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A262959 Positive integers that cannot be written as the sum of a fourth power, a square and a positive triangular number. 4
9, 34, 63, 89, 99, 139, 164, 174, 193, 204, 245, 314, 324, 399, 424, 454, 464, 489, 504, 524, 549, 714, 1049, 1089, 1149, 1174, 1439, 1504, 1539, 1639, 1714, 1799, 1814, 1919, 2164, 2239, 2313, 2374, 2414, 2439, 2764, 2789, 3079, 3319, 3414, 3669, 3774, 3814, 4019, 4114 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Conjecture: (i) Each term is congruent to one of 3, 4, 5, 9 modulo 10.

(ii) a(n+1) - a(n) > 4 for all n > 0.

Part (ii) of this conjecture is stronger than the conjecture in A262956. Note that a(139) - a(138) = 18089 - 18084 = 5.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..500

Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.

EXAMPLE

a(1) = 9 since each of 1..8 can be written as x^4 + y^2 + z*(z+1)/2 with z > 0, but 9 cannot be represented in this way. Clearly, 1 = 0^4 + 0^2 + 1*2/2, 2 = 0^4 + 1^2 + 1*2/2, 3 = 1^4 + 1^2 + 1*2/2, 4 = 0^4 + 1^2 + 2*3/2, 5 = 1^4 + 1^2 + 2*3/2, 6 = 0^4 + 0^2 + 3*4/2, 7 = 0^4 + 1^2 + 3*4/2 and 8 = 1^3 + 1^2 + 3*4/2.

MATHEMATICA

TQ[n_]:=TQ[n]=n>0&&IntegerQ[Sqrt[8n+1]]

n=0; Do[Do[If[TQ[m-x^4-y^2], Goto[aa]], {x, 0, m^(1/4)}, {y, 0, Sqrt[m-x^4]}]; n=n+1; Print[n, " ", m]; Label[aa]; Continue, {m, 1, 5000}]

CROSSREFS

Cf. A000217, A000290, A000583, A262956.

Sequence in context: A044086 A044467 A020163 * A139275 A236370 A273744

Adjacent sequences:  A262956 A262957 A262958 * A262960 A262961 A262962

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Oct 05 2015

STATUS

approved

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Last modified November 26 03:20 EST 2020. Contains 338632 sequences. (Running on oeis4.)