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%I #19 Mar 13 2018 04:09:32
%S 1,2,6,4,3,10,24,14,15,8,54,40,21,22,96,5,26,9,56,900,16,33,34,35,216,
%T 38,39,160,1764,88,135,46,384,7,250,51,104,486,55,224,57,58,7200,62,
%U 189,32,65,4356,136,69,4900,864,74,375,152,77,6084,640,27,82
%N The partial products of a(n) are the distinct values of the exponential of the von Mangoldt function modified by restricting the divisors to prime divisors (A205957).
%C The partial products of a(n) are A216152(n) which are the distinct values of the 'prime lcm(n)' A205957.
%C Let b(n) denote the nonprime numbers A018252(n).
%C If n = 1 then a(n) = b(n) = 1
%C else if a(n) < b(n) then
%C a(n) is a cototient of consecutive pure powers of primes (A053211),
%C b(n) is a prime power with exponent > 1 (A025475),
%C b(n)/a(n) is a prime root of n-th nontrivial prime power (A025476);
%C else if a(n) > b(n) then
%C b(n) is a number which is neither a prime power nor a semiprime (A102467);
%C else if a(n) = b(n) then
%C a(n) is the product of two distinct primes (A006881).
%H Vincenzo Librandi, <a href="/A216153/b216153.txt">Table of n, a(n) for n = 1..1000</a>
%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/VonMangoldtTransformation">The von Mangoldt Transformation.</a>
%F a(n) = A205957(A018252(n))/A205957(A018252(n-1)) for n > 1, a(1) = 1.
%t A205957[n_] := Exp[-Sum[ MoebiusMu[p]*Log[k/p], {k, 1, n}, {p, FactorInteger[k][[All, 1]]}]]; nonPrime[1] = 1; nonPrime[n_] := Which[k0 = k /. FindRoot[ n + PrimePi[k] == k , {k, n}] // Floor; n+PrimePi[k0] == k0, k0 , n+PrimePi[k0+1] == k0+1, k0+1, n+PrimePi[k0+2] == k0+2, k0+2, True, k0]; a[1] = 1; a[n_] := A205957[nonPrime[n]] / A205957[nonPrime[n-1]]; Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Jun 27 2013 *)
%o (Sage)
%o def A216153(n):
%o if n == 1 : return 1
%o return A205957(A018252(n))/A205957(A018252(n-1))
%Y Cf. A205957, A205959, A216152.
%K nonn,easy
%O 1,2
%A _Peter Luschny_, Sep 02 2012
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