A242192 -id:A242192 - cp's OEIS frontend

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.

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

Lars Blomberg, May 06 2014

nonn

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

			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.

Lars Blomberg, Table of n, a(n) for n = 1..2241

Cf. A242184, A242185, A242186.

a242183 n = a242183_list !! (n-1)
a242183_list = concatMap (\(r,x) -> take r [x,x..]) $
                         zip a242192_list [1..]
-- Reinhard Zumkeller, May 07 2014

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 *)

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

A300564 Numbers z such that there is a solution to x^2 + y^3 = z^4 with x, y, z >= 1.

Original entry on oeis.org

6, 9, 15, 35, 36, 42, 48, 57, 63, 71, 72, 75, 78, 90, 98, 100, 120, 135, 141, 147, 162, 195, 196, 204, 208, 215, 225, 243, 252, 260, 279, 280, 288, 289, 295, 300, 306, 336, 363, 364, 384, 405, 441, 450, 456, 462, 504, 510, 525, 537, 550, 568, 576, 600, 624, 630, 713, 720, 722, 735, 750, 784, 800, 819, 828, 841, 845, 847, 867, 875

Offset: 1

M. F. Hasler, Apr 16 2018

nonn

Cf. A242183, A242192, A300565 (z^5 = x^3 + x^4), A300566 (z^6 = x^4 + y^5).

is(z)=for(y=1,sqrtnint(z^4,3),issquare(z^4-y^3,&x)&&x&&return(1))

Equals sequence A242183 with duplicates removed.

A242186 Positive integers, c, such that there is more than one solution to the equation a^2 + b^3 = c^4, with a, b > 0.

Original entry on oeis.org

72, 100, 147, 225, 456, 576, 800, 1050, 1176, 1539, 1800, 1944, 2028, 2645, 2646, 2695, 2700, 3025, 3087, 3275, 3648, 3844, 3969, 4335, 4356, 4500, 4608, 4950, 5412, 6000, 6075, 6400, 7260, 7623, 8225, 8400, 8405, 8450, 8664, 8820, 9000, 9408, 9828, 10108

Offset: 1

Lars Blomberg and Robert G. Wilson v, May 06 2014

nonn

225, 1050, 1176, 2028, 3025, 6075, 7260, 8400, 8450, 8820, 9408, 10890, 12312, 18375, 19494, 21160, 24696, 26775, 28125, 28350, 31752, 31974, 34300, 39600, 43245, 44100, 49923, 53361, 54756, 58080, 64980, 67200, 71415, 75264, 87120, 98496, 131250, 139425, 144150, 145656, 159048, 164025, ... have three solutions;
1800, 11025, 14400, 16224, 38025, 48600, 61347, 67600, 70560, 81675, 88200, 115200, 129792, 147000, 155952, 166419, ... have four solutions;
24200, 77175, ... have five solutions.
A242192(a(n)) > 1. - Reinhard Zumkeller, May 07 2014

			72 is in the sequence since 72^4 = 1728^2 + 288^3 = 4941^2 + 135^3.

Lars Blomberg, Table of n, a(n) for n = 1..202

Cf. A242183, A242184, A242185.

a242186 n = a242186_list !! (n-1)
a242186_list = filter ((> 1) . a242192) [1..]
-- Reinhard Zumkeller, May 07 2014

f[n_] := f[n] = Module[{a}, Array[(a = Sqrt[n^4 - #^3]; If[ IntegerQ@ a && a > 0, {a, #}, Sequence @@ {}]) &, Floor[n^(4/3)]]]; Select[ Range@ 10000, f@# > 1 &]

cp's OEIS Frontend

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.

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A300564 Numbers z such that there is a solution to x^2 + y^3 = z^4 with x, y, z >= 1.

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A242186 Positive integers, c, such that there is more than one solution to the equation a^2 + b^3 = c^4, with a, b > 0.

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Haskell

Mathematica