A045634 -id:A045634 - cp's OEIS frontend

A022549 Sum of a square and a nonnegative cube.

0, 1, 2, 4, 5, 8, 9, 10, 12, 16, 17, 24, 25, 26, 27, 28, 31, 33, 36, 37, 43, 44, 49, 50, 52, 57, 63, 64, 65, 68, 72, 73, 76, 80, 81, 82, 89, 91, 100, 101, 108, 113, 121, 122, 125, 126, 127, 128, 129, 134, 141, 144, 145

Offset: 1

N. J. A. Sloane

It appears that there are no modular constraints on this sequence; i.e., every residue class of every integer has representatives here. - Franklin T. Adams-Watters, Dec 03 2009

A045634(a(n)) > 0. - Reinhard Zumkeller, Jul 17 2010

R. Zumkeller, Table of n, a(n) for n = 1..10000 - _Reinhard Zumkeller_, Jul 17 2010

Complement of A022550; A002760 and A179509 are subsequences.

Cf. A055394, A045634, A055393, A123291, A169618, A123364, A111925.

q=30; imax=q^2; Select[Union[Flatten[Table[x^2+y^3, {y,0,q^(2/3)}, {x,0,q}]]], #<=imax&] (* Vladimir Joseph Stephan Orlovsky, Apr 20 2011 *)

is(n)=for(k=0,sqrtnint(n,3), if(issquare(n-k^3), return(1))); 0 \\ Charles R Greathouse IV, Aug 24 2020

list(lim)=my(v=List(),t); for(k=0,sqrtnint(lim\=1,3), t=k^3; for(n=0,sqrtint(lim-t), listput(v,t+n^2))); Set(v) \\ Charles R Greathouse IV, Aug 24 2020

A022550 Numbers that are not the sum of a square and a nonnegative cube.

Original entry on oeis.org

3, 6, 7, 11, 13, 14, 15, 18, 19, 20, 21, 22, 23, 29, 30, 32, 34, 35, 38, 39, 40, 41, 42, 45, 46, 47, 48, 51, 53, 54, 55, 56, 58, 59, 60, 61, 62, 66, 67, 69, 70, 71, 74, 75, 77, 78, 79

Offset: 1

N. J. A. Sloane

nonn

Complement of A022549; A045634(a(n))=0. - Reinhard Zumkeller, Jul 17 2010

R. Zumkeller, Table of n, a(n) for n = 1..10000

A135911 Number of 4-tuples (x,y,z,t) of nonnegative integers such that x^2+y^3+z^4+t^5 = n.

Original entry on oeis.org

1, 4, 6, 4, 2, 3, 3, 1, 1, 4, 6, 4, 2, 2, 1, 0, 2, 7, 8, 3, 1, 2, 1, 0, 2, 6, 7, 5, 5, 4, 1, 1, 4, 8, 7, 2, 3, 7, 5, 1, 2, 5, 5, 4, 6, 5, 1, 1, 3, 7, 7, 3, 5, 6, 2, 0, 2, 5, 5, 4, 5, 2, 0, 2, 6, 11, 9, 3, 4, 6, 2, 0, 2, 5, 5, 3, 4, 3, 1, 2, 4, 10, 11, 7, 6, 4, 2, 1, 1, 6, 9, 7, 5, 3, 1, 1, 4, 8, 8, 3, 4

Offset: 0

N. J. A. Sloane, Mar 07 2008

nonn

T. D. Noe, Table of n, a(n) for n = 0..10000
K. B. Ford, The representation of numbers as sums of unlike powers II, J. Amer. Math. Soc., 9 (1996), 919-940.

Cf. A045634, A135910, A135912.

Cf. A111151 (n such that a(n)=0).

A135910 Number of triples (x,y,z) of nonnegative integers such that x^2+y^3+z^4 = n.

Original entry on oeis.org

1, 3, 3, 1, 1, 2, 1, 0, 1, 3, 3, 1, 1, 1, 0, 0, 2, 5, 3, 0, 1, 1, 0, 0, 2, 4, 3, 2, 3, 1, 0, 1, 2, 3, 1, 0, 2, 3, 1, 0, 1, 1, 1, 2, 3, 1, 0, 1, 0, 2, 2, 1, 3, 2, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 3, 5, 3, 0, 2, 1, 0, 0, 1, 3, 1, 0, 1, 1, 0, 1, 3, 5, 4, 2, 1, 1, 1, 0, 1, 4, 4, 2, 2, 1, 0, 0, 1, 2, 3, 0, 2

Offset: 0

N. J. A. Sloane, Mar 07 2008

nonn

T. D. Noe, Table of n, a(n) for n = 0..10000
K. B. Ford, The representation of numbers as sums of unlike powers II, J. Amer. Math. Soc., 9 (1996), 919-940.

Cf. A045634, A135911, A135912.

M:=10; M2:=M^2; t0:=array(0..M2);
for i from 0 to M2 do t0[i]:=0; od:
for a from 0 to M do for b from 0 to M do for c from 0 to M do
i:=a^2+b^3+c^4; if i <= M2 then t0[i]:=t0[i]+1; fi;
od: od: od: [seq(t0[i],i=0..M2)];

A135912 Number of 5-tuples (x,y,z,t,u) of nonnegative integers such that x^2+y^3+z^4+t^5+u^6 = n.

