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A366905 The largest exponentially odious divisor of n.

%I A366905 #8 Oct 28 2023 03:47:59
%S A366905 1,2,3,4,5,6,7,4,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,12,25,26,
%T A366905 9,28,29,30,31,16,33,34,35,36,37,38,39,20,41,42,43,44,45,46,47,48,49,
%U A366905 50,51,52,53,18,55,28,57,58,59,60,61,62,63,16,65,66,67,68
%N A366905 The largest exponentially odious divisor of n.
%C A366905 First differs from A353897 at n = 128.
%C A366905 The largest divisor of n that is an exponentially odious number (A270428).
%C A366905 The number of exponentially odious divisors of n is A366901(n) and their sum is A366903(n).
%H A366905 Amiram Eldar, <a href="/A366905/b366905.txt">Table of n, a(n) for n = 1..10000</a>
%F A366905 Multiplicative with a(p^e) = p^max{k=1..e, k odious}.
%F A366905 a(n) <= n, with equality if and only if n is exponentially odious number (A270428).
%F A366905 Sum_{k=1..n} a(k) ~ c*n^2, where c = (1/2) * Product_{p prime} (1 + Sum_{e>=1} (p^f(e) - p^(f(e-1)+1))/p^(2*e)) = 0.4636829525..., f(e) = max{k=1..e, k odious} for e >= 1, and f(0) = 0.
%t A366905 maxOdious[e_] := Module[{k = e}, While[EvenQ[DigitCount[k, 2, 1]], k--]; k]; f[p_, e_] := p^maxOdious[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A366905 (PARI) s(n) = {my(k = n); while(!(hammingweight(k)%2), k--); k;}
%o A366905 a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2]));}
%Y A366905 Cf. A270428, A366901,A366903.
%Y A366905 Similar sequences: A353897, A365683, A366906.
%K A366905 nonn,easy,mult
%O A366905 1,2
%A A366905 _Amiram Eldar_, Oct 27 2023