A390371 a(n) is the largest bi-unitary divisor of n that is an exponentially odd number (A268335).

1, 2, 3, 1, 5, 6, 7, 8, 1, 10, 11, 3, 13, 14, 15, 8, 17, 2, 19, 5, 21, 22, 23, 24, 1, 26, 27, 7, 29, 30, 31, 32, 33, 34, 35, 1, 37, 38, 39, 40, 41, 42, 43, 11, 5, 46, 47, 24, 1, 2, 51, 13, 53, 54, 55, 56, 57, 58, 59, 15, 61, 62, 7, 32, 65, 66, 67, 17, 69, 70, 71

Offset

1,2

Comments

The number of these divisors is A368977(n), and their sum is A390372(n).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

Multiplicative with a(p^2) = 1, and a(p^e) = p^(e + (e mod 2) - 1) for e != 2.

a(n) = 1 if and only if n is the square of a squarefree number (A062503).

a(n) = n if and only if n is an exponentially odd number.

Dirichlet g.f.: zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^(2*s-2) + 1/p^(2*s) + 1/p^(4*s-3) - 1/p^(4*s-2)).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (35/(4*Pi^2)) * Product_{p prime} (1 - 2/p^2 + 1/p^3) = (35/(4*Pi^2)) * A065464 = 0.37966903458268100561... .

Mathematica

f[p_, e_] := p^(e + Mod[e, 2] - 1); f[p_, 2] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 2, 1, f[i, 1]^(f[i, 2] + f[i, 2]%2 - 1))); }

Crossrefs

Cf. A062503, A065464, A222266, A268335, A286324, A331737, A350389, A368977, A390372.

Sequence in context: A214392 A071975 A350389 * A368886 A182659 A359593

Adjacent sequences: A390368 A390369 A390370 * A390372 A390373 A390374

Keyword

nonn,easy,mult

Author

Amiram Eldar, Nov 03 2025

Status

approved