Original entry on oeis.org

1, 5, 10, 10, 6, 5, 6, 4, 2, 5, 10, 10, 6, 4, 3, 1, 2, 9, 15, 11, 4, 3, 3, 1, 2, 8, 13, 12, 10, 9, 5, 2, 5, 12, 15, 9, 5, 10, 12, 6, 3, 7, 10, 9, 10, 11, 6, 2, 4, 10, 14, 10, 8, 11, 8, 2, 2, 7, 10, 9, 9, 7, 2, 2, 9, 21, 26, 16, 9, 13, 11, 3, 3, 11, 16, 12, 9, 9, 5, 3, 8, 21, 29, 21, 14, 12, 7, 3, 4

Offset: 0

N. J. A. Sloane, Mar 07 2008

nonn

a(n) > 0 for n <= 10000. Is there any n for which a(n) = 0?
Note that there are many famous hard problems connected with sequences A045634, A135910, A135911 and the present entry (see the Ford reference).
The graph of this sequence suggests that a(n) is never zero. Checked to 10^5. - T. D. Noe, Mar 07 2008

T. D. Noe, Table of n, a(n) for n = 0..10000
K. B. Ford, The representation of numbers as sums of unlike powers II, J. Amer. Math. Soc., 9 (1996), 919-940.

Cf. A045634, A135910, A135911.

M:=100; M2:=M^2; t0:=array(0..M2); for i from 0 to M2 do t0[i]:=0; od:
for a from 0 to M do na:=a^2; for b from 0 to M do nb:=na+b^3;
if nb <= M2 then for c from 0 to M do nc:=nb+c^4; if nc <= M2 then for d from 0 to M2 do nd:=nc+d^5; if nd <= M2 then for e from 0 to M2 do i:=nd+e^6; if i <= M2 then t0[i]:=t0[i]+1; fi; od: fi; od; fi; od: fi; od: od:
[seq(t0[i],i=0..M2)];
for i from 0 to M2 do if t0[i]=0 then lprint(i); fi; od:

A179509 Numbers that can be written uniquely as sum of a square and a nonnegative cube.

Original entry on oeis.org

0, 2, 4, 5, 8, 10, 12, 16, 24, 25, 26, 27, 28, 31, 33, 37, 43, 44, 49, 50, 52, 57, 63, 68, 72, 73, 76, 80, 81, 82, 91, 101, 113, 121, 122, 125, 126, 127, 128, 134, 141, 144, 148, 150, 152, 161, 164, 169, 170, 171, 174, 177, 185, 189, 197, 204, 206, 208, 216, 217, 220

Offset: 1

Reinhard Zumkeller, Jul 17 2010

nonn

A045634(a(n)) = 1;

subsequence of A022549.

R. Zumkeller, Table of n, a(n) for n = 1..10000

A060861 Least number of the form x^2 + y^3 (x, y nonnegative) in exactly n ways.

Original entry on oeis.org

3, 0, 1, 225, 1025, 92025, 1334025, 5472225, 35964225, 930860225, 1000837225, 4979585600, 38515961025, 88154795025, 2046945411225, 88813460025, 5684061441600, 13052612865600, 64745012358225

Offset: 0

David W. Wilson, May 04 2001

nonn

A045634(a(n)) = n and A045634(m) <> n for m < a(n). [Reinhard Zumkeller, Jul 17 2010]
a(n) > 2*10^14 for n >= 19. [Donovan Johnson, Dec 13 2008]
a(19) <= 4143680790926400 = 64*a(18). [Jon E. Schoenfield, Aug 11 2010]

			a(3)=225: A045634(225) = #{15^2+0^3, 10^2+5^3, 3^2+6^3} = 3;
a(4)=1025: A045634(1025) = #{32^2+1^3, 31^2+4^3, 30^2+5^3, 5^2+10^3} = 4;
a(5)=92025: A045634(92025) = #{303^2+6^3, 255^2+30^3, 213^2+36^3, 152^2+41^3, 30^2+45^3} = 5.

a(14) and a(16)-a(18) from Donovan Johnson, Dec 13 2008

a(0) and examples from Reinhard Zumkeller, Jul 17 2010

cp's OEIS Frontend

A022549 Sum of a square and a nonnegative cube.

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A022550 Numbers that are not the sum of a square and a nonnegative cube.

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A135911 Number of 4-tuples (x,y,z,t) of nonnegative integers such that x^2+y^3+z^4+t^5 = n.

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A135910 Number of triples (x,y,z) of nonnegative integers such that x^2+y^3+z^4 = n.

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A135912 Number of 5-tuples (x,y,z,t,u) of nonnegative integers such that x^2+y^3+z^4+t^5+u^6 = n.

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A179509 Numbers that can be written uniquely as sum of a square and a nonnegative cube.

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A060861 Least number of the form x^2 + y^3 (x, y nonnegative) in exactly n ways.

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