A140706 A054525 * A014683; a(n) = Sum_{d|n} mu(d)*A014683(n/d).

1, 2, 3, 1, 5, 0, 7, 4, 5, 2, 11, 5, 13, 4, 6, 8, 17, 7, 19, 9, 10, 8, 23, 8, 19, 10, 18, 13, 29, 11, 31, 16, 18, 14, 22, 12, 37, 16, 22, 16, 41, 15, 43, 21, 25, 20, 47, 16, 41, 21, 30, 25, 53, 18, 38, 24, 34, 26, 59, 15, 61, 28, 37, 32, 46, 23, 67, 33, 42, 27, 71, 24, 73, 34, 41

Offset

1,2

Comments

a(n) = n iff n is prime.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Formula

Möbius transform of A014683: (1, 3, 4, 4, 6, 6, 8, 8, 9, 10, ...); where A014683(n) = n if n is not prime; but (n+1) if n is prime.

a(n) = Sum_{d|n} A008683(d)*A014683(n/d), where A008683 is Moebius mu function. - Antti Karttunen, Jul 28 2017

Example

a(4) = 1 = (0, -1, 0, 1) dot (1, 3, 4, 4), where (0, -1, 0, 1) = row 4 of triangle A054525.

Maple

read("transforms") : A014683 := proc(n) if isprime(n) then 1+n; else n; fi; end: a014683 := [seq(A014683(n), n=1..150)] ; a140706 := MOBIUS(a014683) ; for i from 1 to nops(a140706) do printf("%d, ", op(i, a140706)) ; od: # R. J. Mathar, Jan 19 2009

Mathematica

Table[Sum[MoebiusMu[d] (# + Boole@ PrimeQ@ #) &[n/d], {d, Divisors@ n}], {n, 75}] (* Michael De Vlieger, Jul 29 2017 *)

Prog

(PARI)
A014683(n) = (n+isprime(n));
A140706(n) = sumdiv(n, d, moebius(d)*A014683(n/d)); \\ Antti Karttunen, Jul 28 2017
(Python)
from sympy import isprime, mobius, divisors
def a014683(n): return n + isprime(n)
def a140706(n): return sum(mobius(d)*a014683(n//d) for d in divisors(n))
print([a140706(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 29 2017

Crossrefs

Cf. A008683, A014683, A054525.

Sequence in context: A080063 A187680 A328731 * A200068 A139764 A371356

Adjacent sequences: A140703 A140704 A140705 * A140707 A140708 A140709

Keyword

nonn

Author

Gary W. Adamson, May 24 2008

Extensions

More terms from R. J. Mathar, Jan 19 2009

Second part added to the name by Antti Karttunen, Jul 28 2017

Status

approved