A140706 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

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