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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. 3

%I

%S 9,34,63,89,99,139,164,174,193,204,245,314,324,399,424,454,464,489,

%T 504,524,549,714,1049,1089,1149,1174,1439,1504,1539,1639,1714,1799,

%U 1814,1919,2164,2239,2313,2374,2414,2439,2764,2789,3079,3319,3414,3669,3774,3814,4019,4114

%N Positive integers that cannot be written as the sum of a fourth power, a square and a positive triangular number.

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

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

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

%H Zhi-Wei Sun, <a href="/A262959/b262959.txt">Table of n, a(n) for n = 1..500</a>

%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.4064/aa127-2-1">Mixed sums of squares and triangular numbers</a>, Acta Arith. 127(2007), 103-113.

%e 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.

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

%t 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}]

%Y Cf. A000217, A000290, A000583, A262956.

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, Oct 05 2015

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Last modified October 19 04:40 EDT 2019. Contains 328211 sequences. (Running on oeis4.)