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%I #11 Nov 02 2023 10:27:43
%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,8,17,18,19,20,21,22,23,24,25,26,
%T 27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,24,49,
%U 50,51,52,53,54,55,56,57,58,59,60,61,62,63,32,65,66,67,68
%N The largest divisor of n that is not a term of A322448.
%C First differs from A365683 at n = 64.
%C The largest divisor of n whose prime factorization has exponents that are all either 1 or primes.
%C The number of these divisors is A366991(n) and their sum is A366992(n).
%H Amiram Eldar, <a href="/A366994/b366994.txt">Table of n, a(n) for n = 1..10000</a>
%F Multiplicative with a(p) = p and a(p^e) = p^A007917(e) for e >= 2.
%F a(n) <= n, with equality if and only if n is not in A322448.
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} f(1/p) = 0.48535795387619596052..., where f(x) = (1 - x) * (1 + Sum_{k>=1} x^(2*k-s(k))), s(k) = A007917(k) for k >= 2, and s(1) = 1.
%t f[p_, e_] := p^If[e == 1, 1, NextPrime[e+1, -1]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, f[i, 1], f[i, 1]^precprime(f[i, 2])));}
%Y Cf. A007917, A322448, A365683, A366989, A366991, A366992.
%K nonn,easy,mult
%O 1,2
%A _Amiram Eldar_, Oct 31 2023
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