A060835 -id:A060835 - cp's OEIS frontend

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

Reinhard Zumkeller, Nov 25 2002

nonn

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
a(3375900) = 6 and a(5472225) = 7 are the first entries with those values. - Robert Israel, Jun 25 2024, [but see A060835. - Hugo Pfoertner, Jun 26 2024]

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

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

Cf. A054402, A055394, A060835, A066650, A078360, A171385.

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

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 *)

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

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

A122054 Least positive number with exactly n partitions into a square and a cube.

Original entry on oeis.org

1, 2, 17, 1737, 1025, 92025, 3375900, 5472225, 35964225, 930860225, 1000837225, 4979585600, 38515961025, 88154795025, 203947076025, 88813460025, 5684061441600, 77806025000000, 64745012358225

Offset: 0

Zak Seidov, Oct 15 2006

The sequence is not monotonic.

Equals 1 followed by A060835.

			a(0)=1 because there's no partition of 1 into a square and a cube;
a(1)=2 because 2 = 1^2 + 1^3, one partition;
a(2)=17 because 17 = 3^2 + 2^3 = 4^2 + 1^3, two partitions;
a(3)=1737 because 1737 = 3^2 + 12^3 = 35^2 + 8^3 = 39^2 + 6^3, three partitions, etc.
Table of partitions:
{m,{a_i,b_i}}
{1,{-}}
{2,{1,1}}
{17,{3,2},{4,1}}
{1737,{3,12},{35,8},{39,6}}
{1025,{5,10},{30,5},{31,4},{32,1}}
{92025,{30,45},{152,41},{213,36},{255,30},{303,6}}
{3375900,{30,150},{1430,110},{1551,99},{1794,54},{1830,30},{1837,11}}
{5472225,{143,176},{990,165},{1935,120},{2035,110},{2251,74},{2277,66},{2321,44}}
{35964225,{165,330},{3167,296},{4035,270},{4191,264},{5610,165},{5885,110},{5973,66},{5997,6}}
{930860225,{13315,910},{23139,734},{26817,596},{29560,385},{30271,244},{30335,220},{30460,145},{30465,140},{30510,5}}
{1000837225,{915,1000},{16921,894},{19665,850},{20960,825},{24735,730},{29465,510},{30221,444},{31608,121},{31635,40},{31636,9}}
{4979585600,{8160,1700},{24512,1636},{40392,1496},{48785,1375},{49640,1360},{64515,935},{65560,880},{66840,800},{69960,440},{70024,424},{70565,55}}
{38515961025,{20330,3365},{41507,3326},{73755,3210},{95084,3089},{115595,2930},{118431,2904},{132255,2760},{153953,2456},{176355,1950},{190305,1320},{195570,645},{196254,69}}

Cf. A060835, A123388.

m=a^2+b^3, a,b>0.

Corrected a(7), added a(8)-a(12) from Lars Blomberg, Feb 15 2016

A273788 Least number k such that k^2 + k^3 is of the form x^2 + y^3 in exactly n ways where x, y > 0.

Original entry on oeis.org

1, 6, 24, 40, 180, 440, 3640, 18480, 137280, 1320, 703560, 15960, 1256640, 1436160, 96360

Offset: 1

Altug Alkan, May 30 2016

			a(2) = 6 because 6^2 + 6^3 = 15^2 + 3^3.

Cf. A011379, A060835, A122054.

a(7)-a(15) from Giovanni Resta, Jun 03 2016

A274053 Least squarefree number that is the sum of a nonzero square and a positive cube in exactly n ways.

Original entry on oeis.org

2, 17, 2089, 27289, 3030569, 6808609, 1632201497

Offset: 1

Altug Alkan, Jun 08 2016

			a(1) = 2 because 2 = 1^2 + 1^3.
a(2) = 17 because 17 = 4^2 + 1^3 = 3^2 + 2^3.
a(3) = 2089 because 2089 = 45^2 + 4^3 = 33^2 + 10^3 = 19^2 + 12^3.
a(4) = 27289 because 27289 = 29*941 = 165^2 + 4^3 = 129^2 + 22^3 = 108^2 + 25^3 = 17^2 + 30^3.

Cf. A005117, A055394, A060835, A122956, A173795.

a(6)-a(7) from Giovanni Resta, Jun 12 2016

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

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A122054 Least positive number with exactly n partitions into a square and a cube.

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Formula

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A273788 Least number k such that k^2 + k^3 is of the form x^2 + y^3 in exactly n ways where x, y > 0.

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A274053 Least squarefree number that is the sum of a nonzero square and a positive cube in exactly n ways.

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