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A272979 Number of ways to write n as x^2 + 2*y^2 + 3*z^3 + 4*w^4 with x,y,z,w nonnegative integers. 4
1, 1, 1, 2, 3, 2, 3, 3, 3, 4, 2, 3, 4, 3, 1, 3, 4, 1, 3, 3, 2, 3, 4, 2, 3, 5, 3, 4, 4, 3, 4, 4, 4, 4, 4, 2, 7, 5, 2, 4, 6, 4, 3, 4, 3, 3, 4, 3, 4, 2, 3, 6, 3, 3, 5, 5, 2, 7, 5, 1, 5, 6, 3, 1, 6, 2, 5, 5, 5, 4, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Conjecture: For positive integers a,b,c,d, any natural number can be written as a*x^2 + b*y^2 + c*z^3 + d*w^4 with x,y,z,w nonnegative integers, if and only if (a,b,c,d) is among the following 49 quadruples: (1,2,1,1), (1,3,1,1), (1,6,1,1), (2,3,1,1), (2,4,1,1), (1,1,2,1), (1,4,2,1), (1,2,3,1), (1,2,4,1), (1,2,12,1), (1,1,1,2), (1,2,1,2), (1,3,1,2), (1,4,1,2), (1,5,1,2), (1,11,1,2), (1,12,1,2), (2,4,1,2), (3,5,1,2), (1,1,4,2), (1,1,1,3), (1,2,1,3), (1,3,1,3), (1,2,4,3), (1,2,1,4), (1,3,1,4), (2,3,1,4), (1,1,2,4), (1,2,2,4), (1,8,2,4), (1,2,3,4), (1,1,1,5), (1,2,1,5), (2,3,1,5), (2,4,1,5), (1,3,2,5), (1,1,1,6), (1,3,1,6), (1,1,2,6), (1,2,1,8), (1,2,4,8), (1,2,1,10), (1,1,2,10), (1,2,1,11), (2,4,1,11), (1,2,1,12), (1,1,2,13), (1,2,1,14),(1,2,1,15).

See also A262824, A262827, A262857 and A273917 for similar conjectures.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

EXAMPLE

a(0) = 1 since 0 = 0^2 + 2*0^2 + 3*0^3 + 4*0^4.

a(1) = 1 since 1 = 1^2 + 2*0^2 + 3*0^3 + 4*0^4.

a(2) = 1 since 2 = 0^2 + 2*1^2 + 3*0^3 + 4*0^4.

a(14) = 1 since 14 = 3^2 + 2*1^2 + 3*1^3 + 4*0^4.

a(17) = 1 since 17 = 3^2 + 2*2^2 + 3*0^3 + 4*0^4.

a(59) = 1 since 59 = 3^2 + 2*5^2 + 3*0^3 + 4*0^4.

a(63) = 1 since 63 = 3^2 + 2*5^2 + 3*0^2 + 4*1^4.

a(287) = 1 since 287 = 11^2 + 2*9^2 + 3*0^2 + 4*1^4.

MATHEMATICA

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]

Do[r=0; Do[If[SQ[n-4w^4-3z^3-2y^2], r=r+1], {w, 0, (n/4)^(1/4)}, {z, 0, ((n-4w^4)/3)^(1/3)}, {y, 0, ((n-4w^4-3z^3)/2)^(1/2)}]; Print[n, " ", r]; Continue, {n, 0, 100}]

CROSSREFS

Cf. A000290, A000578, A000583, A262824, A262827, A262857, A270969, A273429, A273915, A273917.

Sequence in context: A026263 A080098 A083060 * A102351 A078173 A248005

Adjacent sequences:  A272976 A272977 A272978 * A272980 A272981 A272982

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jul 13 2016

STATUS

approved

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Last modified July 29 10:40 EDT 2021. Contains 346344 sequences. (Running on oeis4.)