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A366244 The largest infinitary divisor of n that is a term of A366242.

%I A366244 #13 Jan 09 2025 15:44:16
%S A366244 1,2,3,1,5,6,7,2,1,10,11,3,13,14,15,16,17,2,19,5,21,22,23,6,1,26,3,7,
%T A366244 29,30,31,32,33,34,35,1,37,38,39,10,41,42,43,11,5,46,47,48,1,2,51,13,
%U A366244 53,6,55,14,57,58,59,15,61,62,7,16,65,66,67,17,69,70,71
%N A366244 The largest infinitary divisor of n that is a term of A366242.
%H A366244 Amiram Eldar, <a href="/A366244/b366244.txt">Table of n, a(n) for n = 1..10000</a>
%F A366244 Multiplicative with a(p^e) = p^A063694(e).
%F A366244 a(n) = n / A366245(n).
%F A366244 a(n) >= 1, with equality if and only if n is a term of A366243.
%F A366244 a(n) <= n, with equality if and only if n is a term of A366242.
%F A366244 Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1-1/p)*(Sum_{k>=1} p^(A063694(k)-2*k)) = 0.35319488024808595542... .
%F A366244 From _Peter Munn_, Jan 09 2025: (Start)
%F A366244 a(n) = max({k in A366242 : A059895(k, n) = k}).
%F A366244 a(n) = Product_{k >= 0} A352780(n, 2k).
%F A366244 Also defined by:
%F A366244 - for n in A046100, a(n) = A007913(n);
%F A366244 - a(n^4) = (a(n))^4;
%F A366244 - a(A059896(n,k)) = A059896(a(n), a(k)).
%F A366244 Other identities:
%F A366244 a(n) = sqrt(A366245(n^2)).
%F A366244 a(A059897(n,k)) = A059897(a(n), a(k)).
%F A366244 a(A225546(n)) = A225546(A247503(n)).
%F A366244 (End)
%t A366244 f[p_, e_] := p^BitAnd[e, Sum[2^k, {k, 0, Floor@ Log2[e], 2}]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A366244 (PARI) s(e) = sum(k = 0, e, (-2)^k*floor(e/2^k));
%o A366244 a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^s(f[i,2]));}
%Y A366244 Cf. A063694, A366242, A366243, A366245.
%Y A366244 See the formula section for the relationships with A007913, A046100, A059895, A059896, A059897, A225546, A247503, A352780.
%K A366244 nonn,easy,mult
%O A366244 1,2
%A A366244 _Amiram Eldar_, Oct 05 2023