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A309131
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Triangle read by rows: T(n, k) = (n - k)*prime(1 + k), with 0 <= k < n.
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3
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2, 4, 3, 6, 6, 5, 8, 9, 10, 7, 10, 12, 15, 14, 11, 12, 15, 20, 21, 22, 13, 14, 18, 25, 28, 33, 26, 17, 16, 21, 30, 35, 44, 39, 34, 19, 18, 24, 35, 42, 55, 52, 51, 38, 23, 20, 27, 40, 49, 66, 65, 68, 57, 46, 29, 22, 30, 45, 56, 77, 78, 85, 76, 69, 58, 31
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OFFSET
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1,1
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COMMENTS
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T(n, k) is the k-superdiagonal sum of an n X n Toeplitz matrix M(n) whose first row consists of successive prime numbers prime(1), ..., prime(n).
The h-th subdiagonal of the triangle T gives the primes multiplied by (h + 1).
The k-th column of the triangle T gives the multiples of prime(1 + k).
Also array A(n, k) = n*prime(1 + k) read by ascending antidiagonals, with 0 <= k < n. - Michel Marcus, Jul 15 2019
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LINKS
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FORMULA
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EXAMPLE
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The triangle T(n, k) begins:
---+-----------------------------------------------------
n\k| 0 1 2 3 4 5 6 7 8
---+-----------------------------------------------------
1 | 2
2 | 4 3
3 | 6 6 5
4 | 8 9 10 7
5 | 10 12 15 14 11
6 | 12 15 20 21 22 13
7 | 14 18 25 28 33 26 17
8 | 16 21 30 35 44 39 34 19
9 | 18 24 35 42 55 52 51 38 23
...
For n = 3 the matrix M(3) is
2, 3, 5
M_{2,1}, 2, 3
M_{3,1}, M_{3,2}, 2
and therefore T(3, 0) = 2 + 2 + 2 = 6, T(3, 1) = 3 + 3 = 6, and T(3, 2) = 5.
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MAPLE
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a:=(n, k)->(n-k)*ithprime(1+k): seq(seq(a(n, k), k=0..n-1), n=1..11);
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MATHEMATICA
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Flatten[Table[(n-k)*Prime[1+k], {n, 1, 11}, {k, 0, n-1}]]
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PROG
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(Magma) [[(n-k)*NthPrime(1+k): k in [0..n-1]]: n in [1..11]]; // triangle output
(PARI)
T(n, k) = (n - k)*prime(1 + k);
tabl(nn) = for(n=1, nn, for(k=0, n-1, print1(T(n, k), ", ")); print); \\ triangle output
(Sage) [[(n-k)*Primes().unrank(k) for k in (0..n-1)] for n in (1..11)] # triangle output
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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