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A273917 Number of ordered ways to write n as w^2 + 3*x^2 + y^4 + z^5, where w is a positive integer and x,y,z are nonnegative integers. 3
1, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 1, 3, 3, 1, 2, 5, 3, 1, 4, 4, 2, 2, 1, 2, 3, 1, 4, 8, 4, 1, 4, 4, 1, 1, 5, 8, 5, 3, 3, 3, 2, 1, 6, 6, 1, 1, 4, 7, 5, 3, 8, 10, 5, 2, 1, 3, 3, 2, 5, 5, 2, 3, 8, 8, 4, 2, 7, 8, 1, 1, 1, 3, 3, 2, 7, 7, 4, 3, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 3, 7, 11, 12, 15, 19, 24, 27, 31, 34, 35, 43, 46, 47, 56, 70, 71, 72, 87, 88, 115, 136, 137, 147, 167, 168, 178, 207, 235, 236, 267, 286, 297, 423, 537, 747, 762, 1017.

(ii) Any positive integer n can be written as w^2 + x^4 + y^5 + pen(z), where w is a positive integer, x,y,z are nonnegative integers, and pen(z) denotes the pentagonal number z*(3*z-1)/2.

See also A262813, A262857, A270566, A271106 and A271325 for some other conjectures on representations.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.NT], 2016-2017.

Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190. (See Remark 1.1.)

Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120. (See Conjecture 3.2.)

EXAMPLE

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

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

a(7) = 1 since 7 = 2^2 + 3*1^2 + 0^4 + 0^5.

a(11) = 1 since 11 = 3^2 + 3*0^2 + 1^4 + 1^5.

a(12) = 1 since 12 = 3^2 + 3*1^2 + 0^4 + 0^5.

a(15) = 1 since 15 = 1^2 + 3*2^2 + 1^4 + 1^5.

a(19) = 1 since 19 = 4^2 + 3*1^2 + 0^4 + 0^5.

a(24) = 1 since 24 = 2^2 + 3*1^2 + 2^4 + 1^5.

a(27) = 1 since 27 = 5^2 + 3*0^2 + 1^4 + 1^5.

a(31) = 1 since 31 = 2^2 + 3*3^2 + 0^4 + 0^5.

a(34) = 1 since 34 = 1^2 + 3*0^2 + 1^4 + 2^5.

a(35) = 1 since 35 = 4^2 + 3*1^2 + 2^4 + 0^5.

a(43) = 1 since 43 = 4^2 + 3*3^2 + 0^4 + 0^5.

a(46) = 1 since 46 = 1^2 + 3*2^2 + 1^4 + 2^5.

a(47) = 1 since 47 = 2^2 + 3*3^2 + 2^4 + 0^5.

a(56) = 1 since 56 = 6^2 + 3*1^2 + 2^4 + 1^5.

a(70) = 1 since 70 = 5^2 + 3*2^2 + 1^4 + 2^5.

a(71) = 1 since 71 = 6^2 + 3*1^2 + 0^4 + 2^5.

a(72) = 1 since 72 = 6^2 + 3*1^2 + 1^4 + 2^5.

a(87) = 1 since 87 = 6^2 + 2*1^2 + 2^4 + 2^5.

a(88) = 1 since 88 = 2^2 + 3*1^2 + 3^4 + 0^5.

a(115) = 1 since 115 = 8^2 + 3*1^2 + 2^4 + 2^5.

a(136) = 1 since 136 = 10^2 + 3*1^2 + 1^4 + 2^5.

a(137) = 1 since 137 = 11^2 + 3*0^2 + 2^4 + 0^5.

a(147) = 1 since 147 = 12^2 + 3*1^2 + 0^4 + 0^5.

a(167) = 1 since 167 = 2^2 + 3*7^2 + 2^4 + 0^5.

a(168) = 1 since 168 = 2^2 + 3*7^2 + 2^4 + 1^5.

a(178) = 1 since 178 = 7^2 + 3*4^2 + 3^4 + 0^5.

a(207) = 1 since 207 = 10^2 + 3*5^2 + 0^4 + 2^5.

a(235) = 1 since 235 = 12^2 + 3*5^2 + 2^4 + 0^5.

a(236) = 1 since 236 = 12^2 + 3*5^2 + 2^4 + 1^5.

a(267) = 1 since 267 = 12^2 + 3*5^2 + 2^4 + 2^5.

a(286) = 1 since 286 = 4^2 + 3*3^2 + 0^4 + 3^5.

a(297) = 1 since 297 = 3^2 + 3*0^2 + 4^4 + 2^5.

a(423) = 1 since 423 = 11^2 + 3*10^2 + 1^4 + 1^5.

a(537) = 1 since 537 = 21^2 + 3*4^2 + 2^4 + 2^5.

a(747) = 1 since 747 = 11^2 + 3*0^2 + 5^4 + 1^5.

a(762) = 1 since 762 = 27^2 + 3*0^2 + 1^4 + 2^5.

a(1017) = 1 since 1017 = 27^2 + 3*0^2 + 4^4 + 2^5.

MATHEMATICA

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

Do[r=0; Do[If[SQ[n-3*x^2-y^4-z^5], r=r+1], {x, 0, Sqrt[(n-1)/3]}, {y, 0, (n-1-3x^2)^(1/4)}, {z, 0, (n-1-3x^2-y^4)^(1/5)}]; Print[n, " ", r]; Continue, {n, 1, 80}]

CROSSREFS

Cf. A000118, A000290, A000326, A000583, A000584, A262813, A262827, A262857, A270566, A270969, A271076, A271106, A273429, A273915.

Sequence in context: A059151 A290091 A059149 * A186187 A013943 A164281

Adjacent sequences:  A273914 A273915 A273916 * A273918 A273919 A273920

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jun 04 2016

STATUS

approved

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Last modified November 12 17:06 EST 2019. Contains 329058 sequences. (Running on oeis4.)