A002760 -id:A002760 - cp's OEIS frontend

A349778 Number of ways to write n as x^2 + y^k + 2*z^m, where x,y,z are nonnegative integers with x >= y, and k and m belong to the set {2,3}.

4, 4, 8, 4, 8, 4, 4, 4, 4, 6, 8, 4, 8, 4, 2, 2, 7, 10, 12, 8, 9, 3, 4, 2, 5, 11, 10, 8, 8, 3, 1, 3, 7, 10, 11, 5, 12, 7, 7, 4, 5, 8, 8, 7, 8, 8, 2, 3, 4, 9, 11, 8, 18, 5, 11, 8, 4, 8, 11, 8, 7, 6, 3, 8, 7, 12, 12, 12, 11, 4, 7, 5, 10, 9, 11, 7, 11, 4, 3, 6, 11, 13, 17, 9, 10, 6, 5, 7, 7, 13, 13, 12, 5, 6, 3, 3, 5, 14, 12, 10, 18

Offset: 0

Zhi-Wei Sun, Nov 29 2021

nonn

Conjecture 1: a(n) > 0 for all n >= 0, and a(n) = 1 only for n = 30, 120, 142.
We have verified this for all n <= 10^6.
Conjecture 2: Let S = {x^k: k = 2,3 and x = 0,1,2,...}, and let a be 3 or 4 or 5. Then any nonnegative integer can be written as x + 2*y + a*z, where x,y,z are elements of the set S.
Conjecture 3: Let T = {x^k: k = 2,3,4,... and x = 0,1,2,...}. If (b,c) is among the ordered pairs (1,2), (2,4), (2,5) and (3,2), then each n = 0,1,... can be written as x + b*y + c*z, where x and y are elements of T, and z is a square.

			a(3) = 4. In fact, 3 = 1^2 + 0^2 + 2*1^2 = 1^2 + 0^2 + 2*1^3 = 1^2 + 0^3 + 2*1^2 = 1^2 + 0^3 + 2*1^3 with 1 >= 0.
a(30) = 1 with 30 = 2^2 + 2^3 + 2*3^2 and 2 >= 2.
a(120) = 1 with 120 = 10^2 + 2^2 + 2*2^3 and 10 >= 2.
a(142) = 1 with 142 = 6^2 + 2^3 + 2*7^2 and 6 >= 2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, New Conjectures in Number Theory and Combinatorics (in Chinese), Harbin Institute of Technology Press, 2021.

Cf. A000290, A000578, A001597, A002760, A275168, A275169, A349787.

tab={};Do[r=0;Do[If[IntegerQ[((n-x^2-y^k)/2)^(1/m)],r=r+1],{x,0,Sqrt[n]},{k,2,3},{y,0,Min[x,(n-x^2)^(1/k)]},{m,2,3}];tab=Append[tab,r],{n,0,100}];Print[tab]

A349787 Number of ways to write n as x^2 + y^k + 2^a + 2^b, where x,y,a,b are nonnegative integers with x >= y and a >= b, and k is either 2 or 3.

Original entry on oeis.org

2, 4, 6, 6, 8, 8, 8, 8, 11, 9, 11, 11, 12, 8, 9, 11, 15, 14, 16, 16, 17, 8, 10, 14, 15, 12, 16, 16, 12, 7, 11, 17, 22, 16, 17, 18, 17, 10, 16, 22, 23, 15, 17, 19, 17, 8, 15, 23, 19, 11, 20, 23, 17, 12, 17, 20, 20, 14, 18, 18, 13, 7, 12, 21, 23, 21, 25, 27, 26, 11, 17, 27, 25, 15, 22, 24, 14, 8, 17, 27, 29, 20, 29, 28

Offset: 2

Zhi-Wei Sun, Nov 30 2021

nonn

Conjecture 1: a(n) > 0 for all n >= 2.
Below is our weaker version of Conjecture 1.
Conjecture 2: Each n = 2,3,... can be written as a sum of two perfect powers (including 0 and 1) and two powers of 2 (including 2^0 = 1).
In contrast, R. Crocker proved in 2008 that there are infinitely many positive integers which cannot be written as a sum of two squares and two powers of 2.

			a(2) = 2 with 2 = 0^2 + 0^2 + 2^0 + 2^0 = 0^2 + 0^3 + 2^0 + 2^0.
a(535903) > 0 since 535903 = 336^2 + 31^3 + 2^18 + 2^17 with 336 >= 31 and 18 >= 17.

