A054402 -id:A054402 - 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

A078360 Numbers having a unique representation as sum of a positive square and a positive cube.

Original entry on oeis.org

2, 5, 9, 10, 12, 24, 26, 28, 31, 33, 36, 37, 43, 44, 50, 52, 57, 63, 68, 72, 73, 76, 80, 82, 91, 100, 101, 113, 122, 126, 127, 128, 134, 141, 148, 150, 152, 161, 164, 170, 171, 174, 177, 185, 189, 196, 197, 204, 206, 208, 217, 220, 223, 226, 232, 241, 246, 257

Offset: 1

Reinhard Zumkeller, Nov 25 2002

nonn

			10 is a term, as 10 = 3^2 + 1^3 and all other sums of positive squares and positives cubes are not equal 10.
17 is not a term, as 17 = 3^2 + 2^3 = 4^2 + 1^3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A055394, A054402, A078359.

Select[Range[300], Length[Solve[a^2 + b^3 == # && a > 0 && b > 0, {a, b}, Integers]] == 1 &] (* Amiram Eldar, Mar 27 2025 *)

A078359(a(n))=1.

A206606 Primes that can be written as a sum of a positive square and a positive cube in more than two ways.

Original entry on oeis.org

2089, 4481, 7057, 15193, 15641, 16649, 23417, 34721, 65537, 68489, 69697, 72577, 93241, 118673, 123209, 146161, 173897, 176401, 191969, 199873, 205721, 216233, 239633, 259121, 264169, 271169, 280009, 286289, 296353, 301409, 318313, 342233, 347993, 357569, 381529, 447569, 466273, 477577, 526249, 534577

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 10 2012

nonn

A subset of these, {65537, 93241, 191969, ..} allows this representation in more than 3 ways.

			2089 = 19^2+12^3 = 33^2+10^3 = 45^2+4^3

Robert Israel, Table of n, a(n) for n = 1..3930

Cf. A054402, A123364, A162930

N:= 10^6: # to get all terms <= N
for x from 1 to floor(N^(1/2)) do
  for y from 1 to floor((N-x^2)^(1/3)) do
    p:= x^2 + y^3;
    if isprime(p) then
      if assigned(R[p]) then R[p]:= R[p]+1
      else R[p]:= 1
      fi
    fi
  od
od:
sort(map(op,select(t -> R[op(t)]>2, [indices(R)]))); # Robert Israel, Mar 21 2017

t={}; Do[Do[AppendTo[t,n^2+m^3],{n,300}],{m,300}]; t=Sort[t]; t3={}; Do[If[t[[n]]==t[[n+2]]&&PrimeQ[t[[n]]],AppendTo[t3,t[[n]]]],{n,Length[t]-2}]; t3; f1[l_]:=Module[{t={}},Do[If[l[[n]]!=l[[n+1]],AppendTo[t,l[[n]]]],{n,Length[l]-1}];t]; (*ExtractSingleTermsOnly*) f1[t3] (* or *)
mx = 10^6; First /@ Sort@ Select[ Tally[ Join @@ Reap[(Sow@ Select[#^3 + Range[ Sqrt[mx - #^3]]^2, PrimeQ]) & /@ Range[mx^(1/3)]][[2, 1]]], #[[2]]>2 &] (* faster, Giovanni Resta, Mar 21 2017 *)

More terms from Robert Israel, Mar 21 2017

A162930 Primes that can be written as a sum of a positive square and a positive cube in more than one way.

Original entry on oeis.org

17, 89, 233, 449, 577, 593, 1289, 1367, 1601, 1753, 2089, 2521, 3391, 4481, 4721, 5953, 6121, 6427, 7057, 7577, 8081, 9649, 10313, 10657, 10729, 11969, 12329, 13121, 13457, 15137, 15193, 15641, 15661, 16033, 16649, 18523, 21673, 21961, 23201

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 17 2009

nonn

A subset of these, 2089, 4481, 7057, 15193, 15641, etc., allows this representation in more than two ways (See A206606).

			The prime 17 can be written 1^3 + 4^2 as well as 2^3 + 3^2.

