

A349246


Number of ways to write n as w^8 + x^4 + 2*y^4 + 4*z^4 + t*(t+1), where w, x, y, z, and t are nonnegative integers.


2



1, 2, 3, 4, 4, 4, 5, 6, 5, 4, 3, 2, 3, 4, 3, 2, 3, 3, 4, 4, 4, 4, 5, 5, 4, 4, 3, 3, 3, 2, 2, 3, 4, 5, 5, 5, 5, 5, 5, 5, 3, 1, 3, 5, 4, 4, 4, 3, 5, 6, 4, 2, 3, 4, 3, 2, 2, 4, 5, 4, 4, 4, 4, 5, 5, 4, 4, 6, 5, 3, 2, 2, 5, 6, 5, 5, 5, 5, 7, 8, 4, 2, 4, 5, 4, 5, 5, 6, 7, 6, 6, 5, 6, 8, 9, 8, 6, 7, 5, 3, 3
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OFFSET

0,2


COMMENTS

Conjecture: a(n) > 0 for all n = 0,1,2,....
This has been verified for all n = 0..10^8.
It seems that a(n) = 1 only for n = 0, 41, 131, 141, 145, 225, 251, 297, 591, 621, 916, 1021, 1241, 1431, 2025, 4691.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 0..10000
ZhiWei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no. 2, 97120.


EXAMPLE

a(145) = 1 with 145 = 0^8 + 3^4 + 2*0^4 + 4*2^4 + 0*1.
a(225) = 1 with 225 = 1^8 + 2^4 + 2*3^4 + 4*1^4 + 6*7.
a(916) = 1 with 916 = 2^8 + 2^4 + 2*4^4 + 4*0^4 + 11*12.
a(1021) = 1 with 1021 = 0^8 + 5^4 + 2*0^4 + 4*3^4 + 8*9.
a(1241) = 1 with 1241 = 0^8 + 5^4 + 2*0^4 + 4*2^4 + 23*24.
a(1431) = 1 with 1431 = 1^8 + 6^4 + 2*1^4 + 4*0^4 + 11*12.
a(2025) = 1 with 2025 = 2^8 + 3^4 + 2*2^4 + 4*3^4 + 36*37.
a(4691) = 1 with 4691 = 2^8 + 3^4 + 2*0^4 + 4*2^4 + 65*66.


MATHEMATICA

QQ[n_]:=QQ[n]=IntegerQ[Sqrt[4n+1]];
tab={}; Do[r=0; Do[If[QQ[nw^84z^42y^4x^4], r=r+1], {w, 0, n^(1/8)}, {z, 0, ((nw^8)/4)^(1/4)}, {y, 0, ((nw^84z^4)/2)^(1/4)}, {x, 0, (nw^84z^42y^4)^(1/4)}]; tab=Append[tab, r], {n, 0, 100}]; Print[tab]


CROSSREFS

Cf. A000583, A001016, A002378.
Sequence in context: A060740 A307467 A145339 * A123273 A284359 A167991
Adjacent sequences: A349243 A349244 A349245 * A349247 A349248 A349249


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Mar 26 2022


STATUS

approved



