A275168 -id:A275168 - cp's OEIS frontend

A275169 Positive integers not in the form x^3 + 2*y^2 + z^2 with x,y,z nonnegative integers.

15, 21, 47, 53, 79, 85, 92, 111, 117, 120, 181, 183, 245, 309, 311, 335, 372, 373, 398, 405, 421, 437, 447, 501, 565, 573, 629, 636, 645, 655, 693, 757, 791, 807, 820, 821, 853, 869, 885, 888, 949, 967, 1013, 1045, 1077, 1141, 1205, 1223, 1269, 1271, 1303, 1461, 1555, 1591, 1613, 1653, 2087, 2101, 2255, 2421

Offset: 1

Zhi-Wei Sun, Jul 18 2016

nonn

Conjecture: The sequence has totally 174 terms as listed in the b-file the largest of which is 375565.
This implies the conjecture in A275150. We note that the sequence contains no term greater than 375565 and not exceeding 10^6.
See also A275168 for a similar conjecture.

			a(1) = 15 since 15 is the least positive integer not in the form x^3 + 2*y^2 + z^2 with x,y,z nonnegative integers.

Zhi-Wei Sun, Table of n, a(n) for n = 1..174

Cf. A000290, A000578, A022551, A022552, A262813, A270488, A274274, A275083, A275150, A275168.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
n=0;Do[Do[If[SQ[m-x^3-2*y^2],Goto[aa]],{x,0,m^(1/3)},{y,0,Sqrt[(m-x^3)/2]}];n=n+1;Print[n," ",m];Label[aa];Continue,{m,1,2421}]

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}.

Original entry on oeis.org

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]

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A275169 Positive integers not in the form x^3 + 2*y^2 + z^2 with x,y,z nonnegative integers.

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