cp's OEIS Frontend

A329110 Number of integer sequences 1 <= b_1 < b_2 < ... < b_t <= n such that b_i divides b_(i+1) for all 0 < i < t.

%I A329110 #24 Aug 12 2022 09:23:30
%S A329110 1,3,5,9,11,17,19,27,31,37,39,55,57,63,69,85,87,103,105,121,127,133,
%T A329110 135,175,179,185,193,209,211,237,239,271,277,283,289,341,343,349,355,
%U A329110 395,397,423,425,441,457,463,465,561,565,581,587,603,605,645,651,691
%N A329110 Number of integer sequences 1 <= b_1 < b_2 < ... < b_t <= n such that b_i divides b_(i+1) for all 0 < i < t.
%C A329110 Cumulative sum of A067824.
%H A329110 Peter Kagey, <a href="/A329110/b329110.txt">Table of n, a(n) for n = 1..10000</a>
%H A329110 Mathematics Stack Exchange user Markus Scheuer, <a href="https://math.stackexchange.com/a/3421536/121988">Finding the number of subsets of a set such that an element divides the succeeding element</a>.
%F A329110 From _Vaclav Kotesovec_, Mar 18 2021: (Start)
%F A329110 a(n) ~ -2*n^r/(r*zeta'(r)), where r=A107311 is the root of the equation zeta(r)=2.
%F A329110 a(n) ~ 2*A247667 * n^A107311 / A107311.
%F A329110 a(n) ~ 2*A217598 * n^A107311. (End)
%e A329110 For n = 4 the a(4) = 9 sequences are 1; 1, 2; 1, 2, 4; 1, 3; 1, 4; 2; 2, 4; 3; and 4.
%o A329110 (PARI) s=0; for (n=1, #(z=vector(56)), print1 (s += z[n]=1+sumdiv(n, k, if (k<n, z[k], 0)) ", ")) \\ _Rémy Sigrist_, Nov 08 2019
%Y A329110 Cf. A067824.
%K A329110 nonn
%O A329110 1,2
%A A329110 _Peter Kagey_, Nov 04 2019