A126132 a(n) = number of k's, 1<=k<=n, where d(k) is equal to any divisor of n, where d(k) is the number of positive divisors of k.

1, 2, 1, 3, 1, 5, 1, 7, 3, 5, 1, 12, 1, 7, 3, 12, 1, 12, 1, 15, 3, 9, 1, 23, 2, 10, 4, 19, 1, 19, 1, 23, 4, 12, 2, 33, 1, 13, 4, 31, 1, 22, 1, 29, 6, 15, 1, 45, 1, 18, 5, 32, 1, 31, 2, 40, 5, 17, 1, 53, 1, 19, 6, 45, 2, 33, 1, 41, 5, 23, 1, 69, 1, 22, 6, 45, 2, 39, 1, 59, 6, 23, 1, 70, 3, 24, 5

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Formula

a(n) = Sum_{k=1..n} (1 - ceiling(n/d(k)) + floor(n/d(k))). - Wesley Ivan Hurt, Apr 21 2023

Example

The number of divisors of the integers 1 through 10 form the sequence 1, 2, 2, 3, 2, 4, 2, 4, 3, 4. The divisors of 10 are 1, 2, 5, 10. The terms of the sequence of the first ten d(k)'s which equal any divisor of 10 are the five terms 1, 2, 2, 2, 2. So a(10) = 5.

Mathematica

f[n_] := Length@Select[Table[Length@Divisors[k], {k, n}], MemberQ[Divisors[n], # ] &]; Table[f[n], {n, 87}] (* Ray Chandler, Dec 20 2006 *)

Prog

(PARI) A126132(n) = sum(k=1, n, !(n%numdiv(k))); \\ Antti Karttunen, Apr 01 2021
(PARI) first(n) = { n = min(n, 245044799); qdivs = vector(960); res = vector(n); for(i = 1, n, nd = numdiv(i); qdivs[nd]++; d = select(x -> x <= #qdivs, divisors(i)); res[i] = sum(j = 1, #d, qdivs[d[j]]) ); res } \\ David A. Corneth, Apr 01 2021

Crossrefs

Cf. A000005, A002182, A126131.

Sequence in context: A127203 A227288 A336752 * A129982 A052552 A147000

Adjacent sequences: A126129 A126130 A126131 * A126133 A126134 A126135

Keyword

nonn,look

Author

Leroy Quet, Dec 18 2006

Extensions

Extended by Ray Chandler, Dec 20 2006

Status

approved