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A366245 The largest infinitary divisor of n that is a term of A366243.

%I A366245 #13 Jan 09 2025 15:44:26
%S A366245 1,1,1,4,1,1,1,4,9,1,1,4,1,1,1,1,1,9,1,4,1,1,1,4,25,1,9,4,1,1,1,1,1,1,
%T A366245 1,36,1,1,1,4,1,1,1,4,9,1,1,1,49,25,1,4,1,9,1,4,1,1,1,4,1,1,9,4,1,1,1,
%U A366245 4,1,1,1,36,1,1,25,4,1,1,1,1,1,1,1,4,1
%N A366245 The largest infinitary divisor of n that is a term of A366243.
%C A366245 First differs from A335324 at n = 256.
%H A366245 Amiram Eldar, <a href="/A366245/b366245.txt">Table of n, a(n) for n = 1..10000</a>
%F A366245 Multiplicative with a(p^e) = p^A063695(e).
%F A366245 a(n) = n / A366244(n).
%F A366245 a(n) >= 1, with equality if and only if n is a term of A366242.
%F A366245 a(n) <= n, with equality if and only if n is a term of A366243.
%F A366245 From _Peter Munn_, Jan 09 2025: (Start)
%F A366245 a(n) = max({k in A366243 : A059895(k, n) = k}).
%F A366245 a(n) = Product_{k >= 0} A352780(n, 2k+1).
%F A366245 Also defined by:
%F A366245 - for n in A046100, a(n) = A008833(n);
%F A366245 - a(n^4) = (a(n))^4;
%F A366245 - a(A059896(n,k)) = A059896(a(n), a(k)).
%F A366245 Other identities:
%F A366245 a(n) = sqrt(A366244(n^2)).
%F A366245 a(A059897(n,k)) = A059897(a(n), a(k)).
%F A366245 a(A225546(n)) = A225546(A248101(n)).
%F A366245 (End)
%t A366245 f[p_, e_] := p^BitAnd[e, Sum[2^k, {k, 1, Floor@ Log2[e], 2}]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A366245 (PARI) s(e) = -sum(k = 1, e, (-2)^k*floor(e/2^k));
%o A366245 a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^s(f[i,2]));}
%Y A366245 Cf. A063695, A335324, A366242, A366243, A366244.
%Y A366245 See the formula section for the relationships with A008833, A046100, A059895, A059896, A059897, A225546, A248101, A352780.
%K A366245 nonn,easy,mult
%O A366245 1,4
%A A366245 _Amiram Eldar_, Oct 05 2023