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A350634
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Products of the parts s,t in each partition of k (= 2,3,...) into two parts, ordered by increasing k and then by increasing values of s*t (see example).
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2
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1, 2, 3, 4, 4, 6, 5, 8, 9, 6, 10, 12, 7, 12, 15, 16, 8, 14, 18, 20, 9, 16, 21, 24, 25, 10, 18, 24, 28, 30, 11, 20, 27, 32, 35, 36, 12, 22, 30, 36, 40, 42, 13, 24, 33, 40, 45, 48, 49, 14, 26, 36, 44, 50, 54, 56, 15, 28, 39, 48, 55, 60, 63, 64, 16, 30, 42, 52, 60, 66, 70, 72
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OFFSET
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1,2
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COMMENTS
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If b > 0 and c > 0 are the integer coefficients of a monic quadratic x^2 + b*x + c, it has integer roots if its discriminant d^2 = b^2 - 4c is a perfect square. This sequence is the values of c for increasing b sorted by b then c. The first pair of (b, c) = (2, 1) and has d = 0. The n-th pair of (b, c) = (A027434(n),a(n)) and has d = A082375(n-1). - Frank M Jackson, Jan 22 2024
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LINKS
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FORMULA
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EXAMPLE
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---------------------------------------------------------------------------
The products of the parts start: 1*1, 1*2, 1*3, 2*2, 1*4, 2*3, etc., which are precisely the values of a(n): 1, 2, 3, 4, 4, 6, ...
[1,9]
[1,7] [1,8] [2,8]
[1,5] [1,6] [2,6] [2,7] [3,7]
[1,3] [1,4] [2,4] [2,5] [3,5] [3,6] [4,6]
[1,1] [1,2] [2,2] [2,3] [3,3] [3,4] [4,4] [4,5] [5,5]
k 2 3 4 5 6 7 8 9 10
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MATHEMATICA
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Times@@@Flatten[Table[IntegerPartitions[k, {2}], {k, 2, 100}], 1] (* Frank M Jackson, Jan 22 2024 *)
lst={}; Do[If[IntegerQ[d=Sqrt[b^2-4c]], AppendTo[lst, c]], {b, 1, 100}, {c, 1, b^2/4}]; lst (* Frank M Jackson, Jan 22 2024 *)
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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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