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A364767
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The number of divisors of n that are practical numbers (A005153).
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0
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1, 2, 1, 3, 1, 3, 1, 4, 1, 2, 1, 5, 1, 2, 1, 5, 1, 4, 1, 4, 1, 2, 1, 7, 1, 2, 1, 4, 1, 4, 1, 6, 1, 2, 1, 7, 1, 2, 1, 6, 1, 4, 1, 3, 1, 2, 1, 9, 1, 2, 1, 3, 1, 5, 1, 6, 1, 2, 1, 8, 1, 2, 1, 7, 1, 4, 1, 3, 1, 2, 1, 10, 1, 2, 1, 3, 1, 4, 1, 8, 1, 2, 1, 8, 1, 2, 1, 5
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OFFSET
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1,2
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LINKS
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FORMULA
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a(p) = 1 for p prime, p > 2.
a(2*p) = 2 for p prime, p > 3.
a(2*3^k) = k + 2, k >= 1;
a(2*p^k) = 2, k >= 1, p prime, p >= 5.
a(2^n) = n + 1.
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EXAMPLE
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n = 1 has only one divisor 1 = A005153(1).
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MATHEMATICA
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f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[n_] := (ind = Position[(fct = FactorInteger[n])[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), _?(# > 1 &)]) == {}; a[n_] := DivisorSum[n, 1 &, pracQ[#] &]; Array[a, 100] (* Amiram Eldar, Aug 21 2023 *)
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PROG
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(Magma) sk:=func<n, k|&+[Divisors(n)[i]:i in [1..k]]>; f:=func<n|forall{k: k in [2..#Divisors(n)]|sk(n, k-1) ge Divisors(n)[k]-1}>; [#[d:d in Divisors(n)|f(d)]:n in [1..100]];
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CROSSREFS
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Inverse Möbius transform of A322860.
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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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