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A284118 Sum of nonprime squarefree divisors of n. 4
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
LINKS
FORMULA
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).
a(n) = Sum_{d|n, d nonprime squarefree} d.
a(n) = 1 if n is a prime power.
EXAMPLE
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.
MATHEMATICA
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]]
Table[Total[Select[Divisors[n], #1 == 1 || (SquareFreeQ[#1] && PrimeNu[#1] > 1) &]], {n, 90}]
PROG
(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
(Python)
from sympy import divisors
from sympy.ntheory.factor_ import core
def a(n): return sum([i for i in divisors(n) if core(i)==i and isprime(i)==0]) # Indranil Ghosh, Mar 21 2017
CROSSREFS
Sequence in context: A348281 A317940 A318674 * A165725 A214685 A327670
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Mar 20 2017
STATUS
approved

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Last modified May 23 13:40 EDT 2024. Contains 372763 sequences. (Running on oeis4.)