OFFSET
1,10
COMMENTS
Former name: number of "semidivisors" of n, numbers m < n that do not divide n but divide n^e for some integer e > 1. See ACM Inroads paper.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Exploring Number Bases as Tools, ACM Inroads, March 2012, Vol. 3, No. 1, pp. 4-12.
Michael De Vlieger, Regular and coregular numbers, ResearchGate, 2024.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20
FORMULA
a(n) = (Sum_{1<=k<=n, gcd(n,k)=1} mu(k)*floor(n/k)) - tau(n). - Michael De Vlieger, May 10 2016, after Benoit Cloitre at A010846.
From Michael De Vlieger, Aug 11 2024" (Start)
a(n) = number of terms in row n of A272618.
a(n) = sum of row n of A304570. (End)
EXAMPLE
From Michael De Vlieger, Aug 11 2024: (Start)Let S(n) = row n of A162306 and let D(n) = row n of A027750.a(2) = 0 since S(2) \ D(2) = {1, 2} \ {1, 2} is null.
a(10) = 2 since S(10) \ D(10) = {1, 2, 4, 5, 8, 10} \ {1, 2, 5, 10} = {4, 8}.a(16) = 0 since S(16) \ D(16) = {1, 2, 4, 8, 16} \ {1, 2, 4, 8, 16} is null, etc.Table of a(n) and S(n) \ D(n):
n a(n) row n of A272618.
---------------------------
6 1 {4}
10 2 {4, 8}
12 2 {8, 9}
14 2 {4, 8}
15 1 {9}
18 4 {4, 8, 12*, 16}
20 2 {8, 16}
21 1 {9}
22 3 {4, 8, 16}
24 3 {9, 16, 18*}
26 3 {4, 8, 16}
28 2 {8, 16}
30 10 {4, 8, 9, 12, 16, 18, 20, 24, 25, 27}
MATHEMATICA
Table[Count[Range[n], _?(And[Divisible[n, Times @@ FactorInteger[#][[All, 1]]], ! Divisible[n, #]] &)], {n, 120}] (* Michael De Vlieger, Aug 11 2024 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Jun 11 2014
EXTENSIONS
New name from David James Sycamore, Aug 11 2024
STATUS
approved