A370077 -id:A370077 - cp's OEIS frontend

A370078 a(n) = log_2(A370077(n)).

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

Offset: 1

Amiram Eldar, Feb 09 2024

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

Cf. A005361, A007814, A268335, A367168, A370077.

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

s(n) = my(e = valuation(n, 2)); if(n >> e == 1, e, 0);
a(n) = vecsum(apply(x -> s(x), factor(n)[, 2]));

a(n) = A007814(A005361(A367168(n))).
Additive with a(p^e) = log_2(e) if e is a power of 2, and 0 otherwise.
a(n) = 0 if and only if n is an exponentially odd number (A268335).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} k * (P(2^k)-P(2^k+1)) = 0.36616241880640645934..., where P(s) is the prime zeta function .

A372331 The number of infinitary divisors of the smallest number k such that k*n is a number whose number of divisors is a power of 2 (A036537).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Apr 28 2024

First differs from A370077 and A370080 at n = 32.
The number of divisors d of n that are infinitarily relatively prime to n (see A064379), i.e., d have no common infinitary divisors with n.
Equivalently, the number of divisors d of n such that for each prime divisor p of d, bitand(v_p(n), v_p(d)) = 0, where v_p(k) is the highest power of p that divides k. Note that for infinitary divisors d of n (A077609), bitand(v_p(n), v_p(d)) = v_p(d).

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

Cf. A036537, A037445, A064379, A077609, A080100, A372328.

Cf. A370077, A370080.

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

a(n) = vecprod(apply(x -> 2^(logint(x, 2) + 1 - hammingweight(x)), factor(n)[, 2]));

a(n) = A037445(A372328(n)).
Multiplicative with a(p^e) = 2^A023416(e) = A080100(e).
a(n) = 1 if and only if n is in A036537.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=1} A080100(k)/p^k) = 1.51209151045338102469... .

cp's OEIS Frontend

A370078 a(n) = log_2(A370077(n)).

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