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A155457 a(n) = exp(Lambda(n)), where Lambda(n) is the von Mangoldt function for odd (!) primes.

%I A155457 #10 Apr 06 2025 09:24:50
%S A155457 1,1,3,1,5,1,7,1,3,1,11,1,13,1,1,1,17,1,19,1,1,1,23,1,5,1,3,1,29,1,31,
%T A155457 1,1,1,1,1,37,1,1,1,41,1,43,1,1,1,47,1,7,1,1,1,53,1,1,1,1,1,59,1,61,1,
%U A155457 1,1,1,1,67,1,1,1,71,1,73,1,1,1,1,1,79,1,3,1,83,1,1,1,1,1,89
%N A155457 a(n) = exp(Lambda(n)), where Lambda(n) is the von Mangoldt function for odd (!) primes.
%C A155457 a(n) = p if n = p^k and p odd prime, k >= 1, otherwise 1.
%D A155457 Tom M. Apostol, Introduction to analytic number theory, Springer-Verlag, 1976.
%H A155457 Manjul Bhargava, <a href="http://dx.doi.org/10.2307/2695734">The factorial function and generalizations</a>, Amer. Math. Monthly, 107 (Nov. 2000), 783-799.
%H A155457 Angelo B. Mingarelli, <a href="https://nntdm.net/volume-19-2013/number-4/43-76/">Abstract factorials</a>, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, No. 4, 43-76, (page 44).
%H A155457 Wikipedia, <a href="http://en.wikipedia.org/wiki/Von_Mangoldt_function">Von Mangoldt function</a>
%F A155457 a(n) = 1 + Sum_{k=3..n} (k-1)*A010051(k)*(floor(k^n/n)-floor((k^n -1)/n)). - _Anthony Browne_, Jun 16 2016
%e A155457 a(8) = 1 because 8 = 2^3 is not the power of an odd prime, a(49) = 7 because 49 = 7^2.
%p A155457 a := proc(n) local lcm; lcm := n -> ilcm(seq(i,i = 1..n)); if type(n,even) then 1 else lcm(n)/lcm(n-1) fi end;
%t A155457 a[n_] := If[IntegerQ[Log[2, n]], 1, Exp[MangoldtLambda[n]]]; Table[a[n], {n, 1, 89}] (* _Jean-François Alcover_, Jan 27 2014 *)
%Y A155457 Cf. A014963, A010051.
%K A155457 nonn
%O A155457 1,3
%A A155457 _Peter Luschny_, Jan 22 2009, Jan 25 2009