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A336041
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Number of refactorable divisors of n.
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7
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1, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 1, 3, 1, 4, 1, 2, 1, 2, 1, 5, 1, 2, 2, 2, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 4, 1, 2, 1, 2, 2, 2, 1, 5, 1, 2, 1, 2, 1, 4, 1, 4, 1, 2, 1, 4, 1, 2, 2, 3, 1, 2, 1, 2, 1, 2, 1, 9, 1, 2, 1, 2, 1, 2, 1, 5, 2, 2, 1, 4, 1, 2, 1, 4, 1, 4, 1, 2
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OFFSET
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1,2
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COMMENTS
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Inverse Möbius transform of A336040. - Antti Karttunen, Nov 24 2021
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 1..65537
Eric Weisstein's World of Mathematics, Refactorable Number
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FORMULA
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a(n) = Sum_{d|n} c(d), where c(n) is the refactorable characteristic of n (A336040).
a(n) = Sum_{d|n} (1 - ceiling(d/tau(d)) + floor(d/tau(d))), where tau(n) is the number of divisors of n (A000005).
a(n) = A000005(n) - A349658(n). - Antti Karttunen, Nov 24 2021
a(p) = 1 for odd primes p. - Wesley Ivan Hurt, Nov 28 2021
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EXAMPLE
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a(6) = 2; The divisors of 6 are {1,2,3,6}. Only two of these divisors are refactorable since d(1) = 1|1 and d(2) = 2|2, but d(3) = 2 does not divide 3 and d(6) = 4 does not divide 6.
a(7) = 1; The divisors of 7 are {1,7} and d(1) = 1|1, but d(7) = 2 does not divide 7, so a(7) = 1.
a(8) = 3; The divisors of 8 are {1,2,4,8}. 1, 2 and 8 are refactorable since d(1) = 1|1, d(2) = 2|2 and d(8) = 4|8 but d(4) = 3 does not divide 4, so a(8) = 3.
a(9) = 2; The divisors of 9 are {1,3,9}. 1 and 9 are refactorable since d(1) = 1|1 and d(9) = 3|9 but d(3) = 2 does not divide 3. Thus, a(9) = 2.
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MAPLE
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A336041 := proc(n)
local a ;
a := 0 ;
for d in numtheory[divisors](n) do
if type(d/numtheory[tau](d), integer) then
a := a+1 ;
end if;
end do:
a ;
end proc:
seq(A336041(n), n=1..30) ; # R. J. Mathar, Nov 24 2020
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MATHEMATICA
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a[n_] := DivisorSum[n, 1 &, Divisible[#, DivisorSigma[0, #]] &]; Array[a, 100] (* Amiram Eldar, Jul 08 2020 *)
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PROG
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(PARI) a(n) = sumdiv(n, d, d%numdiv(d) == 0); \\ Michel Marcus, Jul 07 2020
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CROSSREFS
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Cf. A000005 (tau), A033950 (refactorable numbers), A336040 (refactorable characteristic), A349658 (number of nonrefactorable divisors).
Cf. also A335182, A335665.
Sequence in context: A167970 A126433 A237271 * A176725 A085029 A185318
Adjacent sequences: A336038 A336039 A336040 * A336042 A336043 A336044
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KEYWORD
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nonn,easy
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AUTHOR
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Wesley Ivan Hurt, Jul 07 2020
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STATUS
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approved
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