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A286324
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a(n) is the number of bi-unitary divisors of n.
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37
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1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 8, 2, 6, 4, 4, 4, 4, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 8, 4, 8, 4, 4, 2, 8, 2, 4, 4, 6, 4, 8, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 4, 8, 2, 8, 4, 4, 2, 8, 4, 4, 4, 8, 2, 8
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OFFSET
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1,2
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COMMENTS
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a(n) is the number of terms of the n-th row of A222266.
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LINKS
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FORMULA
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Multiplicative with a(p^e) = e + (e mod 2). - Andrew Howroyd, Aug 05 2018
a(n) = A000005(A350390(n)) (the number of divisors of the largest exponentially odd number dividing n). - Amiram Eldar, Sep 01 2023
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - (p^s - 1)/((p^s + 1)*p^(2*s))).
Let f(s) = Product_{p prime} (1 - (p^s - 1)/((p^s + 1)*p^(2*s))).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - (p-1)/((p+1)*p^2)) = A306071 = 0.80733082163620503914865427993003113402584582508155664401800520770441381...,
f'(1) = f(1) * Sum_{p prime} 2*(p^2 - p - 1) * log(p) /(p^4 + 2*p^3 + 1) = f(1) * 0.40523703144422392508596509911218523410441417240419849262346362977537989... = f(1) * A306072
and gamma is the Euler-Mascheroni constant A001620. (End)
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EXAMPLE
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a(1) = 1 since 1 is the empty product; all divisors of 1 (i.e., 1) have a greatest common unitary divisor that is 1. 1 is a unitary divisor of all numbers n.
a(p) = 2 since 1 and p have greatest common unitary divisor 1.
a(6) = 4 since the divisor pairs {1, 6} and {2, 3} have greatest common unitary divisor 1.
a(24) = 8 since {1, 24}, {2, 12}, {3, 8}, {4, 6} have greatest unitary divisors {1, {1, 3, 8, 24}}, {{1, 2}, {1, 3, 4, 12}}, {{1, 3}, {1, 8}}, {1, 4}, {1, 2, 3, 6}}: 1 is the greatest common unitary divisor among all 4 pairs.
(End)
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MATHEMATICA
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f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; Table[DivisorSum[n, 1 &, Last@ Intersection[f@ #, f[n/#]] == 1 &], {n, 90}] (* Michael De Vlieger, May 07 2017 *)
f[p_, e_] := If[OddQ[e], e + 1, e]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 120] (* Amiram Eldar, Dec 19 2018 *)
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PROG
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(PARI) udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));
a(n) = #biudivs(n);
(PARI) a(n)={my(f=factor(n)[, 2]); prod(i=1, #f, my(e=f[i]); e + e % 2)} \\ Andrew Howroyd, Aug 05 2018
(PARI) for(n=1, 100, print1(direuler(p=2, n, (X^3 - X^2 + X + 1) / ((X-1)^2 * (X+1)))[n], ", ")) \\ Vaclav Kotesovec, Jan 11 2024
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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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