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 A254595 Least positive integer m such that m can be written as x*(x+1) + y*(3*y+1)/2 + z*(3*z-1)/2 in exactly n ways, where x,y,z are nonnegative integers. 3
 1, 2, 9, 7, 14, 37, 64, 68, 57, 119, 112, 168, 194, 147, 267, 259, 222, 477, 427, 404, 519, 652, 567, 497, 512, 749, 722, 719, 952, 1209, 904, 1139, 1267, 1184, 1069, 1737, 1594, 1667, 1734, 2077, 1799, 1659, 1729, 1814, 1762, 1862, 2577, 2444, 2997, 2072, 2457, 2842, 3029, 3249, 3094, 3589, 3999, 4208, 3479, 3232 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Conjecture: (i) a(n) exists for any n > 0. Moreover, n^2 is the main term of a(n) as n tends to the infinity. (ii) a(n) is never divisible by 5. Also, for any n > 1 the term a(n) is not congruent to 1 modulo 5. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..200 Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015. EXAMPLE a(3) = 9 since 9 is the first positive integer m with A254573(m)=3. Note that 9 = 2*3 +1*(3*1+1)/2 + 1*(3*1-1)/2 = 1*2 + 1*(3*1+1)/2 + 2*(3*2-1)/2 = 1*2 + 2*(3*2+1)/2 + 0*(3*0-1)/2. MATHEMATICA sQ[n_]:=IntegerQ[Sqrt[4n+1]] Do[Do[m=0; Label[aa]; m=m+1; r=0; Do[If[sQ[m-y(3y+1)/2-z(3z-1)/2], r=r+1; If[r>n, Goto[aa]]], {y, 0, (Sqrt[24m+1]-1)/6}, {z, 0, (Sqrt[24(m-y(3y+1)/2)+1]+1)/6}]; If[r==n, Print[n, " ", m]; Goto[bb], Goto[aa]]]; Label[bb]; Continue, {n, 1, 60}] CROSSREFS Cf. A000326, A002378, A005449, A254573, A254617. Sequence in context: A121837 A160439 A245287 * A137453 A063381 A019078 Adjacent sequences:  A254592 A254593 A254594 * A254596 A254597 A254598 KEYWORD nonn AUTHOR Zhi-Wei Sun, Feb 02 2015 STATUS approved

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Last modified October 21 14:20 EDT 2019. Contains 328301 sequences. (Running on oeis4.)