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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A106267 Least c such that A106265(n) + b^2 = c^3 for some b (given in A106266).

Original entry on oeis.org

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, 7, 4, 18, 4, 5, 4, 4, 23, 8, 6, 99, 5, 20, 13, 27, 7, 5, 6, 5, 9, 11, 143, 5, 8, 5, 7, 5, 5, 5, 15, 16, 12, 6, 47, 14, 195, 7, 197, 175, 8, 6, 9, 51, 10, 6, 59, 13, 7, 11, 6
Offset: 1

Views

Author

Zak Seidov, Apr 28 2005

Keywords

Links

Crossrefs

Cf. A106265 (main entry), A106266 (the b-values).

Programs

  • PARI
    for(a=1, 999, (c=is_A106265(a))&& print1(c", ")) \\ M. F. Hasler, Oct 04 2013, edited Aug 10 2024: main program moved to A106265, as function is_A106265 which returns the (least) c-value a(n), always nonzero for terms of A106265, or 0 if the argument isn't a term of A106265.

Formula

a(n) = (A106265(n) + A106266(n)^2)^(1/3). - M. F. Hasler, Oct 04 2013

Extensions

More terms from Robert G. Wilson v, Apr 28 2005
All values recomputed from A106266 and A106265 by R. J. Mathar, Aug 24 2008
Initial term 1 and other solutions with b=0 added by M. F. Hasler, Oct 04 2013

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
Showing 1-2 of 2 results.

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