A100271 -id:A100271 - cp's OEIS frontend

A331523 a(n) is the least positive k such that A100271(n) - k^3 is a fourth power.

1, 1, 3, 2, 1, 3, 7, 8, 7, 1, 13, 6, 15, 15, 12, 16, 16, 2, 10, 20, 18, 21, 7, 16, 9, 22, 13, 22, 23, 24, 20, 10, 25, 12, 25, 22, 26, 27, 11, 26, 28, 30, 27, 31, 22, 31, 34, 30, 35, 36, 28, 31, 37, 8, 37, 16, 27, 36, 35, 40, 1, 3, 13, 31, 37, 28, 33, 42, 21

Offset: 1

Rémy Sigrist, Jan 19 2020

nonn

			The first terms, alongside A100271(n), are:
  n   a(n)  A100271(n)
  --  ----  ----------------
   1     1     2 = 1^3 + 1^4
   2     1    17 = 1^3 + 2^4
   3     3    43 = 3^3 + 2^4
   4     2    89 = 2^3 + 3^4
   5     1   257 = 1^3 + 4^4
   6     3   283 = 3^3 + 4^4
   7     7   359 = 7^3 + 2^4
   8     8   593 = 8^3 + 3^4
   9     7   599 = 7^3 + 4^4
  10     1  1297 = 1^3 + 6^4

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A331523

See A331435 for similar sequences.

Cf. A000583, A100271.

PARI
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A100291 Numbers of the form a^4 + b^3 with a, b > 0.

Original entry on oeis.org

2, 9, 17, 24, 28, 43, 65, 80, 82, 89, 108, 126, 141, 145, 206, 217, 232, 257, 264, 283, 297, 320, 344, 359, 381, 424, 472, 513, 528, 593, 599, 626, 633, 652, 689, 730, 745, 750, 768, 810, 841, 968, 985, 1001, 1016, 1081, 1137, 1256, 1297, 1304, 1323, 1332

Offset: 1

T. D. Noe, Nov 18 2004

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Gian Cordana Sanjaya and Xiaoheng Wang, On the squarefree values of a^4+b^3, arXiv:2107.10380 [math.NT], 2021.

Cf. A100271 (primes of the form a^4 + b^3).
Cf. A055394 (a^2 + b^3: contains this as subsequence), A111925 (a^2 + b^4), A100291 (a^4 + b^3), A100292 (a^5 + b^2), A100293 (a^5 + b^3), A100294 (a^5 + b^4), A303372 (a^2 + b^6), A303373 (a^3 + b^6), A303374 (a^4 + b^6), A303375 (a^5 + b^6).
Roots of 5th powers are listed in A300565 (z^5 = x^3 + y^4); see also A300564 (z^4 = x^2 + y^3) and A242183, A300566 (z^6 = x^4 + y^5), A300567 (z^7 = x^6 + y^5), A302174.

lst={}; Do[p=a^4+b^3; If[p<2000, AppendTo[lst, p]], {a, 64}, {b, 256}]; Union[lst]
With[{nn=20},Select[Union[#[[1]]^4+#[[2]]^3&/@Tuples[Range[20],2]],#<= nn^3+1&]] (* Harvey P. Dale, May 27 2020 *)

is(n)=for(a=1, sqrtnint(n-1, 4), ispower(n-a^4, 3) && return(a)) \\ Returns a > 0 if n is in the sequence, or 0 otherwise. - M. F. Hasler, Apr 25 2018

list(lim)=my(v=List());for(b=1,sqrtnint(lim\=1,3), my(b3=b^3); for(a=1,sqrtnint(lim-b3,4), listput(v,a^4+b3))); Set(v) \\ Charles R Greathouse IV, Jul 26 2021

Edited by M. F. Hasler, Apr 25 2018

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A331523 a(n) is the least positive k such that A100271(n) - k^3 is a fourth power.

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A100291 Numbers of the form a^4 + b^3 with a, b > 0.

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