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A055654 Difference between n and the result of "Phi-summation" over unitary divisors of n.

%I A055654 #27 Oct 04 2024 10:02:05
%S A055654 0,0,0,1,0,0,0,3,2,0,0,3,0,0,0,7,0,4,0,5,0,0,0,9,4,0,8,7,0,0,0,15,0,0,
%T A055654 0,15,0,0,0,15,0,0,0,11,10,0,0,21,6,8,0,13,0,16,0,21,0,0,0,15,0,0,14,
%U A055654 31,0,0,0,17,0,0,0,37,0,0,12,19,0,0,0,35,26,0,0,21,0,0,0,33,0,20,0,23
%N A055654 Difference between n and the result of "Phi-summation" over unitary divisors of n.
%C A055654 Squarefree numbers are roots of a(n)=0 equation, while Min n for which a(n)=k is k^2. See also A000188, A008833.
%H A055654 Reinhard Zumkeller, <a href="/A055654/b055654.txt">Table of n, a(n) for n = 1..10000</a>
%F A055654 a(n) = n - Sum_{u|n, gcd(u,n/u) = 1} phi(u), i.e. when u is a unitary divisor of n.
%F A055654 a(n) = n - A055653(n). - _Sean A. Irvine_, Mar 30 2022
%F A055654 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 - A065465 = 0.11848616... . - _Amiram Eldar_, Oct 04 2024
%t A055654 Table[n - DivisorSum[n, EulerPhi[#] &, CoprimeQ[#, n/#] &], {n, 92}] (* _Michael De Vlieger_, Oct 26 2017 *)
%t A055654 f[p_, e_] := p^e - p^(e-1) + 1; a[1] = 0; a[n_] := n - Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 04 2024 *)
%o A055654 (Haskell)
%o A055654 a055654 n = a055654_list !! (n-1)
%o A055654 a055654_list = zipWith (-) [1..] a055653_list
%o A055654 -- _Reinhard Zumkeller_, Mar 11 2012
%o A055654 (PARI) a(n) = n - sumdiv(n, d, if (gcd(d, n/d)==1, eulerphi(d))); \\ _Michel Marcus_, Oct 27 2017
%o A055654 (PARI) a(n) = {my(f = factor(n)); n - prod(k = 1, #f~, f[k,1]^f[k,2] - f[k,1]^(f[k,2] - 1) + 1);} \\ _Amiram Eldar_, Oct 04 2024
%Y A055654 Cf. A000188, A008833, A055653, A053570, A053571, A055631, A055632, A055653, A065465.
%K A055654 nonn,easy
%O A055654 1,8
%A A055654 _Labos Elemer_, Jun 07 2000