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A378877 Let k = A379336(n). Then a(n) = sum of divisors d | k such that d neither divides nor is coprime to k/d, and k/d does not divide d.

%I A378877 #26 Feb 05 2025 21:56:58
%S A378877 10,14,14,15,18,16,22,18,20,26,21,42,30,21,22,82,27,28,24,38,24,26,25,
%T A378877 42,32,54,33,106,30,55,50,30,66,39,54,40,34,121,32,66,44,62,45,150,66,
%U A378877 35,65,36,154,123,42,52,146,78,35,78,42,91,46,57,36,178,36,78
%N A378877 Let k = A379336(n). Then a(n) = sum of divisors d | k such that d neither divides nor is coprime to k/d, and k/d does not divide d.
%H A378877 Michael De Vlieger, <a href="/A378877/b378877.txt">Table of n, a(n) for n = 1..10000</a>
%H A378877 Michael De Vlieger, <a href="/A378877/a378877.png">Log log scatterplot of a(n)</a> n = 1..21639, showing primes in red, proper prime powers in gold, squarefree composites in green (primorials > 2 with large dots), and numbers neither squarefree nor prime powers in blue or magenta, with magenta also representing powerful numbers that are not prime powers.
%e A378877 Define quality Q(k,m) regarding necessarily composite numbers k and m that neither divide nor are coprime to one another. The examples show only those divisor pairs d, k/d, such that Q(d, k/d) is true.
%e A378877 Let s = A379336.
%e A378877 a(1) = 10 since s(1) = 24 = 4*6.
%e A378877 a(2) = 14 since s(2) = 40 = 4*10.
%e A378877 a(3) = 14 since s(3) = 48 = 6*8.
%e A378877 a(4) = 15 since s(4) = 54 = 6*9.
%e A378877 a(5) = 18 since s(5) = 56 = 4*14.
%e A378877 a(6) = 16 since s(6) = 60 = 6*10.
%e A378877 a(12) = 42 since s(12) = 96 = 6*16 = 8*12, etc.
%t A378877 nn = 2^10; mm = Floor@ Sqrt[nn]; p = 2; q = 3;
%t A378877 s = Complement[
%t A378877   Select[Range[nn], And[#2 > #1 > 1, #2 > 3] & @@ {PrimeNu[#], PrimeOmega[#]} &],
%t A378877   Union[Reap[
%t A378877     While[p <= mm, q = NextPrime[p];
%t A378877       While[p*q <= mm,
%t A378877         If[p != q, Sow[p*q]]; q = NextPrime[q]];
%t A378877           p = NextPrime[p]] ][[-1, 1]] ]^2];
%t A378877 Map[Function[n, DivisorSum[n, # &, 1 < GCD[#1, #2] < Min[#1, #2] & @@ {#, n/#} &]], s]
%Y A378877 Cf. A376281, A379336.
%K A378877 nonn
%O A378877 1,1
%A A378877 _Michael De Vlieger_, Jan 27 2025