A078359 Number of ways to write n as sum of a positive square and a positive cube.

0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0

Offset

1,17

Comments

a(A066650(n))=0, a(A055394(n))>0, a(A078360(n))=1, a(A054402(n))>1.

Earliest entries with a(n)=3 are n=1737, 2089, 2628, 2817. Earliest entries with a(n)=4 are n=1025, 19225, 27289, 29025, 39329, 48025, 54225. Earliest entries with a(n)=5 are n=92025, 540900, 567225, 747225. There are no a(n)>=6 in the range n=1..700000. - R. J. Mathar, Aug 16 2006

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

G.f.: (Sum_{k>=1} x^(k^2)) * (Sum_{k>=1} x^(k^3)). - Seiichi Manyama, Jun 17 2023

Example

a(1025)=4, as 1025 = 5^2 + 10^3 = 30^2 + 5^3 = 31^2 + 4^3 = 32^2 + 1^3.

Maple

interface(prettyprint=0) : A078359 := proc(n) local resul, isq, icu ; resul := 0 ; icu := 1 ; while icu^3 < n do if issqr(n-icu^3) then resul := resul+1 ; fi ; icu := icu+1 ; od ; RETURN(resul) ; end: for n from 1 to 100000 do printf("%d %d ", n, A078359(n)) ; od ; # R. J. Mathar, Aug 16 2006

Mathematica

a[n_] := Which[r = Reduce[x > 0 && y > 0 && n == x^2 + y^3, {x, y}, Integers]; r === False, 0, r[[0]] === And, 1, r[[0]] === Or, Length[r], True, Print["error: ", r]];
Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Feb 13 2018 *)

Prog

(Python)
from collections import Counter
from itertools import count, takewhile, product
def aupto(lim):
  sqs = list(takewhile(lambda x: x<=lim-1, (i**2 for i in count(1))))
  cbs = list(takewhile(lambda x: x<=lim-1, (i**3 for i in count(1))))
  cts = Counter(sum(p) for p in product(sqs, cbs))
  return [cts[i] for i in range(1, lim+1)]
print(aupto(105)) # Michael S. Branicky, May 29 2021

Crossrefs

Cf. A054402, A055394, A066650, A078360, A171385.

Sequence in context: A096562 A096563 A216512 * A107329 A263717 A230279

Adjacent sequences: A078356 A078357 A078358 * A078360 A078361 A078362

Keyword

nonn

Author

Reinhard Zumkeller, Nov 25 2002

Status

approved