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

%I A140706 #22 May 15 2020 08:42:08
%S A140706 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,
%T A140706 11,31,16,18,14,22,12,37,16,22,16,41,15,43,21,25,20,47,16,41,21,30,25,
%U A140706 53,18,38,24,34,26,59,15,61,28,37,32,46,23,67,33,42,27,71,24,73,34,41
%N A140706 A054525 * A014683; a(n) = Sum_{d|n} mu(d)*A014683(n/d).
%C A140706 a(n) = n iff n is prime.
%H A140706 Antti Karttunen, <a href="/A140706/b140706.txt">Table of n, a(n) for n = 1..10000</a>
%F A140706 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.
%F A140706 a(n) = Sum_{d|n} A008683(d)*A014683(n/d), where A008683 is Moebius mu function. - _Antti Karttunen_, Jul 28 2017
%e A140706 a(4) = 1 = (0, -1, 0, 1) dot (1, 3, 4, 4), where (0, -1, 0, 1) = row 4 of triangle A054525.
%p A140706 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
%t A140706 Table[Sum[MoebiusMu[d] (# + Boole@ PrimeQ@ #) &[n/d], {d, Divisors@ n}], {n, 75}] (* _Michael De Vlieger_, Jul 29 2017 *)
%o A140706 (PARI)
%o A140706 A014683(n) = (n+isprime(n));
%o A140706 A140706(n) = sumdiv(n,d,moebius(d)*A014683(n/d)); \\ _Antti Karttunen_, Jul 28 2017
%o A140706 (Python)
%o A140706 from sympy import isprime, mobius, divisors
%o A140706 def a014683(n): return n + isprime(n)
%o A140706 def a140706(n): return sum(mobius(d)*a014683(n//d) for d in divisors(n))
%o A140706 print([a140706(n) for n in range(1,51)]) # _Indranil Ghosh_, Jul 29 2017
%Y A140706 Cf. A008683, A014683, A054525.
%K A140706 nonn
%O A140706 1,2
%A A140706 _Gary W. Adamson_, May 24 2008
%E A140706 More terms from _R. J. Mathar_, Jan 19 2009
%E A140706 Second part added to the name by _Antti Karttunen_, Jul 28 2017