A385129 The number of divisors of n whose maximum exponent in their prime factorization is odd.
0, 1, 1, 1, 1, 3, 1, 2, 1, 3, 1, 3, 1, 3, 3, 2, 1, 3, 1, 3, 3, 3, 1, 5, 1, 3, 2, 3, 1, 7, 1, 3, 3, 3, 3, 3, 1, 3, 3, 5, 1, 7, 1, 3, 3, 3, 1, 5, 1, 3, 3, 3, 1, 5, 3, 5, 3, 3, 1, 7, 1, 3, 3, 3, 3, 7, 1, 3, 3, 7, 1, 6, 1, 3, 3, 3, 3, 7, 1, 5, 2, 3, 1, 7, 3, 3, 3
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
q[n_] := OddQ[Max[FactorInteger[n][[;; , 2]]]]; q[1] = False; a[n_] := DivisorSum[n, 1 &, q[#] &]; Array[a, 100] (* second program: *) a[n_] := Module[{e = FactorInteger[n][[;;, 2]], emax, kmax}, emax = Max[e]; kmax = emax + Mod[emax, 2]; Sum[(-1)^k * Product[Min[e[[i]], k-1] + 1, {i, 1, Length[e]}], {k, 1, kmax}]]; a[1] = 0; Array[a, 100] -
PARI
q(n) = if(n == 1, 0, vecmax(factor(n)[,2]) % 2); a(n) = sumdiv(n, d, q(d));
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PARI
a(n) = if(n == 1, 0, my(e = factor(n)[,2], emax = vecmax(e), kmax = emax + emax % 2); sum(k = 1, kmax, (-1)^k * prod(i = 1, #e, min(e[i], k-1)+1)));
Formula
a(n) = Sum_{d|n} (A051903(d) mod 2).
a(n) = Sum_{k=1..kmax(n)} (-1)^k * Product_{i=1..r} (min(e_i, k-1) + 1), for n >= 2; if n = Product_{i=1..r} p_i^e_i, r = omega(n) = A001221(n), then emax(n) = max(e_i) = A051903(n), and kmax(n) = emax(n) if emax(n) is even, and emax(n)+1 otherwise.
Sum_{k=1..n} a(k) ~ c1 * n * (log(n) + 2*gamma - 1) - c2 * n, where gamma is Euler's constant (A001620), c1 = 1 - Sum_{k>=2} (-1)^k * (1-1/zeta(k)) = 0.7240832794017729923099..., and c2 = 1 + Sum_{k>=2} (-1)^k * k * zeta'(k)/zeta(k)^2 = 0.56812633046434345687... .
Comments