383, 386, 406, 419, 428, 436, 461, 467, 479, 484, 486
Offset: 1

Views

Author

T. D. Noe, Oct 12 2002

Keywords

Comments

This is the classic Diophantine equation of S. S. Pillai, who conjectured that there are only a finite number of solutions for each k. A generalization of Catalan's conjecture that a^x - b^y = 1 has only one solution. See A076427 for the number of solutions for each k. Interestingly, the unique solutions (k,a,x,b,y) fall into two groups: (A076439) those in which x and y are even numbers, so that k is the difference of squares, and (A076440) those requiring an odd power. This sequence was found by examining all perfect powers (A001597) less than 2^63-1. By examining a larger set of perfect powers, we may discover that some of these numbers do not have a unique representation.

References

  • R. K. Guy, Unsolved Problems in Number Theory, D9.
  • T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986.

Links

Crossrefs

Cf. A001597, A053289, A074981, A076427, A076438, A076439, A076440, A207079.

A076439 Numbers k which appear to have a unique representation as the difference of two perfect powers where those powers are both 2; that is, there is only one solution to Pillai's equation a^x - b^y = k, with a > 0, b > 0, x > 1, y > 1 and that solution has x = y = 2.

Original entry on oeis.org

29, 43, 52, 59, 173, 181, 263, 283, 317, 332, 347, 349, 361, 379, 383, 419, 428, 436, 461, 467, 479, 484, 491, 509, 523, 529, 569, 571, 607, 613, 619, 641, 643, 653, 661, 677, 691, 709, 733, 773, 787, 788, 811, 827, 839, 853, 877, 881, 883, 907, 911, 941
Offset: 1

Views

Author

T. D. Noe, Oct 12 2002

Keywords

Comments

There are two types of unique solutions. See A076438 for the general case. The k for which the unique solution can be written as k = a^2 - b^2 seems to have the following properties: (1) b = a-1 for odd k and b = a-2 for even k and (2) k = 4^r * p^s, where r is in {0,1}, p is an odd prime and s is in {1,2}. This sequence was found by examining all perfect powers (A001597) less than 2^63-1. By examining a larger set of perfect powers, we may discover that some of these numbers do not have a unique representation.

References

  • R. K. Guy, Unsolved Problems in Number Theory, D9.
  • T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986.

Links

Crossrefs

Cf. A001597, A076438, A076440.
Showing 1-3 of 3 results.

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