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

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, 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, 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

Offset: 1

Author

Peter Luschny, Jan 22 2009, Jan 25 2009

Keywords

• nonn

Comments

a(n) = p if n = p^k and p odd prime, k >= 1, otherwise 1.

Examples

			a(8) = 1 because 8 = 2^3 is not the power of an odd prime, a(49) = 7 because 49 = 7^2.
		

References

• Tom M. Apostol, Introduction to analytic number theory, Springer-Verlag, 1976.

Links

• Manjul Bhargava, The factorial function and generalizations, Amer. Math. Monthly, 107 (Nov. 2000), 783-799.
• Angelo B. Mingarelli, Abstract factorials, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, No. 4, 43-76, (page 44).
• Wikipedia, Von Mangoldt function

Crossrefs

Cf. A014963, A010051.

Programs

Maple

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;

Mathematica

a[n_] := If[IntegerQ[Log[2, n]], 1, Exp[MangoldtLambda[n]]]; Table[a[n], {n, 1, 89}] (* Jean-François Alcover, Jan 27 2014 *)

Formula

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