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A280444
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Least positive integer m such that n - p(m) = x*(3x-1)/2 + y*(3y+1)/2 for some nonnegative integers x and y, or 0 if no such m exists, where p(.) is the partition function given by A000041.
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2
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1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 4, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 3, 6, 1, 2, 1, 1, 2, 1, 1, 1, 2, 3, 1, 2, 3, 1, 2, 1, 1, 1, 1, 2, 3, 4, 1, 1, 2, 3, 1, 1, 2, 3, 4, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 1, 1, 1
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OFFSET
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1,5
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COMMENTS
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The conjecture in A280455 asserts that a(n) > 0 for all n > 0.
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LINKS
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Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28--Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
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EXAMPLE
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a(12) = 4 since 12 - p(4) = 12 - 5 = 7 = 0*(3*0-1)/2 + 2*(3*2+1)/2.
a(35) = 6 since 35 - p(6) = 35 - 11 = 24 = 4*(3*4-1)/2 + 1*(3*1+1)/2.
a(4327) = 15 since 4327 - p(15) = 4327 - 176 = 4151 = 16*(3*16-1)/2 + 50*(3*50+1)/2.
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MATHEMATICA
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SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
p[n_]:=p[n]=PartitionsP[n];
Pen[n_]:=Pen[n]=SQ[24n+1]&&Mod[Sqrt[24n+1], 6]==1;
Do[m=1; Label[bb]; If[p[m]>n, Goto[cc]]; Do[If[Pen[n-p[m]-x(3x-1)/2], Print[n, " ", m]; Goto[aa]], {x, 0, (Sqrt[24(n-p[m])+1]+1)/6}]; m=m+1; Goto[bb]; Label[cc]; Print[n, " ", 0]; Label[aa]; Continue, {n, 1, 80}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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