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A380144 Sum of divisors d | k such that rad(d) = rad(k/d) where k is in A001694 and rad = A007947.

%I A380144 #24 Jan 19 2025 11:03:39
%S A380144 1,2,6,3,14,5,12,30,6,7,62,18,39,10,24,11,30,126,42,13,14,30,72,15,
%T A380144 120,254,90,17,78,56,19,42,70,168,21,22,60,510,23,186,155,234,60,26,
%U A380144 363,98,150,29,360,30,31,66,240,180,1022,33,90,378,34,35,546,84,132
%N A380144 Sum of divisors d | k such that rad(d) = rad(k/d) where k is in A001694 and rad = A007947.
%C A380144 Alternatively, sum of divisors d | k such that omega(d) = omega(k/d) = omega(k) for powerful k in A001694.
%C A380144 Not a permutation of natural numbers: 4 does not appear.
%C A380144 This sequence is A364988 without zeros. See A364988 for multiplicative properties.
%H A380144 Michael De Vlieger, <a href="/A380144/b380144.txt">Table of n, a(n) for n = 1..10000</a>
%H A380144 Michael De Vlieger, <a href="/A380144/a380144.png">Log log scatterplot of a(n)</a>, n = 1..10^5.
%F A380144 Let s = A001694.
%F A380144 a(n) = A364988(A001694(n)).
%F A380144 a(n) = m*rad(n) with m >= 1.
%F A380144 a(n) = sqrt(s(n)) = rad(s(n)) for s(n) = rad(s(n))^2 (i.e., s(n) in A062503).
%F A380144 a(n) > sqrt(s(n)) and a(n) = m*rad(s(n)), m > 1, for s(n) in A320966.
%F A380144 The sequence of record setters is A151821.
%F A380144 a(n) = s(n)-2 for s(n) in A151821.
%e A380144 a(1) = 1 since s(1) = 1 = 1*1. (We count divisor 1 only once, since it is a square root.)
%e A380144 a(2) = 2 since s(2) = 4 = 2*2, and omega(2) = omega(4). Alternatively, rad(2) = rad(4).
%e A380144 a(3) = 6 since s(3) = 8 = 2*4, omega(2) = omega(4) = omega(8), i.e., rad(2) = rad(4) = rad(8), and 2+4 = 6.
%e A380144 a(4) = 3 since s(4) = 9 = 3*3, and omega(3) = omega(9); rad(3) = rad(9).
%e A380144 a(5) = 14 since s(5) = 16 = 2*8 = 4*4; 2+4+8 = 14.
%e A380144 a(6) = 5 since s(6) = 25 = 5*5.
%e A380144 a(7) = 12 since s(7) = 27 = 3*9; 3+9 = 12.
%e A380144 a(8) = 30 since s(8) = 32 = 2*16 = 4*8; 2+4+8+16 = 30.
%e A380144 a(9) = 6 since s(9) = 36 = 6*6, etc.
%t A380144 nn = 1500;
%t A380144 s = Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}];
%t A380144 Map[Function[{n, r},
%t A380144   DivisorSum[n, # &, PrimeNu[#] == PrimeNu[n/#] == r &]] @@
%t A380144   {#, PrimeNu[#]} &, s]
%Y A380144 Cf. A001221, A001694, A007947, A062503, A151821, A320966, A364988.
%K A380144 nonn
%O A380144 1,2
%A A380144 _Michael De Vlieger_, Jan 15 2025