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A055076 Multiplicity of Max{gcd(d, n/d)} when d runs over divisors of n.

%I A055076 #59 May 08 2025 14:26:00
%S A055076 1,2,2,1,2,4,2,2,1,4,2,2,2,4,4,1,2,2,2,2,4,4,2,4,1,4,2,2,2,8,2,2,4,4,
%T A055076 4,1,2,4,4,4,2,8,2,2,2,4,2,2,1,2,4,2,2,4,4,4,4,4,2,4,2,4,2,1,4,8,2,2,
%U A055076 4,8,2,2,2,4,2,2,4,8,2,2,1,4,2,4,4,4,4,4,2,4,4,2,4,4,4,4,2,2,2,1,2,8,2,4,8
%N A055076 Multiplicity of Max{gcd(d, n/d)} when d runs over divisors of n.
%C A055076 Number of distinct values of gcd(d, n!/d) if d runs over divisors of n! seems to be A046951(n).
%C A055076 a(n) = 1 iff n is a square. - _Bernard Schott_, Oct 22 2019
%C A055076 a(n) is the number of the unitary divisors (cf. A077610) of n that are exponentially odd (A268335). - _Amiram Eldar_, Nov 11 2022
%C A055076 The number of infinitary divisors of n that are squarefree (A005117). - _Amiram Eldar_, Jan 09 2024
%H A055076 Alois P. Heinz, <a href="/A055076/b055076.txt">Table of n, a(n) for n = 1..10000</a>
%H A055076 Vaclav Kotesovec, <a href="/A055076/a055076.jpg">Graph - the asymptotic ratio (100000 terms)</a>.
%H A055076 Christian Krause, <a href="https://github.com/ckrause/loda">LODA, an assembly language, a computational model and a tool for mining integer sequences</a>.
%H A055076 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A055076 Multiplicative with a(p^e) = 2^(e mod 2). - _Vladeta Jovovic_, Dec 13 2002
%F A055076 a(n) = 2^A162642(n). - _Antti Karttunen_, Dec 02 2017
%F A055076 a(n) = A034444(A007913(n)). [Found by LODA miner, see C. Krause link. Essentially the same formula as the above ones] - _Antti Karttunen_, Apr 05 2021
%F A055076 From _Amiram Eldar_, Sep 09 2023: (Start)
%F A055076 a(n) = A034444(A350389(n)).
%F A055076 Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 2/p^s). (End)
%F A055076 From _Vaclav Kotesovec_, Sep 09 2023: (Start)
%F A055076 Let f(s) = Product_{p prime} (1 - 3/p^(2*s) + 2/p^(3*s)).
%F A055076 Dirichlet g.f.: zeta(s)^2 * zeta(2*s) * f(s).
%F A055076 Sum_{k=1..n} a(k) ~ (Pi^2 * f(1) * n / 6) * (log(n) + 2*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)), where
%F A055076 f(1) = A065473 = Product_{primes p} (1 - 3/p^2 + 2/p^3) = 0.286747428434478734107892712789838446434331844097056995641477859336652243...,
%F A055076 f'(1) = f(1) * Sum_{primes p} 6*log(p) / (p^2 + p - 2) = f(1) * 2.798014228561519243358371276385174449737670294137200281334256087932048625...
%F A055076 and gamma is the Euler-Mascheroni constant A001620. (End)
%e A055076 n=120, the set of gcd(d, 120/d) values for the 16 divisors of 120 is {1,2,1,2,1,2,1,2,2,1,2,1,2,1,2,1}. The max is 2 and it occurs 8 times, so a(120)=8. This sequence seems to consist of powers of 2.
%p A055076 with(numtheory):
%p A055076 a:= n->(p->coeff(p, x, degree(p)))(add(x^igcd(d, n/d), d=divisors(n))):
%p A055076 seq(a(n), n=1..105);  # _Alois P. Heinz_, Jul 21 2015
%t A055076 a[n_] := With[{g = GCD[#, n/#]& /@ Divisors[n]}, Count[g, Max[g]]];
%t A055076 Array[a, 105] (* _Jean-François Alcover_, Mar 28 2017 *)
%t A055076 f[p_, e_] := 2^Mod[e, 2]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* _Amiram Eldar_, Nov 11 2022 *)
%o A055076 (PARI) A055076(n) = if(1==n,n,my(es=factor(n)[,2]~); prod(i=1,#es,2^(es[i]%2))); \\ _Antti Karttunen_, Apr 05 2021
%o A055076 (Scheme)
%o A055076 ;; With memoization-macro definec.
%o A055076 (definec (A055076 n) (if (= 1 n) n (* (+ 1 (A000035 (A067029 n))) (A055076 (A028234 n))))) ;; _Antti Karttunen_, Dec 02 2017
%Y A055076 Cf. A000188, A005117, A007913, A034444, A077610, A162642, A268335, A295316, A350389.
%K A055076 nonn,easy,mult
%O A055076 1,2
%A A055076 _Labos Elemer_, Jun 13 2000