A007622 Consider Leibniz's harmonic triangle (A003506) and look at the non-boundary terms. Sequence gives numbers appearing in denominators, sorted.

6, 12, 20, 30, 42, 56, 60, 72, 90, 105, 110, 132, 140, 156, 168, 182, 210, 240, 252, 272, 280, 306, 342, 360, 380, 420, 462, 495, 504, 506, 552, 600, 630, 650, 660, 702, 756, 812, 840, 858, 870, 930, 992, 1056, 1092, 1122, 1190, 1260, 1320, 1332

Offset: 1

Author

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

Keywords

• nonn

Comments

No term is prime, about 80% are abundant, but the first few deficient are: 105, 110, 182, 495, 506, 1365, 1406, 1892, 2162, 2756, 2907, 3422, 3782, 4556, 5313, .... - Robert G. Wilson v, Aug 16 2010

A002943 = (6, 20, 42, 72, 110, 156, 210, 272, 342, 420, 506, 600, 702, ...) is a subsequence: indeed, this is every second denominator of the first differences of the sequence 1/n. - M. F. Hasler, Oct 11 2015

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 35.

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..1217.
• Eric Weisstein's World of Mathematics, Leibniz Harmonic Triangle.

Programs

Mathematica

L[n_, 1] := 1/n; L[n_, m_] := L[n, m] = L[n - 1, m - 1] - L[n, m - 1]; Take[ Union[ Flatten[ Table[ 1/L[n, m], {n, 3, 150}, {m, 2, Floor[n/2 + .5]}]]], 65]
t[n_, k_] := Denominator[n!*k!/(n + k + 1)!]; Take[ DeleteDuplicates@ Rest@ Sort@ Flatten@ Table[t[n - k, k], {n, 2, 150}, {k, n/2 + 1}], 65] (* Robert G. Wilson v, Jun 12 2014 *)

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jul 25 2000. Rechecked Jun 27 2003.