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

1, 3, 5, 9, 11, 15, 17, 23, 27, 31, 33, 40, 42, 46, 50, 59, 61, 68, 70, 77, 81, 85, 87, 97, 101, 105, 111, 118, 120, 128, 130, 141, 145, 149, 153, 165, 167, 171, 175, 185, 187, 195, 197, 204, 211, 215, 217, 231, 235, 242, 246, 253, 255, 265, 269, 279, 283, 287

Offset

1,2

Comments

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).

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Formula

a(n) = n + Sum_{r > 1, j > 0} floor(n/r^j).

Example

a(1): [1]; a(2): [1],[2],[1,2]; a(3): [1],[2],[3],[1,2],[1,3].

Maple

g := (n, b) -> local i; add(iquo(n, b^i), i = 1..floor(log(n, b))):
a := n -> local b; n + add(g(n, b), b = 2..n):
seq(a(n), n = 1..58);  # Peter Luschny, Apr 03 2025

Mathematica

Accumulate[1 + Table[Total[IntegerExponent[n, Rest[Divisors[n]]]], {n, 100}]] (* Paolo Xausa, Aug 27 2025 *)

Prog

(PARI) A078651(n) = {my(s=0, k=2); while(k<=n, s+=(n - sumdigits(n, k))/(k-1); k=k+1); n + s} \\ Zhuorui He, Aug 28 2025

Crossrefs

a(n) = n + A078632(n).

See A366471 for rational ratios.

See A078567 for APs.

Partial sums of A169594.

Sequence in context: A164121 A383297 A333171 * A268732 A101114 A120696

Adjacent sequences: A078648 A078649 A078650 * A078652 A078653 A078654

Keyword

nonn,easy

Author

Robert E. Sawyer (rs.1(AT)mindspring.com), Jan 08 2003

Status

approved