This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A190867 #47 Jul 05 2025 11:24:24
%S A190867 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,4,1,1,
%T A190867 1,1,1,1,1,2,1,1,1,1,1,1,1,3,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1,5,1,1,1,1,
%U A190867 1,1,1,2,1,1,1,1,1,1,1,3,3,1,1,1,1,1,1,2,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,2,1,1,1,2,1,1
%N A190867 Count of the 3-full divisors of n.
%C A190867 a(n) is the number of divisors d of n with d an element of A036966.
%C A190867 This is the 3-full analog of the 2-full case A005361.
%H A190867 Antti Karttunen, <a href="/A190867/b190867.txt">Table of n, a(n) for n = 1..10000</a>
%H A190867 Aleksandar Ivić, <a href="https://eudml.org/doc/257502">On the asymptotic formulas for powerful numbers</a>, Publications de l'Institut Mathématique, Vol. 23, No. 37 (1978), pp. 85-94; <a href="http://elib.mi.sanu.ac.rs/files/journals/publ/43/12.pdf">alternative link</a>.
%H A190867 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A190867 a(n) = Sum_{d|n, d in A036966} 1.
%F A190867 a(n) <= A005361(n).
%F A190867 Multiplicative with a(p^e) = max(1,e-1).
%F A190867 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/(p^2*(p-1))) (A065483). (Ivić, 1978). - _Amiram Eldar_, Jul 23 2022
%F A190867 Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 1/p^(3*s)). - _Amiram Eldar_, Sep 21 2023
%e A190867 a(16)=3 because the divisors of 16 are {1,2,4,8,16}, and three of these are 3-full: 1, 8=2^3 and 16=2^4.
%p A190867 f:= n -> convert(map(t -> max(1,t[2]-1), ifactors(n)[2]),`*`):
%p A190867 map(f, [$1..200]); # _Robert Israel_, Jul 19 2017
%t A190867 Table[Product[Max[{1, i - 1}], {i, FactorInteger[n][[All, 2]]}], {n, 1, 200}] (* _Geoffrey Critzer_, Feb 12 2015 *)
%t A190867 Table[1 + DivisorSum[n, 1 &, AllTrue[FactorInteger[#][[All, -1]], # > 2 &] &], {n, 120}] (* _Michael De Vlieger_, Jul 19 2017 *)
%o A190867 (PARI) A190867(n) = { my(f = factor(n), m = 1); for (k=1, #f~, m *= max(1,f[k, 2]-1); ); m; } \\ _Antti Karttunen_, Jul 19 2017
%o A190867 (Python)
%o A190867 from functools import reduce
%o A190867 from sympy import factorint
%o A190867 from operator import mul
%o A190867 def a(n): return 1 if n==1 else reduce(mul, [max(1, e - 1) for e in factorint(n).values()])
%o A190867 print([a(n) for n in range(1, 101)]) # _Indranil Ghosh_, Jul 19 2017
%Y A190867 Cf. A005361, A036966, A065483.
%K A190867 nonn,mult,easy
%O A190867 1,8
%A A190867 _R. J. Mathar_, May 27 2011