David A. Corneth, Table of n, a(n) for n = 1..10808 (first 500 terms from Seiichi Manyama, terms <= 8*10^7)

Cf. A000040, A054402, A123364, A123388, A173795, A206606.

isA162930 := proc(n) if isprime(n) then wa := 0 ; for y from 1 to n/2 do if issqr(n-y^3) then if n -y^3 > 0 then wa := wa+1 ; fi; fi; od: RETURN( wa>1) ; else false; fi; end:
for i from 1 to 2700 do if isA162930 ( ithprime(i)) then printf("%d,",ithprime(i)) ; fi; od: # R. J. Mathar, Jul 21 2009

lst={};Do[Do[AppendTo[lst,n^2+m^3],{n,2*5!}],{m,2*5!}];lst=Sort[lst]; lst2={};Do[If[lst[[n]]==lst[[n+1]]&&PrimeQ[lst[[n]]],AppendTo[lst2, lst[[n]]]],{n,Length[lst]-1}];lst2;

upto(n) = {my(res = List(), v = vector(n), i, j, i2); for(i = 1, sqrtint(n), i2 = i^2; for(j = 1, sqrtnint(n - i^2, 3), v[i2 + j^3]++)); forprime(p = 2, n, if(v[p] > 1, listput(res, p))); kill(v); res} \\ David A. Corneth, Jun 20 2023

A000040 INTERSECT A054402.

Slightly edited by R. J. Mathar, Jul 21 2009

A171385 Sum of a positive square and a positive cube in at least three ways.

Original entry on oeis.org

1025, 1737, 2089, 2628, 2817, 3033, 3664, 4481, 4825, 4977, 5841, 7057, 7785, 7948, 8225, 8289, 8900, 10025, 11025, 11665, 12393, 14049, 14400, 14724, 15193, 15345, 15641, 15689, 15884, 16649, 16857, 18201, 18369, 19225, 19664, 19908, 20224

Offset: 1

Alan Frank, Dec 07 2009

Four or more representations have 1025, 19225, 27289, 29025, 39329, 48025, 54225, 65537, 65600, 79129, 92025, 93241, 105192, 105633, 111681, ... - R. J. Mathar, Dec 11 2009

			1025 = 32^2 + 1^3, 31^2 + 4^3, 30^2 + 5^3, and 5^2 + 10^3;
1737 = 35^2 + 8^3, 3^2 + 12^3, and 39^2 + 6^3.

Cf. A054402, integers with two or more such representations.

isA171385 := proc(n) a := 0 ; for c from 1 do if c^3 >= n then break; else if issqr(n-c^3) then a := a+1 ; end if; end if; end do ; a >= 3 ; end proc: for n from 2 to 120000 do if isA171385(n) then printf("%d,",n) ; fi; end do ; # R. J. Mathar, Dec 11 2009

More terms from R. J. Mathar, Dec 11 2009

A273508 Values of A272701 that are the sum of a positive square and a positive cube in more than one way.

Original entry on oeis.org

36998208, 449519625, 2367885312, 8016025680, 9563569561, 14753560033, 26971693632, 28769256000, 61358997609, 151544659968, 225128651328, 278450575201, 282429583137, 310289733000, 310289733000, 327699806625, 498700534033, 513025643520, 578097000000

Offset: 1

Altug Alkan, May 23 2016

nonn

Taxi-cab numbers (A001235) that are the sum of two nonzero squares in more than one way and also the sum of a positive square and a positive cube in more than one way.
Subsequence of A273498.
A001235(293) = 6^3*A001235(16) = 6^3*171288 = 36998208 is the least number with this property.
14753560033 = 1453*2677*3793 is the first term that is in A272935.
Obviously, in this sequence there are perfect powers infinitely many times.

			36998208 is a term because 36998208 = 102^3 + 330^3 = 144^3 + 324^3 = 1728^2 + 324^3 = 5832^2 + 144^3 = 648^2 + 6048^2 = 1728^2 + 5832^2.

Cf. A001235, A007692, A054402, A272701, A273498.

a(2)-a(19) from Giovanni Resta, May 24 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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A078360 Numbers having a unique representation as sum of a positive square and a positive cube.

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A206606 Primes that can be written as a sum of a positive square and a positive cube in more than two ways.

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A162930 Primes that can be written as a sum of a positive square and a positive cube in more than one way.

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A171385 Sum of a positive square and a positive cube in at least three ways.

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A273508 Values of A272701 that are the sum of a positive square and a positive cube in more than one way.

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