A366073 -id:A366073 - cp's OEIS frontend

A366074 The number of "Fermi-Dirac primes" (A050376) that are unitary divisors of n.

0, 1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 0, 2, 1, 3, 1, 0, 2, 2, 2, 2, 1, 2, 2, 1, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2, 2, 1, 3, 1, 2, 2, 0, 2, 3, 1, 2, 2, 3, 1, 1, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2

Offset: 1

Amiram Eldar, Sep 28 2023

First differs from A293439 at n = 128.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor.

Cf. A050376, A064547, A077610, A077761, A209229, A366073.

f[p_, e_] := If[e == 2^IntegerExponent[e, 2], 1, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecsum(apply(x -> (x == 1 << valuation(x, 2)), factor(n)[, 2]));

Additive with a(p^e) = A209229(e).

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = -P(2) + Sum_{k>=1} (P(2^k) - P(2^k+1)) = -0.13145993422430119364..., where P(s) is the prime zeta function.

A362412 The number of prime factors of the square root of the largest square dividing n, counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0

Offset: 1

Amiram Eldar, Sep 28 2023

First differs from A366073 at n = 64.

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

Cf. A000188, A000290, A001222, A004526, A005117, A008833, A154945, A162642, A349326, A366073.

f[p_, e_] := Floor[e/2]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI
```
a(n) = vecsum(factor(n)[, 2]\2);
```

a(n) = A001222(A000188(n)).
a(n) = A001222(A008833(n))/2.
Additive with a(p^e) = floor(e/2) = A004526(e).
a(n) >= 0, with equality if and only if n is squarefree (A005117).
a(n) <= A001222(n)/2, with equality if and only if n is square (A000290).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p^2-1) = 0.551693... (A154945).
a(n) = (A001222(n) - A162642(n))/2. - Ridouane Oudra, Apr 19 2025

cp's OEIS Frontend

A366074 The number of "Fermi-Dirac primes" (A050376) that are unitary divisors of n.

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