A140668 a(n) = n + A140664(n).

2, 0, 0, 4, 0, 7, 0, 8, 9, 11, 0, 12, 0, 15, 16, 16, 0, 18, 0, 20, 22, 23, 0, 24, 25, 27, 27, 28, 0, 29, 0, 32, 34, 35, 36, 36, 0, 39, 40, 40, 0, 41, 0, 44, 45, 47, 0, 48, 49, 50, 52, 52, 0, 54, 56, 56, 58, 59, 0, 60, 0, 63, 63, 64, 66, 65, 0, 68, 70, 69, 0, 72, 0, 75, 75, 76, 78, 77, 0, 80, 81, 83, 0, 84, 86, 87, 88, 88, 0

Offset: 1

Author

Gary W. Adamson and Mats Granvik, May 20 2008

Keywords

• nonn

Comments

For n>1, a(n) = n iff mu(n) = 0; 0 if n is prime; and (n+1) iff mu(n) = 1.

Examples

			a(4) = 4 = 4 + A140664(4) = 4 + 0.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A140664, A008683.

Programs

Mathematica

Table[Exp[MangoldtLambda[n]]*MoebiusMu[n] + n, {n, 100}] (* G. C. Greubel, Feb 13 2019 *)

PARI

A014963(n) = ispower(n, , &n); if(isprime(n), n, 1); \\ From A014963
A140664(n) = (moebius(n)*A014963(n));
A140668(n) = (n+A140664(n)); \\ Antti Karttunen, Feb 13 2019

Sage

def A140668(n): return simplify(exp(add(moebius(d)*log(n/d) for d in divisors(n))))*moebius(n) + n
[A140668(n) for n in (1..100)] # G. C. Greubel, Feb 13 2019

Formula

a(n) = n + A140664(n), where A140664 = (1, -2, -3, 0, -5, 1, -7, 0, 0, 1,...)

Extensions

More terms from Antti Karttunen, Feb 13 2019