This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A380691 #5 Feb 16 2025 23:02:18
%S A380691 1,1,1,2,1,2,2,1,1,3,1,1,2,2,2,1,1,3,1,1,3,2,2,2,1,4,1,1,2,2,3,3,1,1,
%T A380691 4,1,2,2,2,2,2,4,1,1,2,2,1,2,4,3,1,4,1,1,1,3,5,2,1,2,5,2,2,3,2,1,3,1,
%U A380691 4,2,4,2,2,2,2,1,6,1,1,1,2,2,5,2,1,4,1
%N A380691 Number of divisors d | k, d < k/d, such that (d, k/d) are neither unitary nor both coreful, where k is neither squarefree nor prime power (in A126706).
%C A380691 A divisor d | k is said to be coreful if rad(d) = rad(k), where rad = A007947.
%C A380691 In other words, half the number of divisors d | k such that both gcd(d, k/d) > 1 and rad(d) != rad(k/d).
%C A380691 Divisors d and k/d have at least 1 prime factor in common and at least one prime factor q divides one but not the other divisor. Thus, the reference domain S is the intersection of nonsquarefree numbers (k in A013929) and numbers that are not prime powers (k in A024619).
%C A380691 Let S = { prime p : p | d } and let T = { prime p : p | k/d }. Then this sequence counts divisor pairs (d, k/d), d < k/d, such that the symmetric difference of S and T is not empty. For instance, for k = 24 = 2*12 = 4*6, where, in both cases, the product P of the symmetric difference is 3. For k = 180 = 2*90 = 3*60 = 6*30 = 10*18 = 12*15, the products of symmetric differences are 15, 10, 5, 15, and 10, respectively. In the case of 10*18, it is evident that neither rad(10) = rad(180) nor rad(18) = rad(30).
%H A380691 Michael De Vlieger, <a href="/A380691/b380691.txt">Table of n, a(n) for n = 1..10000</a>
%F A380691 Let tau = A000005, omega = A001221, rad = A007947, and S = A126706.
%F A380691 a(n) = card({ d | k : d < k/d, gcd(d, k/d) > 1, rad(d) != rad(k/d) }), k = S(n).
%F A380691 For k in S(n), a(n) = 1/2 * tau(k) - 2^omega(k) - Product_{p|k} m-1, where p^m | k but p^(m-1) does not divide k.
%F A380691 For k = S(n), a(n) = 1/2 * (A000005(k) - A034444(k) - A361430(k)).
%e A380691 Table of n, a(n) listing divisors d and S(n)/d for select values of n:
%e A380691 n S(n) a(n) d*S(n)/d
%e A380691 ---------------------------------------------------------------------
%e A380691 1 12 1 2*6
%e A380691 2 18 1 3*6
%e A380691 3 20 1 2*10
%e A380691 4 24 2 2*12, 4*6
%e A380691 5 28 1 2*14
%e A380691 6 36 2 2*18, 3*12
%e A380691 7 40 2 2*20, 4*10
%e A380691 10 48 3 2*24, 4*12, 6*8
%e A380691 26 96 4 2*48, 4*24, 6*16, 8*12
%e A380691 57 180 5 2*90, 3*60, 6*30, 10*18, 12*15
%e A380691 77 240 6 2*120, 4*60, 6*40, 8*30, 10*24, 12*20
%e A380691 123 360 8 2*180, 3*120, 4*90, 6*60, 10*36, 12*30, 15*24, 18*20
%t A380691 Table[1/2*(DivisorSigma[0, k] - 2^PrimeNu[k] - Apply[Times, FactorInteger[k][[All, -1]] - 1]), {k, Select[Range[12, 240], Nor[PrimePowerQ[#], SquareFreeQ[#]] &] }]
%Y A380691 Cf. A000005, A001221, A007947, A013929, A024619, A034444, A126706, A361430.
%K A380691 nonn,easy
%O A380691 1,4
%A A380691 _Michael De Vlieger_, Feb 09 2025