cp's OEIS Frontend

A360906 Numbers with the same number of cubefree divisors and 3-full divisors.

%I A360906 #12 Feb 26 2023 03:26:47
%S A360906 1,16,81,384,625,640,896,1296,1408,1664,2176,2401,2432,2944,3712,3968,
%T A360906 4374,4736,5248,5504,6016,6784,7552,7808,8576,9088,9216,9344,10000,
%U A360906 10112,10624,10935,11392,12416,12928,13184,13696,13952,14464,14641,15309,16256,16768
%N A360906 Numbers with the same number of cubefree divisors and 3-full divisors.
%C A360906 Numbers k such that A073184(k) = A190867(k).
%C A360906 Numbers whose largest cubefree divisor (A007948) and cubefull part (A360540) have the same number of divisors (A000005).
%C A360906 If k and m are coprime terms, then k*m is also a term.
%C A360906 The characteristic function of this sequence depends only on prime signature.
%C A360906 1 is the only cubefree (A004709) term.
%C A360906 Includes the 4th powers of squarefree numbers (1 and A113849).
%C A360906 The 4th powers of primes (A030514) are the only terms that are prime powers (A246655).
%C A360906 Numbers of the for m*p^(3*2^k+1), where m is squarefree, p is prime, gcd(m, p) = 1 and omega(m) = k, are all terms. In particular, this sequence includes numbers of the form p^7*q, where p != q are primes (A179664), and numbers of the form p^13*q*r where p, q, and r are distinct primes.
%C A360906 The corresponding numbers of cubefree (or 3-full) divisors are 1, 3, 3, 6, 3, 6, 6, 9, 6, 6, 6, 3, 6, 6, ... .
%H A360906 Amiram Eldar, <a href="/A360906/b360906.txt">Table of n, a(n) for n = 1..10000</a>
%t A360906 q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Times @@ (Min[#, 3] & /@ (e + 1)) == Times @@ (Max[#, 1] & /@ (e - 1))]; q[1] = True; Select[Range[10^4], q]
%o A360906 (PARI) is(k) = {my(e = factor(k)[,2]); prod(i = 1, #e, min(e[i] + 1, 3)) == prod(i = 1, #e, max(e[i] - 1, 1)); }
%Y A360906 Cf. A000005, A004709, A007948, A073184, A190867, A246655, A360540, A360902.
%Y A360906 Subsequences: A030514, A113849, A179664, A360907.
%K A360906 nonn
%O A360906 1,2
%A A360906 _Amiram Eldar_, Feb 25 2023