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 A056622 #45 Jan 27 2025 05:40:43
%S A056622 1,1,1,2,1,1,1,1,3,1,1,2,1,1,1,4,1,3,1,2,1,1,1,1,5,1,1,2,1,1,1,2,1,1,
%T A056622 1,6,1,1,1,1,1,1,1,2,3,1,1,4,7,5,1,2,1,1,1,1,1,1,1,2,1,1,3,8,1,1,1,2,
%U A056622 1,1,1,3,1,1,5,2,1,1,1,4,9,1,1,2,1,1,1,1,1,3,1,2,1,1,1,2,1,7,3,10,1,1,1,1
%N A056622 a(n) = A000188(n)/A055229(n).
%C A056622 Previous name: "Square root of largest unitary square divisor of n." The previous name was incorrect for numbers that have an odd exponent in their prime factorization that is larger than 3. For the correct square root of largest unitary square divisor of n see A071974. - _Amiram Eldar_, Jul 26 2024
%C A056622 Multiplicative because quotient of two multiplicative sequences. - _Christian G. Bower_, May 16 2005
%H A056622 Antti Karttunen, <a href="/A056622/b056622.txt">Table of n, a(n) for n = 1..16384</a>
%H A056622 Vaclav Kotesovec, <a href="/A056622/a056622.jpg">Graph - the asymptotic ratio (100000 terms)</a>
%F A056622 Multiplicative with a(p^e) = p^(e/2) if e even, a(p) = 1, and a(p^e) = p^((e-3)/2) for odd e > 1. - _Amiram Eldar_, Sep 14 2020
%F A056622 Dirichlet g.f.: zeta(2*s-1) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s-1) + 1/p^(3*s)). - _Amiram Eldar_, Dec 18 2023
%F A056622 a(n) = sqrt(A056623(n)). - _Amiram Eldar_, Jul 26 2024
%F A056622 From _Vaclav Kotesovec_, Jan 27 2025: (Start)
%F A056622 Dirichlet g.f.: zeta(s) * zeta(2*s-1) * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s) - 1/p^(3*s-1) - 1/p^(4*s) + 1/p^(4*s-1)).
%F A056622 Let f(s) = Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s) - 1/p^(3*s-1) - 1/p^(4*s) + 1/p^(4*s-1)).
%F A056622 Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 3*gamma - 1 + f'(1)/f(1)) / 2, where
%F A056622 f(1) = Product_{p prime} (1 - 2/p^2 + 2/p^3 - 1/p^4) = 0.490798286634728225909154323920711804307234495196201399106047774...,
%F A056622 f'(1) = f(1) * Sum_{p prime} (5*p^2 - 7*p + 4) * log(p) / (p^4 - 2*p^2 + 2*p - 1) = f(1) * 1.94788222046256567576552118452630646598176999674201755783...
%F A056622 and gamma is the Euler-Mascheroni constant A001620. (End)
%e A056622 For n = 125: A000188(125) = 5, A055229(125) = 5, so a(125) = 1.
%e A056622 For n = 360: A000188(360) = 6, A055229(360) = 2, so a(360) = 3.
%t A056622 f[p_, e_] := If[EvenQ[e], p^(e/2), If[e == 1, 1, p^((e - 3)/2)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 14 2020 *)
%o A056622 (PARI)
%o A056622 A000188(n) = core(n, 1)[2]; \\ _Michel Marcus_, Feb 27 2013
%o A056622 A055229(n) = { my(c=core(n)); gcd(c, n/c); }; \\ _Charles R Greathouse IV_, Nov 20 2012
%o A056622 A056622(n) = (A000188(n)/A055229(n)); \\ _Antti Karttunen_, Nov 19 2017
%Y A056622 Cf. A000188, A055229, A034444, A056623, A071974.
%K A056622 nonn,easy,mult
%O A056622 1,4
%A A056622 _Labos Elemer_, Aug 08 2000
%E A056622 Name replaced with a formula by _Amiram Eldar_, Jul 26 2024