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A127348
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Coefficient of x^2 in the polynomial (x-p(n))*(x-p(n+1))*(x-p(n+2))*(x-p(n+3)), where p(k) is the k-th prime.
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7
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101, 236, 466, 838, 1330, 1918, 2862, 3856, 5350, 7096, 8622, 10558, 12654, 15228, 18090, 21550, 24916, 27702, 31500, 35068, 39298, 45322, 51240, 56980, 62398, 66130, 69958, 77854, 86230, 96618, 106888, 115842, 124342, 133122, 144090
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OFFSET
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1,1
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LINKS
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FORMULA
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a(n) = p(n)*p(n+1) + p(n)*p(n+2) + p(n)*p(n+3) + p(n+1)*p(n+2) + p(n+1)*p(n+3) + p(n+2)*p(n+3), where p(k) is the k-th prime (by Viete's formula relating the zeros and the coefficients of a polynomial). - Emeric Deutsch, Jan 20 2007
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EXAMPLE
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a(1)=101 because (x-2)*(x-3)*(x-5)*(x-7) = x^4 - 17x^3 + 101x^2 - 247x + 210.
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MAPLE
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a:=n->coeff(expand((x-ithprime(n))*(x-ithprime(n+1))*(x-ithprime(n+2))*(x-ithprime(n+3))), x, 2): seq(a(n), n=1..45); # Emeric Deutsch, Jan 20 2007
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MATHEMATICA
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Table[Prime[x] Prime[x + 1] + Prime[x] Prime[x + 2] + Prime[x] Prime[x + 3] + Prime[x + 1] Prime[x + 2] + Prime[x + 1] Prime[x + 3] + Prime[x + 2] Prime[x + 3], {x, 1, 100}]
Total[Times@@@Subsets[#, {2}]]&/@Partition[Prime[Range[40]], 4, 1] (* Harvey P. Dale, Apr 15 2019 *)
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PROG
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(PARI) 1. {m=35; k=3; for(n=1, m, print1(sum(i=n, n+k-1, sum(j=i+1, n+k, prime(i)*prime(j))), ", "))} 2. {m=35; k=3; for(n=1, m, print1(abs(polcoeff(prod(j=0, k, (x-prime(n+j))), 2)), ", "))} // Klaus Brockhaus, Jan 21 2007
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CROSSREFS
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KEYWORD
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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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