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 A334086 Positive numbers not of the form 2*x^4 + y*(y+1)/2 + z*(z+1)/2 with x,y,z nonnegative integers 3
 19, 82, 109, 118, 145, 149, 271, 280, 296, 349, 350, 371, 392, 454, 491, 643, 670, 692, 754, 755, 923, 937, 986, 989, 1021, 1031, 1150, 1189, 1210, 1294, 1346, 1372, 1610, 1682, 1699, 1720, 1819, 1913, 2050, 2065, 2141, 2227, 2479, 2524, 2753, 2996, 3184, 3451, 3590, 3805, 3968, 4129, 4139, 4199, 4261, 4706 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Conjecture: The sequence has totally 216 terms as listed in the b-file. As none of the 216 terms in the b-file is divisible by 3, the conjecture implies that for each nonnegative integer n we can write 3*n as 2*x^4 + y*(y+1)/2 + z*(z+1)/2 and hence 12*n+1 = 8*x^4 + (y+z+1)^2 + (y-z)^2, where x,y,z are integers. Our computation indicates that after the 216-th term 4592329 there are no other terms below 10^8. It is known that each n = 0,1,2,... can be written as the sum of two triangular numbers and twice a square. a(217) > 10^9, if it exists. - Giovanni Resta, Apr 14 2020 LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..216 Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113. Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no. 2, 97-120. (Cf. Conjecture 1.4(ii).) EXAMPLE a(1) = 19 since 19 is the first nonnegative integer which cannot be written as the sum of two triangular numbers and twice a fourth power. MATHEMATICA TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]]; tab={}; Do[Do[If[TQ[n-2x^4-y(y+1)/2], Goto[aa]], {x, 0, (n/2)^(1/4)}, {y, 0, (Sqrt[4(n-2x^4)+1]-1)/2}]; tab=Append[tab, n]; Label[aa], {n, 0, 5000}]; Print[tab] CROSSREFS Cf. A000217, A000583, A115160, A290491, A306227, A334113. Sequence in context: A212056 A290224 A071633 * A217052 A044206 A044587 Adjacent sequences:  A334083 A334084 A334085 * A334087 A334088 A334089 KEYWORD nonn AUTHOR Zhi-Wei Sun, Apr 14 2020 STATUS approved

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Last modified November 26 07:15 EST 2020. Contains 338632 sequences. (Running on oeis4.)