cp's OEIS Frontend

A362852 The number of divisors of n that are both bi-unitary and exponential.

%I A362852 #10 Feb 16 2025 08:34:05
%S A362852 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,2,1,1,
%T A362852 1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1,3,1,1,1,1,
%U A362852 1,1,1,2,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1
%N A362852 The number of divisors of n that are both bi-unitary and exponential.
%C A362852 First differs from A061704 at n = 128, and from A304327 and abs(A307428) at n = 64.
%C A362852 If e > 0 is the exponent of the highest power of p dividing n (where p is a prime), then for each divisor d of n that is both a bi-unitary and an exponential divisor, the exponent of the highest power of p dividing d is a number k such that k | e but k != e/2.
%C A362852 The least term that is higher than 2 is a(64) = 3.
%C A362852 This sequence is unbounded. E.g., a(2^(2^prime(n))) = prime(n).
%H A362852 Amiram Eldar, <a href="/A362852/b362852.txt">Table of n, a(n) for n = 1..10000</a>
%H A362852 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BiunitaryDivisor.html">Biunitary Divisor</a>.
%H A362852 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/e-Divisor.html">e-Divisor</a>.
%F A362852 Multiplicative with a(p^e) = d(e) if e is odd, and d(e)-1 if e is even, where d(k) is the number of divisors of k (A000005).
%F A362852 a(n) = 1 if and only if n is cubefree (A004709).
%F A362852 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=1} (d(k)+(k mod 2)-1)/p^k) = 1.1951330849... .
%e A362852 a(8) = 2 since 8 has 2 divisors that are both bi-unitary and exponential: 2 and 8.
%t A362852 f[p_, e_] := DivisorSigma[0, e] - If[OddQ[e], 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A362852 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, numdiv(f[i, 2]) - !(f[i, 2] % 2));}
%Y A362852 Cf. A000005, A004709, A049419, A188999, A222266, A286324, A322791, A359411, A362853 (indices of records), A362854.
%Y A362852 Cf. A061704, A304327, A307428.
%K A362852 nonn,mult
%O A362852 1,8
%A A362852 _Amiram Eldar_, May 05 2023