Zhi-Wei Sun, Table of n, a(n) for n = 2..10000
R. C. Crocker, On the sum of two squares and two powers of k, Colloq. Math. 112(2008), 235-267.
Dave Platt and Tim Trudgian, On the sum of two squares and at most two powers of 2, arXiv:1610.01672 [math.NT], 2016.

Cf. A000079, A000290, A000578, A001597, A002760, A273812, A303363, A303637, A349778.

PowQ[n_]:=PowQ[n]=IntegerQ[Log[2,n]];
tab={};Do[r=0;Do[If[PowQ[n-x^2-y^k-2^a],r=r+1],{x,0,Sqrt[n-2]},{k,2,3},{y,0,Min[x,(n-2-x^2)^(1/k)]},{a,0,Log[2,n-x^2-y^k]-1}];tab=Append[tab,r],{n,2,85}];Print[tab]

A233444 Primes that are exactly halfway between the nearest square and the nearest cube.

Original entry on oeis.org

2203, 90863, 185477, 388573, 613607, 912349, 1293899, 1600919, 2146457, 30661333, 35608189, 81190429, 105823093, 122753857, 204341747, 338602837, 368601707, 374788121, 426958673, 498675409, 586371239, 656232799, 665360321, 674509487, 693132527, 1102304669, 1180942709

Offset: 1

Alex Ratushnyak, Dec 09 2013

nonn

A subsequence of A233075.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A000290, A000578, A233075.

Cf. A002760 (Squares and cubes).

Res:= NULL:
for x from 3 to 2000 do
  x3:= x^3;
  y:= floor(sqrt(x3));
  p:= (x3+y^2)/2;
  if p::integer and x3-p <= p - (x-1)^3 and p - y^2 <= (y+1)^2-p and isprime(p) then
    Res:= Res, p;
  fi;
  p:= (x3+(y+1)^2)/2;
  if p::integer and p-x3 <= (x+1)^3-p and (y+1)^2-p <= p - y^2 and isprime(p) then
      Res:= Res, p;
  fi
od:
Res; # Robert Israel, May 01 2018

A354761 Least number of squares and cubes that add up to n.

Original entry on oeis.org

1, 2, 3, 1, 2, 3, 4, 1, 1, 2, 3, 2, 2, 3, 4, 1, 2, 2, 3, 2, 3, 3, 4, 2, 1, 2, 1, 2, 2, 3, 2, 2, 2, 2, 2, 1, 2, 3, 3, 2, 2, 3, 2, 2, 2, 3, 3, 3, 1, 2, 3, 2, 2, 2, 3, 3, 2, 2, 3, 3, 2, 3, 2, 1, 2, 3, 3, 2, 3, 3, 3, 2, 2, 2, 3, 2, 3, 3, 3, 2, 1, 2, 3, 3, 2, 3

Offset: 1

Peter Schorn, Jun 06 2022

nonn

a(n) <= 4 since any number can be written as a sum of 4 squares (Lagrange's theorem).

Sequence first differs from A063274, A225926 and A274459 at n = 32 since 32 is a powerful number, a prime power and a perfect power but neither a square nor a cube.

			a(1) = 1, a(4) = 1 (4 = 2^2), a(7) = 4 (7 = 2^2 + 1^2 + 1^2 + 1^2), a(8) = 1 (8 = 2^3), a(12) = 2 (12 = 2^3 + 2^2), a(17) = 2 (17 = 4^2 + 1^2), a(32) = 2 (32 = 4^2 + 4^2).

Cf. A002760, A063274, A225926, A274459, A002376, A002828, A000290, A000578.

lista(n) = {my(v = vector(n)); for(j = 2, 3, for(i = 2, sqrtnint(n, j), v[i^j] = 1)); v[1]=1; v[2]=2; for(i=3, #v, if(v[i]==0, v[i] = vecmin(vector(i\2, k, v[k] + v[i-k])))); v}

cp's OEIS Frontend

A349778 Number of ways to write n as x^2 + y^k + 2*z^m, where x,y,z are nonnegative integers with x >= y, and k and m belong to the set {2,3}.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A349787 Number of ways to write n as x^2 + y^k + 2^a + 2^b, where x,y,a,b are nonnegative integers with x >= y and a >= b, and k is either 2 or 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A233444 Primes that are exactly halfway between the nearest square and the nearest cube.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

A354761 Least number of squares and cubes that add up to n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI