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A100251
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The square root of A100252; the index of the least square number greater than 1 that is also an n-gonal number, or 0 if none exists.
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7
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6, 2, 99, 35, 9, 15, 3, 0, 14, 8, 6, 21, 55, 4, 133, 10, 22, 0, 51, 27, 261, 15, 5, 85, 161, 9, 35, 451, 21, 33, 69, 14, 124, 6, 44, 715, 28, 24, 7421, 217, 34, 16, 23001, 54, 1065, 36, 7, 76, 156, 0, 245
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OFFSET
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3,1
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COMMENTS
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Let j be the smallest integer for which 1 + (1+1*n) + (1+2*n) + ... + (1+j*n) = k^2 = s. Then a(n)=k; if no such j exists, then a(n)=0. Basis for sequence is shortest arithmetic series with initial term 1 and difference n that sums to a perfect square.
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LINKS
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FORMULA
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a(n)^2 = 1 + (1+1*n) + (1+2*n) + ... + (1+A100254(n)*n) = 1 + (1+1*n) +(1+2*n) + ... + A100253(n).
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EXAMPLE
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a(3)=99 since 1 + 4 + 7 + ... + (1+80*3) = 99^2 = 9801 and no other arithmetic series with initial term 1, difference 3 and fewer terms sums to a perfect square.
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MATHEMATICA
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NgonIndex[n_, v_] := (-4 + n + Sqrt[16 - 8*n + n^2 - 16*v + 8*n*v])/(n - 2)/2; Table[k = 2; While[sqr = k^2; i = NgonIndex[n, sqr]; k < 25000 && ! IntegerQ[i], k++]; If[k == 25000, k = sqr = i = 0]; k, {n, 3, 64}] (* T. D. Noe, Apr 19 2011 *)
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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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