cp's OEIS Frontend

A382063 Numbers whose number of coreful divisors is divisible by their number of exponential divisors.

%I A382063 #9 Mar 14 2025 21:16:56
%S A382063 1,2,3,4,5,6,7,9,10,11,12,13,14,15,17,18,19,20,21,22,23,25,26,28,29,
%T A382063 30,31,33,34,35,36,37,38,39,41,42,43,44,45,46,47,49,50,51,52,53,55,57,
%U A382063 58,59,60,61,62,63,65,66,67,68,69,70,71,73,74,75,76,77,78,79
%N A382063 Numbers whose number of coreful divisors is divisible by their number of exponential divisors.
%C A382063 First differs from A377019 at n = 55: A377019(55) = 64 is not a term of this sequence.
%C A382063 First differs from A344742 at n = 62: A344742(62) = 72 is not a term of this sequence.
%C A382063 All the cubefree numbers (A004709) are terms. The least term that is not cubefree is a(215) = 256 = 2^8. The cubefree numbers are the terms whose number of coreful divisors is equal to their number of exponential divisors.
%C A382063 All the exponentially refactorable numbers (A382065) are terms. The least term that is not in A382065 is a(362) = 432 = 2^4 * 3^3. The next terms that are not in A382065 are 648, 2000, 2160, 3024, 3240, 4536, 4752, 5000, ... .
%C A382063 For a number k whose prime factorization is Product_{i} p_i^e_i, a coreful divisor d of k has the prime factorization Product_{i} p_i^f_i with f_i >= 1 for all i. An exponential divisor of k is a coreful divisor with the additional condition that f_i | e_i for all i.
%C A382063 Numbers k such that A049419(k) | A005361(k).
%C A382063 The criterion according to which a number belongs to this sequence depends only on the prime signature of this number: if {e_1, e_2, ... } are the exponents in the prime factorization of k then k is a term if and only if A005361(k)/A049419(k) = Product_{i} e_i/A000005(e_i) is an integer.
%C A382063 A number k is a term if and only if the cubefull part of k, A360540(k), is a term. Therefore, the primitive terms of this sequence are the cubefull terms, A382064.
%C A382063 The asymptotic density of this sequence is Sum_{n>=1} f(A382064(n)) = 0.83697905945047..., where f(n) = (1/(zeta(3)*n)) * Product_{prime p|n} (p^2/(p^2+p+1)).
%H A382063 Amiram Eldar, <a href="/A382063/b382063.txt">Table of n, a(n) for n = 1..10000</a>
%e A382063 2 is a term since A005361(2) = A049419(2) = 2, so 2 | 2.
%e A382063 256 is a term since A005361(256) = 8, A049419(256) = 4, and 4 | 8.
%t A382063 q[k_] := Module[{e = FactorInteger[k][[;;, 2]]}, Divisible[Times @@ e, Times @@ DivisorSigma[0, e]]]; Select[Range[100], # == 1 || q[#] &]
%o A382063 (PARI) isok(k) = {my(e = factor(k)[, 2]); !(vecprod(e) % vecprod(apply(x -> numdiv(x), e)));}
%Y A382063 Cf. A000005, A002117, A005361, A036966, A049419, A344742, A360540, A377019, A382061.
%Y A382063 Subsequences: A004709, A382064, A382065.
%K A382063 nonn,easy
%O A382063 1,2
%A A382063 _Amiram Eldar_, Mar 14 2025