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 1-5 of 5 results.

A242183 Integers c, listed with multiplicity, such that there is a solution to the equation a^2 + b^3 = c^4, with integers a, b > 0.

Original entry on oeis.org

6, 9, 15, 35, 36, 42, 48, 57, 63, 71, 72, 72, 75, 78, 90, 98, 100, 100, 120, 135, 141, 147, 147, 162, 195, 196, 204, 208, 215, 225, 225, 225, 243, 252, 260, 279, 280, 288, 289, 295, 300, 306, 336, 363, 364, 384, 405, 441, 450, 456, 456, 462, 504, 510, 525, 537
Offset: 1

Views

Author

Lars Blomberg, May 06 2014

Keywords

Comments

A242192(k) gives number of occurrences of k. - Reinhard Zumkeller, May 07 2014
See A300564 for the list of values without duplicates. - M. F. Hasler, Apr 16 2018

Examples

			6 is in the sequence because 6^4 = 28^2 + 8^3.
72 is in the sequence twice because 72^4 = 1728^2 + 288^3 = 4941^2 + 135^3.

Links

Crossrefs

Cf. A242184, A242185, A242186.

Programs

  • Haskell
    a242183 n = a242183_list !! (n-1)
a242183_list = concatMap (\(r,x) -> take r [x,x..]) $
                         zip a242192_list [1..]
-- Reinhard Zumkeller, May 07 2014
  • Mathematica
    f[n_] := f[n] = Module[{a}, Array[(a = Sqrt[n^4 - #^3]; If[ IntegerQ@ a && a > 0, {a, #}, Sequence @@ {}]) &, Floor[n^(4/3)]]];; k = 1; lst = {}; While[k < 3001, If[ f[k] != {}, AppendTo[lst, k]; Print[{k, f[k]}]]; k++]; s = Select[ Range[3000], f@# != {} &]; l = Length@ f@ # & /@ s; Flatten[ Table[ s[[#]], {l[[#]]}] & /@ Range@ Length@ s] (* Robert G. Wilson v, May 06 2014 *)

Formula

c = sqrt(sqrt(a^2+b^3)) is an integer.

A242184 Integers, a, which are the solutions to the equation a^2 + b^3 = c^4, with integers a, b > 0, and indexed off of A242183.

Original entry on oeis.org

27, 28, 63, 433, 648, 1176, 1728, 1792, 2925, 3807, 4032, 4500, 4785, 4941, 6000, 6083, 6875, 7203, 7452, 7902, 8100, 10000, 10125, 12005, 13328, 14703, 15525, 19683, 20412, 21266, 26775, 27712, 32507, 33750, 35672, 40572, 40797, 41328, 41472, 45927, 49375
Offset: 1

Views

Author

Lars Blomberg, May 06 2014

Keywords

Examples

			a(1)=28 since A242183(1)=6 and 6^4 = 28^2 + 8^3.

Links

Crossrefs

Cf. A242183, A242185, A242186.

Programs

  • Mathematica
    (* after running the Mmca coding in A242183 *) k = 1; alst = {}; While[k < 6501, If[f@ k != {}, AppendTo[ alst, Table[#1, {1}] & @@@ f[k]]]; k++]; alst // Flatten (* Robert G. Wilson v, May 06 2014 *)

Formula

a = sqrt(c^4-b^3) is an integer.

A242185 Integers b which are the solution to the equation a^2 + b^3 = c^4, with integers a, b > 0, and indexed from A242183.

Original entry on oeis.org

8, 18, 23, 36, 49, 108, 108, 126, 128, 135, 136, 143, 216, 225, 245, 288, 288, 300, 343, 368, 375, 400, 450, 500, 576, 588, 600, 648, 686, 693, 784, 900, 1026, 1098, 1125, 1156, 1183, 1215, 1350, 1350, 1440, 1458, 1568, 1628, 1638, 1681, 1728, 1728, 1863, 2000
Offset: 1

Views

Author

Lars Blomberg, May 06 2014

Keywords

Examples

			a(1)=8 since A242183(1)=6 and 6^4 = 28^2 + 8^3.

Links

Crossrefs

Cf. A242183, A242184, A242186.

Programs

  • Mathematica
    (* after running the Mmca coding in A242183 *) k = 1; blst = {}; While[k < 6501, If[f@ k != {}, AppendTo[ blst, Table[#2, {1}] & @@@ f[k]]]; k++]; blst // Flatten (* Robert G. Wilson v, May 06 2014 *)

Formula

b = (c^4 - a^2)^(1/3) is an integer.

A242192 Number of ways to write n^4 as sum of a square and a cube, both > 0.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Reinhard Zumkeller, May 07 2014

Keywords

Comments

a(n) = number of occurrences of n in A242183;
a(A242186(n)) > 1.

Examples

			a(6) = #{28^2 + 8^3} = 1;
a(72) = #{4941^2 + 135^3, 1728^2 + 288^3} = 2;
a(225) = #{49375^2 + 500^3, 33750^2 + 1125^3, 10125^2 + 1350^3} = 3;
a(1800) = #{3160000^2 + 8000^3, 2835000^2 + 13500^3, 2160000^2 + 18000^3, 648000^2 + 21600^3} = 4;
a(24200) = #{582914112^2 + 147136^3, 564344000^2 + 290400^3, 479160000^2 + 484000^3, 219615000^2 + 665500^3, 42092875^2 + 698775^3} = 5.

Links

Crossrefs

Cf. A010052, A000578, A000583.

Programs

  • Haskell
    a242192 n = sum $ map (a010052 . (n ^ 4 -)) $
                      takeWhile (< n ^ 4) $ map (^ 3) [1..]
-- Reinhard Zumkeller, May 07 2014

A242381 Positive integers, c, such that there are more than two solutions to the equation a^2 + b^3 = c^4, with a, b > 0.

Original entry on oeis.org

225, 1050, 1176, 1800, 2028, 3025, 6075, 7260, 8400, 8450, 8820, 9408, 10890, 11025, 12312, 14400, 16224, 18375, 19494, 21160, 24200, 24696, 26775, 28125, 28350, 31752, 31974, 34300, 38025, 39600, 43245, 44100, 48600, 49923, 53361, 54756, 58080, 61347, 64980, 67200, 67600, 70560, 71415, 75264, 77175, 81675, 87120, 88200, 98496, 115200, 129792, 131250, 139425, 144150, 145656, 147000, 155952, 159048, 164025, 166419
Offset: 1

Views

Author

Lars Blomberg and Robert G. Wilson v, May 12 2014

Keywords

Comments

Submitted at the request of Ralf Stephan, dated 12 May 2014.
Subsequence of A242183.

Crossrefs

Cf. A242183, A242186.
Showing 1-5 of 5 results.

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

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