cp's OEIS Frontend

A078632 Number of geometric subsequences of [1,...,n] with integral successive-term ratio and length > 1.

%I A078632 #17 Aug 27 2025 12:04:41
%S A078632 0,1,2,5,6,9,10,15,18,21,22,28,29,32,35,43,44,50,51,57,60,63,64,73,76,
%T A078632 79,84,90,91,98,99,109,112,115,118,129,130,133,136,145,146,153,154,
%U A078632 160,166,169,170,183,186,192,195,201,202,211,214,223,226,229,230,242
%N A078632 Number of geometric subsequences of [1,...,n] with integral successive-term ratio and length > 1.
%C A078632 The number of geometric 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 A078632 Zhuorui He, <a href="/A078632/b078632.txt">Table of n, a(n) for n = 1..10000</a>
%F A078632 a(n) = Sum_{r > 1, j > 0} floor(n/r^j).
%e A078632 a(2): [1,2]; a(3): [1,2],[1,3]; a(4): [1,2],[1,3],[1,4],[2,4],[1,2,4].
%p A078632 g := (n, b) -> local i; add(iquo(n, b^i), i = 1..floor(log(n, b))):
%p A078632 a := n -> local b; add(g(n, b), b = 2..n):
%p A078632 seq(a(n), n = 1..60);  # _Peter Luschny_, Apr 03 2025
%t A078632 Accumulate[Table[Total[IntegerExponent[n, Rest[Divisors[n]]]], {n, 100}]] (* _Paolo Xausa_, Aug 27 2025 *)
%o A078632 (PARI) A078632(n) = {my(s=0, k=2); while(k<=n, s+=(n - sumdigits(n, k))/(k-1); k=k+1); s} \\ _Zhuorui He_, Aug 26 2025
%Y A078632 Cf. A078651.
%Y A078632 Row sums of triangle A090623.
%Y A078632 Partial sums of A309891.
%K A078632 nonn,easy,changed
%O A078632 1,3
%A A078632 Robert E. Sawyer (rs.1(AT)mindspring.com)