A055394 -id:A055394 - cp's OEIS frontend

A303373 Numbers of the form a^3 + b^6, with integers a, b > 0.

2, 9, 28, 65, 72, 91, 126, 128, 189, 217, 280, 344, 407, 513, 576, 730, 737, 756, 793, 854, 945, 1001, 1064, 1072, 1241, 1332, 1395, 1458, 1729, 1792, 2060, 2198, 2261, 2457, 2745, 2808, 2926, 3376, 3439, 3473, 4097, 4104, 4123, 4160, 4221, 4312, 4439, 4608, 4825, 4914

Offset: 1

M. F. Hasler, Apr 22 2018

A subsequence of the numbers of the form a^3 + b^2, A055394.

Although it is easy to produce many terms of this sequence, it is nontrivial to check whether a very large number is of this form.

			The first terms are 1^3 + 1^6 = 2, 2^3 + 1^6 = 9, 3^3 + 1^6 = 28, 4^3 + 1^6 = 65, 2^3 + 2^6 = 72, 3^3 + 2^6 = 91, 5^3 + 1^6 = 126, 4^3 + 2^6 = 128, ...

Cf. A055394 (a^2 + b^3), 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).

Cf. A303372 (a^2 + b^6), A303374 (a^4 + b^6), A303375 (a^5 + b^6).

is(n,k=3,m=6)=for(b=1,sqrtnint(n-1,m),ispower(n-b^m,k)&&return(b)) \\ Returns b > 0 if n is in the sequence, else 0.
A303373_vec(L=10^5,k=3,m=6,S=List())={for(a=1,sqrtnint(L-1,m),for(b=1,sqrtnint(L-a^m,k),listput(S,a^m+b^k)));Set(S)} \\ List of all terms up to limit L

A054402 Numbers that are the sum of a positive square and a positive cube in more than one way.

Original entry on oeis.org

17, 65, 89, 108, 129, 145, 225, 233, 252, 297, 316, 388, 449, 464, 505, 537, 548, 577, 593, 633, 730, 737, 745, 792, 793, 801, 873, 1025, 1088, 1090, 1116, 1289, 1304, 1305, 1367, 1412, 1441, 1452, 1529, 1585, 1601

Offset: 1

Henry Bottomley, May 12 2000

			a(1)=17 since 17 = 3^2 + 2^3 = 4^2 + 1^3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A055393, A055394.

lst={};Do[Do[AppendTo[lst,n^2+m^3],{n,5!}],{m,5!}];lst=Sort[lst]; lst2={};Do[If[lst[[n]]==lst[[n+1]],AppendTo[lst2,lst[[n]]]],{n,Length[lst]-1}];lst2; Take[Union[lst2],123] (* Vladimir Joseph Stephan Orlovsky, Jul 17 2009 *)

list(lim)=my(v=List(),u=List());for(n=1,sqrtint(lim\1-1), for(m=1, sqrtnint(lim\1-n^2,3), listput(v,n^2+m^3))); v=vecsort(v); for(i=2,#v, if(v[i]==v[i-1], listput(u,v[i]))); Set(u) \\ Charles R Greathouse IV, May 15 2015

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

Original entry on oeis.org

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

A078377 Number of distinct prime factors of numbers which can be written as sum of a positive square and a positive cube.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 1, 2, 3, 1, 1, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 1, 3, 2, 2, 2, 3, 1, 2, 2, 2, 1, 1, 3, 3, 1, 3, 3, 2, 1, 3, 1, 1, 3, 3, 2, 2, 2, 2, 2, 2, 1, 2, 1, 3, 2, 2, 1, 1, 3, 2, 2, 2, 1, 2, 3, 2, 2, 1, 2, 3, 2, 3, 2

Offset: 1

Reinhard Zumkeller, Nov 25 2002

nonn

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

Cf. A001221, A055394, A078378, A078379.

PrimeNu[Select[Range[200], Length[Reduce[a^2 + b^3 == # && a > 0 && b > 0, {a, b}, Integers]] > 0 &]] (* Amiram Eldar, Mar 27 2025 *)

a(n) = A001221(A055394(n)).

A078378 Total number of prime factors of numbers which can be written as sum of a positive square and a positive cube.

Original entry on oeis.org

1, 1, 2, 2, 3, 1, 4, 2, 3, 1, 2, 4, 1, 1, 3, 3, 3, 2, 3, 2, 3, 5, 1, 3, 5, 2, 1, 2, 4, 1, 5, 1, 2, 4, 1, 7, 2, 2, 2, 2, 3, 4, 4, 2, 3, 3, 3, 3, 2, 2, 4, 4, 1, 4, 2, 5, 2, 4, 1, 4, 2, 4, 1, 1, 3, 5, 1, 4, 5, 2, 1, 5, 1, 2, 3, 4, 4, 3, 7, 2, 3, 3, 1, 4, 1, 4, 4, 6, 1, 1, 6, 2, 5, 3, 1, 2, 3, 3, 5, 1, 2, 5, 3, 4, 4

Offset: 1

Reinhard Zumkeller, Nov 25 2002

nonn

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

Cf. A001222, A055394, A078377, A078379.

