

A238146


Triangle read by rows: T(n,k) is coefficient of x^(nk) in consecutive prime rooted polynomial of degree n, P(x) = Product_{k=1..n} (xp(k)) = 1*x^n + T(n,1)*x^(n1)+ ... + T(n,k1)*x + T(n,k), for 1 <= k <= n.


6



2, 5, 6, 10, 31, 30, 17, 101, 247, 210, 28, 288, 1358, 2927, 2310, 41, 652, 5102, 20581, 40361, 30030, 58, 1349, 16186, 107315, 390238, 716167, 510510, 77, 2451, 41817, 414849, 2429223, 8130689, 14117683, 9699690
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OFFSET

1,1


COMMENTS

The coefficient of first polynomial term with highest degree is always 1.
Each number in triangle is the sum of radicals of integers.
The absolute value of the entry in the kth column is the kth elementary symmetric function of the first n+(k1) primes.


LINKS



EXAMPLE

Triangle begins:
================================================
\k  1 2 3 4 5 6 7
n\ 
================================================
1  2;
2  5, 6;
3  10, 31, 30;
4  17, 101, 247, 210;
5  28, 288, 1358, 2927, 2310;
6  41, 652, 5102, 20581, 40361, 30030;
7  58,1349,16186,107315,390238,716167,510510;
So equation x^7 58*x^6 + 1349*x^5 16186*x^4 + 107315*x^3 390238*x^2+ 716167*x 510510 = 0 has 7 consecutive prime roots: 2,3,5,7,11,13,17


MAPLE

T:= n> (p> seq(coeff(p, x, ni), i=1..n))(mul(xithprime(i), i=1..n)):


MATHEMATICA

a = 1
For [i = 1, i < 10, i++,
a *= (x  Prime[i]);
Print[Drop[Reverse[CoefficientList[Expand[a], x]], 1]]
]


CROSSREFS

A006939 is the determinant of triangle matrix, considering T(n,k) k>n = 0;


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AUTHOR



EXTENSIONS



STATUS

approved



