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 A283367 Irregular triangle read by rows: T(n,k) = number of horizontal positions for the vertical legs of the Dyck paths for the symmetric representation of sigma(n) that are listed in the corresponding positions of the triangle of A259177. 2
 1, 2, 2, 3, 3, 4, 3, 5, 4, 5, 6, 4, 5, 7, 5, 6, 8, 5, 7, 9, 6, 7, 8, 10, 6, 7, 8, 11, 7, 9, 10, 12, 7, 9, 10, 13, 8, 9, 11, 14, 8, 10, 11, 12, 15, 9, 11, 12, 13, 16, 9, 11, 12, 13, 17, 10, 12, 13, 15, 18, 10, 12, 13, 15, 19, 11, 13, 15, 16, 20, 11, 14, 15, 16, 17, 21 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The dot product of the n-th row of this triangle and the n-th row of triangle A259177 equals A024916(n), the sum of all divisors of numbers 1 through n (true for all n <= 20000); the value of a(n) is the sum of the rectangles between the y-axis and the vertical legs of the symmetric representation of sigma(n). This is the companion computation to A283368. LINKS Table of n, a(n) for n=1..76. FORMULA T(n,k) = sum_{i=1..k} f(n, 2*i-1) where f is defined in A237593. A024916(n) = sum_{i=1..row(n)} (T(n,i))*S(n,i)) where S(n,i) refers to the triangle of A259177 and row(n) = floor(sqrt(8*n+1)-1)/2). EXAMPLE The first vertical leg of the symmetric representation of sigma(15) is at x-coordinate 8 and has length 3, and row 15 has 5 entries so that T(15,1) = 8 and T(15,5) = 15. The first 16 rows of the irregular triangle: 1: 1 2: 2 3: 2 3 4: 3 4 5: 3 5 6: 4 5 6 7: 4 5 7 8: 5 6 8 9: 5 7 9 10: 6 7 8 10 11: 6 7 8 11 12: 7 9 10 12 13: 7 9 10 13 14: 8 9 11 14 15: 8 10 11 12 15 16: 9 11 12 13 16 MATHEMATICA (* function f[n, k] and its support functions are defined in A237593 *) a283367[n_, k_] := Sum[f[n, 2*i-1], {i, k}] TableForm[Table[a283367[n, k], {n, 1, 16}, {k, 1, row[n]}]] (* triangle *) Flatten[Table[a283367[n, k], {n, 1, 21}, {k, 1, row[n]}]] (* sequence data *) CROSSREFS Cf. A024916, A237593, A259176, A259177, A283368. Sequence in context: A309067 A205561 A323636 * A322528 A258756 A127431 Adjacent sequences: A283364 A283365 A283366 * A283368 A283369 A283370 KEYWORD nonn,tabf AUTHOR Hartmut F. W. Hoft, Mar 06 2017 STATUS approved

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Last modified February 22 06:21 EST 2024. Contains 370240 sequences. (Running on oeis4.)