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 A309131 Triangle read by rows: T(n, k) = (n - k)*prime(1 + k), with 0 <= k < n. 3
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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 LINKS Table of n, a(n) for n=1..66. Wikipedia, Toeplitz matrix FORMULA T(n, k) = A025581(n, k)*A000040(1 + k). EXAMPLE 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. MAPLE a:=(n, k)->(n-k)*ithprime(1+k): seq(seq(a(n, k), k=0..n-1), n=1..11); MATHEMATICA Flatten[Table[(n-k)*Prime[1+k], {n, 1, 11}, {k, 0, n-1}]] PROG (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 CROSSREFS Cf. A025581, A318173, A325516. Cf. A000040: diagonal; A001747: 1st subdiagonal; A001748: 2nd subdiagonal; A001749: 3rd subdiagonal; A001750: 4th subdiagonal; A005843: 0th column; A008585: 1st column; A008587: 2nd column; A008589: 3rd column; A008593: 4th column; A008595: 5th column; A008599: 6th column; A008601: 7th column; A014148: row sums; A138636: 5th subdiagonal; A272470: 6th subdiagonal. Sequence in context: A375615 A143273 A141429 * A162953 A235451 A348915 Adjacent sequences: A309128 A309129 A309130 * A309132 A309133 A309134 KEYWORD nonn,easy,tabl AUTHOR Stefano Spezia, Jul 14 2019 STATUS approved

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Last modified September 8 22:33 EDT 2024. Contains 375759 sequences. (Running on oeis4.)