cp's OEIS Frontend

A360480 a(n) = number of numbers k < n, with gcd(k, n) > 1, such that there is at least one prime divisor p | k that does not divide n, and at least one prime divisor q | n that does not divide k.

%I A360480 #16 Mar 06 2023 13:22:44
%S A360480 0,0,0,0,0,0,0,0,0,1,0,1,0,3,3,0,0,3,0,5,5,6,0,6,0,8,0,9,0,5,0,0,8,11,
%T A360480 7,10,0,13,10,13,0,12,0,16,13,17,0,16,0,18,14,20,0,19,11,21,16,23,0,
%U A360480 19,0,25,19,0,13,25,0,27,20,27,0,27,0,30,25,31,13,32,0,32,0,34,0,33,17,36,25,37
%N A360480 a(n) = number of numbers k < n, with gcd(k, n) > 1, such that there is at least one prime divisor p | k that does not divide n, and at least one prime divisor q | n that does not divide k.
%C A360480 a(n) = 0 for prime powers, since the definition implies omega(n) >= 2.
%C A360480 a(n) is the cardinality of k in the cototient of n such that rad(k) mod rad(n) != 0 and rad(n) mod rad(k) != 0.
%H A360480 Michael De Vlieger, <a href="/A360480/b360480.txt">Table of n, a(n) for n = 1..10000</a>
%H A360480 Michael De Vlieger, <a href="/A360480/a360480.png">Log log scatterplot of a(n)</a>, n = 1..2^16, ignoring zeros. Scatterplot resembles that of A051953.
%H A360480 Michael De Vlieger, <a href="/A360480/a360480_1.png">Log log scatterplot of a(n)</a>, n = 1..2^16, ignoring zeros, showing composite squarefree numbers (in A120944) in green, numbers neither prime powers nor squarefree (in A126706) in blue, highlighting numbers that are products of composite prime powers (in A286708) in large light blue, and primorials (in A002110) in magenta.
%H A360480 Michael De Vlieger, <a href="/A360480/a360480_2.png">Plot k = 1..n</a> for n = 1..54, showing k counted by this sequence in large black, highlighting all k in row n otherwise in medium red for n in A024619, and for n that are prime powers, k in A024619 in dark blue.
%F A360480 a(n) <= A243823(n).
%F A360480 a(n) = | { k < n : gcd(k,n) > 1 AND (rad(k)|rad(n) NOR rad(n)|rad(k)) } |.
%e A360480 a(6) = 0 since k = 1..5 are prime powers.
%e A360480 a(10) = 1 since only k = 6 is such that p = 3, q = 5, but gcd(6, 10) = 2.
%e A360480 a(14) = 3 since for k = 6, we have p = 3, q = 7, for k = 10, we have p = 5, q = 7, and for k = 12, we have the same situation as we have with 6, while 6, 10, 12, and 14 are even.
%e A360480 Table listing k <= n counted by row n = 10..22 of this sequence:
%e A360480 a(10) = 1:  6  .  .  .  .
%e A360480 a(11) = 0:  .  .  .  .  .  .
%e A360480 a(12) = 1:  .  .  .  .  10 .  .
%e A360480 a(13) = 0:  .  .  .  .  .  .  .  .
%e A360480 a(14) = 3:  6  .  .  .  10 .  12 .  .
%e A360480 a(15) = 3:  6  .  .  .  10 .  12 .  .  .
%e A360480 a(16) = 0:  .  .  .  .  .  .  .  .  .  .  .
%e A360480 a(17) = 0:  .  .  .  .  .  .  .  .  .  .  .  .
%e A360480 a(18) = 3:  .  .  .  .  10 .  .  .  14 15 .  .  .
%e A360480 a(19) = 0:  .  .  .  .  .  .  .  .  .  .  .  .  .  .
%e A360480 a(20) = 5:  6  .  .  .  .  .  12 .  14 15 .  .  18 .  .
%e A360480 a(21) = 5:  6  .  .  .  .  .  12 .  14 15 .  .  18 .  .  .
%e A360480 a(22) = 6:  6  .  .  .  10 .  12 .  14 .  .  .  18 .  20 .  .
%t A360480 rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[k = rad[n]; Count[Range[n], _?(Nor[CoprimeQ[#1, n], Divisible[#2, k], Divisible[k, #2]] & @@ {#, rad[#]} &)], {n, 88}]
%Y A360480 Cf. A007947, A010846, A024619, A051953, A243823, A272619.
%K A360480 nonn
%O A360480 1,14
%A A360480 _Michael De Vlieger_, Feb 28 2023