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%I #59 Aug 11 2024 10:32:39
%S 0,0,0,0,0,1,0,0,0,2,0,2,0,2,1,0,0,4,0,2,1,3,0,3,0,3,0,2,0,10,0,0,2,4,
%T 1,5,0,4,2,3,0,11,0,3,2,4,0,5,0,6,2,3,0,8,1,3,2,4,0,14,0,4,2,0,1,14,0,
%U 4,2,12,0,6,0,5,3,4,1,15,0,4,0,5,0,16,1,5,3,3,0,20,1,4,3,5,1,8,0,7,2,6
%N Number of k < n such that rad(k) | n but k does not divide n, where rad = A007947.
%C 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.
%H Michael De Vlieger, <a href="/A243822/b243822.txt">Table of n, a(n) for n = 1..10000</a>
%H Michael De Vlieger, <a href="https://doi.org/10.1145/2077808.2077809">Exploring Number Bases as Tools</a>, ACM Inroads, March 2012, Vol. 3, No. 1, pp. 4-12.
%H Michael De Vlieger, <a href="https://doi.org/10.13140/RG.2.2.28311.18084">Regular and coregular numbers</a>, ResearchGate, 2024.
%H Michael De Vlieger, <a href="/A243822/a243822.png">Log log scatterplot of a(n)</a>, n = 1..2^20
%F a(n) = A010846(n) - A000005(n) = card({row n of A162306} \ {row n of A027750}).
%F a(n) = A045763(n) - A243823(n).
%F 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.
%F From _Michael De Vlieger_, Aug 11 2024" (Start)
%F a(n) = 0 for n in A000961, a(n) > 0 for n in A024619.
%F a(n) = A051953(n) - A000005(n) + 1 = n - A000010(n) - A000005(n) - A243823(n) + 1.
%F a(n) = A355432(n) + A361235(n).
%F a(n) = A355432(n) for n in A360768.
%F a(n) = A361235(n) for n not in A360768.
%F a(n) = number of terms in row n of A272618.
%F a(n) = sum of row n of A304570. (End)
%e 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.
%e 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):
%e n a(n) row n of A272618.
%e ---------------------------
%e 6 1 {4}
%e 10 2 {4, 8}
%e 12 2 {8, 9}
%e 14 2 {4, 8}
%e 15 1 {9}
%e 18 4 {4, 8, 12*, 16}
%e 20 2 {8, 16}
%e 21 1 {9}
%e 22 3 {4, 8, 16}
%e 24 3 {9, 16, 18*}
%e 26 3 {4, 8, 16}
%e 28 2 {8, 16}
%e 30 10 {4, 8, 9, 12, 16, 18, 20, 24, 25, 27}
%e Terms in A272618 marked with an asterisk are counted by A355432. All other terms are counted by A361235. (End)
%t Table[Count[Range[n], _?(And[Divisible[n, Times @@ FactorInteger[#][[All, 1]]], ! Divisible[n, #]] &)], {n, 120}] (* _Michael De Vlieger_, Aug 11 2024 *)
%Y Cf. A000005, A000961, A024619, A027750, A010846, A045763, A162306, A243823, A272618, A304570, A355432, A361235.
%K nonn
%O 1,10
%A _Michael De Vlieger_, Jun 11 2014
%E New name from _David James Sycamore_, Aug 11 2024
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