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A056674
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Number of squarefree divisors which are not unitary. Also number of unitary divisors which are not squarefree.
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3
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0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 2, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 2, 1, 2, 0, 2, 0, 2, 0, 2, 0, 0, 0, 4, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 2, 2, 0, 0, 0, 2, 1, 0, 0, 4, 0, 0, 0, 2, 0, 4, 0, 2, 0, 0, 0, 2, 0, 2, 2, 3, 0, 0, 0, 2, 0
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OFFSET
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1,12
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COMMENTS
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Numbers of unitary and of squarefree divisors are identical, although the 2 sets are usually different, so sizes of parts outside overlap are also equal to each other.
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LINKS
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FORMULA
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EXAMPLE
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n=252, it has 18 divisors, 8 are unitary, 8 are squarefree, 2 belong to both classes, so 6 are squarefree but not unitary, thus a(252)=6. Set {2,3,14,21,42} forms squarefree but non-unitary while set {4,9,36,28,63,252} of same size gives the set of not squarefree but unitary divisors.
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MATHEMATICA
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Table[DivisorSum[n, 1 &, And[SquareFreeQ@ #, ! CoprimeQ[#, n/#]] &], {n, 105}] (* Michael De Vlieger, Jul 19 2017 *)
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PROG
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(PARI)
\\ Or:
(Python)
from sympy import gcd, primefactors, divisor_count
from sympy.ntheory.factor_ import core
def a055229(n):
c=core(n)
return gcd(c, n//c)
def a056674(n): return 2**len(primefactors(n)) - divisor_count(core(n)//a055229(n))
print([a056674(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 19 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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