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A380085 The largest unitary divisor of n that is a term in A276078.

%I A380085 #10 Jan 12 2025 03:52:10
%S A380085 1,2,3,1,5,6,7,1,9,10,11,3,13,14,15,1,17,18,19,5,21,22,23,3,25,26,1,7,
%T A380085 29,30,31,1,33,34,35,9,37,38,39,5,41,42,43,11,45,46,47,3,49,50,51,13,
%U A380085 53,2,55,7,57,58,59,15,61,62,63,1,65,66,67,17,69,70,71
%N A380085 The largest unitary divisor of n that is a term in A276078.
%H A380085 Amiram Eldar, <a href="/A380085/b380085.txt">Table of n, a(n) for n = 1..10000</a>
%F A380085 Multiplicative with a(p^e) = p^e if e <= pi(p) = A000720(p), and 1 otherwise.
%F A380085 a(n) = 1 if and only if n is in A325127.
%F A380085 a(n) = n if and only if n is in A276078.
%F A380085 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{k>=1} (p(k)^(2*(k+1)) + p(k)^(2*k+1) - p(k)^(k+1) - p(k)^k + 1)/(p(k)^(2*k+1) * (p(k)+1)) = 0.76189494803691349595..., where p(k) = prime(k).
%t A380085 f[p_, e_] := p^If[e <= PrimePi[p], e, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A380085 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^if(f[i,2] <= primepi(f[i,1]), f[i,2], 0));}
%Y A380085 Cf. A000720, A077610, A276078, A325127, A377515, A380086, A380087, A380088.
%K A380085 nonn,easy,mult
%O A380085 1,2
%A A380085 _Amiram Eldar_, Jan 11 2025