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A114364
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a(n) = n*(n+1)^2.
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3
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4, 18, 48, 100, 180, 294, 448, 648, 900, 1210, 1584, 2028, 2548, 3150, 3840, 4624, 5508, 6498, 7600, 8820, 10164, 11638, 13248, 15000, 16900, 18954, 21168, 23548, 26100, 28830, 31744, 34848, 38148, 41650, 45360, 49284, 53428, 57798, 62400
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OFFSET
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1,1
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COMMENTS
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Former name was "Numbers k such that k*x^3 + x + 1 is not prime."
Theorem: y = k*x^3 + x + 1 is not prime for k = 4, 18, 48, ..., n*(n+1)^2. Proof: n*(n+1)^2*x^3 + x + 1 = ((n+1)*x + 1)*((n^2+n)*x^2 - n*x + 1). Thus (n+1)*x + 1 divides y. This could possibly be used as a pre-test for compositeness. This sequence is the same as beginning with the third term of A045991.
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LINKS
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FORMULA
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a(n) = n*(n+1)^2.
Sum_{n>=1} 1/a(n) = 2 - Pi^2/6.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 +2*log(2) - 2. (End)
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MAPLE
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MATHEMATICA
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CoefficientList[Series[(2 (2 + x))/(-1 + x)^4, {x, 0, 38}], x] (* or *)
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PROG
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(PARI) g2(n) = for(x=1, n, y=x*(x+1)^2; print1(y", "))
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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