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A365170
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The sum of divisors d of n such that gcd(d, n/d) is squarefree.
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1
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1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 27, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 51, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 108, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 99, 84, 144
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OFFSET
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1,2
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COMMENTS
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The number of these divisors is A252505(n).
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LINKS
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FORMULA
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Multiplicative with a(p) = 1 + p, a(p^2) = 1 + p + p^2, and a(p^e) = (1 + p)*(1 + p^(e - 1)) if e >= 3.
a(n) >= A034448(n), with equality if and only if n is squarefree number (A005117).
a(n) <= A000203(n), with equality if and only if n is biquadratefree number (A046100).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 315/(4*Pi^4) = A157292 / 2 = 0.808446... . - Amiram Eldar, Aug 25 2023
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MATHEMATICA
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f[p_, e_] := Switch[e, 1, 1 + p, 2, 1 + p + p^2, _, (1 + p)*(1 + p^(e - 1))]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
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PROG
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(PARI) a(n) = {my(f = factor(n), p , e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; if(e == 1, 1 + p, if(e == 2, 1 + p + p^2, (1 + p)*(1 + p^(e - 1))))); }
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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