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A367931
a(n) is the smallest number k such that k*n is an exponentially odious number (A270428).
4
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
OFFSET
1,8
LINKS
FORMULA
Multiplicative with a(p^e) = p^s(e), where s(e) = min{k >= e, k is odious} - e.
a(n) = A367933(n)/n.
a(n) >= 1, with equality if and only if n is an exponentially odious number (A270428).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.30023300..., where f(x) = (1-x) * (1 + Sum_{k>=1} x*(k-s(k))), and s(k) is defined above.
MATHEMATICA
f[p_, e_] := Module[{k = e}, While[! OddQ[DigitCount[k, 2 , 1]], k++]; p^(k-e)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) s(e) = {my(k = e); while(!(hammingweight(k)%2), k++); k - e; };
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2])); }
CROSSREFS
Similar sequences: A365298, A365685, A367932.
Sequence in context: A325989 A055229 A367698 * A365297 A270419 A275216
KEYWORD
nonn,easy,mult,base
AUTHOR
Amiram Eldar, Dec 05 2023
STATUS
approved