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A284118 Sum of nonprime squarefree divisors of n.

%I A284118 #26 Jun 05 2025 00:41:56
%S A284118 1,1,1,1,1,7,1,1,1,11,1,7,1,15,16,1,1,7,1,11,22,23,1,7,1,27,1,15,1,62,
%T A284118 1,1,34,35,36,7,1,39,40,11,1,84,1,23,16,47,1,7,1,11,52,27,1,7,56,15,
%U A284118 58,59,1,62,1,63,22,1,66,128,1,35,70,130,1,7,1,75,16,39,78,150,1,11,1,83,1,84,86,87,88,23,1,62
%N A284118 Sum of nonprime squarefree divisors of n.
%H A284118 G. C. Greubel, <a href="/A284118/b284118.txt">Table of n, a(n) for n = 1..5000</a>
%H A284118 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>.
%F A284118 G.f.: x/(1 - x) + Sum_{k>=2} sgn(omega(k)-1)*mu(k)^2*k*x^k/(1 - x^k), where omega(k) is the number of distinct primes dividing k (A001221) and mu(k) is the Moebius function (A008683).
%F A284118 a(n) = Sum_{d|n, d nonprime squarefree} d.
%F A284118 a(n) = 1 if n is a prime power.
%F A284118 a(n) = A048250(n) - A008472(n). - _Amiram Eldar_, Jun 05 2025
%e A284118 a(30) = 62 because 30 has 8 divisors {1, 2, 3, 5, 6, 10, 15, 30} among which 5 are nonprime squarefree {1, 6, 10, 15, 30} therefore 1 + 6 + 10 + 15 + 30 = 62.
%t A284118 nmax = 90; Rest[CoefficientList[Series[x/(1 - x) + Sum[Sign[PrimeNu[k] - 1] MoebiusMu[k]^2 k x^k/(1 - x^k), {k, 2, nmax}], {x, 0, nmax}], x]]
%t A284118 Table[Total[Select[Divisors[n], #1 == 1 || (SquareFreeQ[#1] && PrimeNu[#1] > 1) &]], {n, 90}]
%o A284118 (PARI) Vec((x/(1 - x)) + sum(k=2, 90, sign(omega(k) - 1) * moebius(k)^2 * k * x^k/(1 - x^k)) + O(x^91)) \\ _Indranil Ghosh_, Mar 21 2017
%o A284118 (Python)
%o A284118 from sympy import divisors
%o A284118 from sympy.ntheory.factor_ import core
%o A284118 def a(n): return sum([i for i in divisors(n) if core(i)==i and isprime(i)==0]) # _Indranil Ghosh_, Mar 21 2017
%Y A284118 Cf. A000469, A001221, A008472, A008683, A048250, A259936.
%K A284118 nonn
%O A284118 1,6
%A A284118 _Ilya Gutkovskiy_, Mar 20 2017