A330285 The maximum number of arithmetic progressions in a sequence of length n.

0, 0, 1, 3, 7, 12, 20, 29, 41, 55, 72, 90, 113, 137, 164, 194, 228, 263, 303, 344, 390, 439, 491, 544, 604, 666, 731, 799, 872, 946, 1027, 1109, 1196, 1286, 1379, 1475, 1579, 1684, 1792, 1903, 2021, 2140, 2266, 2393, 2525, 2662, 2802, 2943, 3093, 3245, 3402, 3562, 3727

Offset: 1

Author

Joseph Wheat, Dec 21 2019

Keywords

• nonn

Comments

The partial arithmetic density D_n(A) up to n is merely the number of arithmetic progressions, A(s(n)), divided by the total number of nonempty subsets of {s(1), s(2), ..., s(n)}, i.e., A(s(n))/(2^n - 1). As n approaches infinity, D_n(A) converges to zero. Furthermore, the infinite sum of the partial densities for any sequence always converges to the total density D(A). Every infinite arithmetic progression has the same total density, Sum_{n >= 1} a(n)/(2^n - 1) = alpha ~ 1.25568880818612911696845537; sequences with a finite number of progressions have D(A) < alpha; and sequences without any arithmetic progressions have D(A) = 0.

Links

• Encyclopedia of Mathematics, Density of a sequence
• Eric Weisstein's World of Mathematics, Arithmetic Progression

Crossrefs

Cf. A049988, A130518, A104429, A065825, A092482.

Partial sums of A002541.

Programs

PARI

a(n) = sum(i=1, n, sum(j=1, i, floor((i - 1)/(j + 1))))

Formula

a(n) = Sum_{i=1..n} Sum_{j=1..i} floor((i - 1)/(j + 1)).