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A065395 Commutator of sigma and phi functions.

%I A065395 #32 Dec 18 2024 22:41:46
%S A065395 0,-1,1,-3,5,-1,8,-1,0,1,14,-5,22,4,7,-15,25,-12,31,3,12,6,28,-1,12,
%T A065395 16,23,4,48,-9,56,-5,26,13,44,-44,73,23,36,7,78,-4,76,18,36,12,56,-29,
%U A065395 60,-18,39,18,80,7,66,28,59,32,74,-17,138,40,43,-63,100,-6
%N A065395 Commutator of sigma and phi functions.
%C A065395 Golomb (1993) proved that the terms are both positive and negative infinitely often. - _Amiram Eldar_, Feb 27 2024
%D A065395 Solomon W. Golomb, Equality among number-theoretic functions, Abstracts Amer. Math. Soc., Vol. 14 (1993), pp. 415-416.
%H A065395 Harry J. Smith, <a href="/A065395/b065395.txt">Table of n, a(n) for n = 1..1000</a>
%H A065395 Jean-Marie De Koninck and Florian Luca, <a href="https://eudml.org/doc/284003">On the composition of the Euler function and the sum of divisors function</a>, Colloquium Mathematicum, Vol. 108, No. 1 (2007), pp. 31-51.
%H A065395 Solomon W. Golomb, <a href="/A006872/a006872_1.pdf">Equality among number-theoretic functions</a>, Unpublished manuscript. (Annotated scanned copy)
%F A065395 a(n) = sigma(phi(n)) - phi(sigma(n)) = A000203(A000010(n)) - A000010(A000203(n)).
%F A065395 a(n) = A062402(n) - A062401(n). - _Amiram Eldar_, Feb 27 2024
%e A065395 n = 13: sigma(13) = 14, phi(14) = 6, phi(13) = 12, sigma(12) = 28, a(13) = 28-6 = 22.
%p A065395 with(numtheory); A065395:=n->sigma(phi(n))-phi(sigma(n)); seq(A065395(n), n=1..100); # _Wesley Ivan Hurt_, Dec 26 2013
%t A065395 Table[DivisorSigma[1, EulerPhi[n]] - EulerPhi[DivisorSigma[1, n]], {n, 100}] (* _T. D. Noe_, Nov 04 2013 *)
%o A065395 (PARI) a(n) = { sigma(eulerphi(n)) - eulerphi(sigma(n)) } \\ _Harry J. Smith_, Oct 18 2009
%o A065395 (Magma) [DivisorSigma(1, EulerPhi(n))-EulerPhi(DivisorSigma(1, n)): n in [1..70]]; // _Bruno Berselli_, Oct 20 2015
%Y A065395 Cf. A000010, A000203, A033632 (positions of 0's), A062401, A062402.
%K A065395 sign,easy
%O A065395 1,4
%A A065395 _Labos Elemer_, Nov 05 2001