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A350102 Number of self-measuring subsets of the initial segment of the natural numbers strictly below n. Number of subsets S of [n] with S = distset(S).

%I A350102 #24 Jan 31 2024 22:23:42
%S A350102 1,2,3,5,7,10,12,16,18,22,25,29,31,37,39,43,47,52,54,60,62,68,72,76,
%T A350102 78,86,89,93,97,103,105,113,115,121,125,129,133,142,144,148,152,160,
%U A350102 162,170,172,178,184,188,190,200,203,209,213,219,221,229,233,241,245
%N A350102 Number of self-measuring subsets of the initial segment of the natural numbers strictly below n. Number of subsets S of [n] with S = distset(S).
%C A350102 We use the notation [n] = {0, 1, ..., n-1}. If S is a subset of [n] then we define the distset of S (set of distances of S) as {|x - y|: x, y in S}. We call a subset S of the natural numbers self-measuring if and only if S = distset(S).
%H A350102 Winston de Greef, <a href="/A350102/b350102.txt">Table of n, a(n) for n = 0..10000</a>
%H A350102 Peter Luschny, <a href="/A350102/a350102.png">Illustrating self-measuring subsets of {0, 1, 2, 3}</a>.
%F A350102 a(n) = a(n - 1) + tau(n - 1) for n >= 2, tau = A000005.
%F A350102 a(n) = 2 + Sum_{k=1..n-1} floor((n - 1)/k) for n >= 1.
%F A350102 a(n) = 2 + A006218(n - 1) for n >= 1.
%F A350102 a(n) = 1 + A054519(n - 1) for n >= 1.
%F A350102 Row sums of A350103.
%F A350102 a(n) >= n + floor(n/2) + floor(n/3).
%e A350102 a(0) = 1 = card({}).
%e A350102 a(4) = 7 = card({}, {0}, {0, 1}, {0, 2}, {0, 3}, {0, 1, 2}, {0, 1, 2, 3}).
%e A350102 a(6) = 12 = card({}, {0}, {0, 1}, {0, 2}, {0, 3}, {0, 4}, {0, 5}, {0, 1, 2}, {0, 2, 4}, {0, 1, 2, 3}, {0, 1, 2, 3, 4}, {0, 1, 2, 3, 4, 5}).
%p A350102 A350102 := n -> ifelse(n = 0, 1, 2 + add(iquo(n-1, k), k = 1 .. n-1)):
%p A350102 seq(A350102(n), n = 0 .. 58);
%t A350102 a[0] = 1; a[1] = 2; a[n_] := a[n] = a[n - 1] + DivisorSigma[0, n - 1];
%t A350102 Table[a[n], {n, 0, 58}]
%Y A350102 Cf. A350103, A349976, A006218, A054519, A000005.
%K A350102 nonn
%O A350102 0,2
%A A350102 _Peter Luschny_, Dec 14 2021