PrimeOmega[Select[Range[200], Length[Reduce[a^2 + b^3 == # && a > 0 && b > 0, {a, b}, Integers]] > 0 &]] (* Amiram Eldar, Mar 27 2025 *)

a(n) = A001222(A055394(n)).

A055393 Sum of a square and a nonnegative cube in more than one way.

Original entry on oeis.org

1, 9, 17, 36, 64, 65, 89, 100, 108, 129, 145, 196, 225, 233, 252, 289, 297, 316, 388, 441, 449, 464, 505, 512, 537, 548, 576, 577, 593, 633, 729, 730, 737, 745, 784, 792, 793, 801, 841, 873, 1000, 1025, 1088, 1090, 1116, 1225, 1289, 1296, 1304, 1305, 1367

Offset: 1

Henry Bottomley, May 12 2000

			a(12) = 225 since 225 = 6^3 + 3^2 = 5^3 + 10^2 = 0^3 + 15^2.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A022549, A055394.

nn = 2000; t = Table[0, {nn}]; Do[n = i^2 + j^3; If[0 < n <= nn, t[[n]]++], {i, 0, Sqrt[nn]}, {j, 0, nn^(1/3)}]; Flatten[Position[t, ?(# > 1 &)]] (* _T. D. Noe, Dec 08 2012 *)

Extended by T. D. Noe, Dec 08 2012

A066650 Numbers not of the form a^2 + b^3 with a, b > 0.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 11, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 25, 27, 29, 30, 32, 34, 35, 38, 39, 40, 41, 42, 45, 46, 47, 48, 49, 51, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 69, 70, 71, 74, 75, 77, 78, 79, 81, 83, 84, 85, 86, 87, 88, 90, 92, 93, 94, 95, 96, 97, 98, 99

Offset: 1

Reinhard Zumkeller, Dec 17 2001

nonn

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

Cf. A066649. Complement of A055394.

q[n_] := Length[Reduce[a^2 + b^3 == n && a > 0 && b > 0, {a, b}, Integers]] == 0; Select[Range[100], q] (* Amiram Eldar, Mar 20 2025 *)

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.

A078379 Minimum exponent in prime factorization of numbers which can be written as sum of a positive square and a positive cube.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Nov 25 2002

nonn

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

Cf. A051904, A055394, A078378, A078379, A078380.

Min[FactorInteger[#][[;; , 2]]] & /@ Select[Range[300], Length[Reduce[a^2 + b^3 == # && a > 0 && b > 0, {a, b}, Integers]] > 0 &] (* Amiram Eldar, Mar 27 2025 *)

a(n) = A051904(A055394(n)).

A078382 Sum of divisors of numbers which can be written as sum of a positive square and a positive cube.

Original entry on oeis.org

3, 6, 13, 18, 28, 18, 60, 42, 56, 32, 48, 91, 38, 44, 84, 93, 98, 80, 104, 84, 126, 195, 74, 140, 186, 126, 90, 112, 217, 102, 280, 114, 186, 312, 128, 255, 176, 204, 192, 180, 266, 372, 300, 192, 294, 324, 260, 360, 240, 228, 320, 399, 198, 504, 312, 434, 256

Offset: 1

Reinhard Zumkeller, Nov 25 2002

nonn

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

Cf. A000203, A055394, A078381, A078383, A078384.

DivisorSigma[1, Select[Range[200], Length[Reduce[a^2 + b^3 == # && a > 0 && b > 0, {a, b}, Integers]] > 0 &]] (* Amiram Eldar, Mar 27 2025 *)

a(n) = A000203(A055394(n)).

cp's OEIS Frontend

A303373 Numbers of the form a^3 + b^6, with integers a, b > 0.

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A054402 Numbers that are the sum of a positive square and a positive cube in more than one way.

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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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A078377 Number of distinct prime factors of numbers which can be written as sum of a positive square and a positive cube.

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A078378 Total number of prime factors of numbers which can be written as sum of a positive square and a positive cube.

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A055393 Sum of a square and a nonnegative cube in more than one way.

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Extensions

A066650 Numbers not of the form a^2 + b^3 with a, b > 0.

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Mathematica

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

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Mathematica

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A078379 Minimum exponent in prime factorization of numbers which can be written as sum of a positive square and a positive cube.