cp's OEIS Frontend

A078651 Number of increasing geometric-progression subsequences of [1,...,n] with integral successive-term ratio and length >= 1.

%I A078651 #20 Aug 27 2025 12:03:30
%S A078651 1,3,5,9,11,15,17,23,27,31,33,40,42,46,50,59,61,68,70,77,81,85,87,97,
%T A078651 101,105,111,118,120,128,130,141,145,149,153,165,167,171,175,185,187,
%U A078651 195,197,204,211,215,217,231,235,242,246,253,255,265,269,279,283,287
%N A078651 Number of increasing geometric-progression subsequences of [1,...,n] with integral successive-term ratio and length >= 1.
%C A078651 The number of geometric-progression subsequences of [1,...,n] with integral successive-term ratio r and length k is floor(n/r^(k-1))(n > 0, r > 1, k > 0).
%H A078651 Paolo Xausa, <a href="/A078651/b078651.txt">Table of n, a(n) for n = 1..10000</a>
%F A078651 a(n) = n + Sum_{r > 1, j > 0} floor(n/r^j).
%e A078651 a(1): [1]; a(2): [1],[2],[1,2]; a(3): [1],[2],[3],[1,2],[1,3].
%p A078651 g := (n, b) -> local i; add(iquo(n, b^i), i = 1..floor(log(n, b))):
%p A078651 a := n -> local b; n + add(g(n, b), b = 2..n):
%p A078651 seq(a(n), n = 1..58);  # _Peter Luschny_, Apr 03 2025
%t A078651 Accumulate[1 + Table[Total[IntegerExponent[n, Rest[Divisors[n]]]], {n, 100}]] (* _Paolo Xausa_, Aug 27 2025 *)
%Y A078651 a(n) = n + A078632(n).
%Y A078651 See A366471 for rational ratios.
%Y A078651 See A078567 for APs.
%K A078651 nonn,easy,changed
%O A078651 1,2
%A A078651 Robert E. Sawyer (rs.1(AT)mindspring.com), Jan 08 2003