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%I #72 Sep 08 2022 08:45:04
%S 2,4,6,9,10,14,14,19,19,22,22,32,26,30,32,39,34,45,38,50,44,46,46,68,
%T 51,54,58,68,58,80,62,79,68,70,72,103,74,78,80,106,82,108,86,104,102,
%U 94,94,140,99,113,104,122,106,138,112,144,116,118,118,184,122,126,140
%N a(n) = sigma(n) + phi(n).
%C a(n) = 2n for n listed in A008578, the prime numbers at the beginning of the 20th century. When a(n) = a(n + 1), n is probably listed in A066198, numbers n where phi changes as fast as sigma (the only exceptions below 10000 are 2 and 854). - _Alonso del Arte_, Nov 16 2005
%C A. Makowski proved that n is prime if and only if a(n) = n * d(n), where d is A000005. - _Charles R Greathouse IV_, Mar 19 2012
%C If n is semiprime, a(n) = 2n+1+ceiling(sqrt(n))-floor(sqrt(n)). - _Wesley Ivan Hurt_, May 05 2015
%C Atanassov proves that a(n) >= n + A001414(n). - _Charles R Greathouse IV_, Dec 06 2016
%C a(n) = 2*n+1 iff n is square of prime (A001248), a(n) = 2*(n+1) iff n is squarefree semiprime (A006881). - _Bernard Schott_, Feb 09 2020
%D K. Atanassov, New integer functions, related to ψ and σ functions. IV., Bull. Number Theory Related Topics 12 (1988), pp. 31-35.
%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 149.
%H N. J. A. Sloane, <a href="/A065387/b065387.txt">Table of n, a(n) for n = 1..10000</a> (First 1000 terms from T. D. Noe.)
%H A. Makowski, <a href="https://www.e-periodica.ch/digbib/view?pid=edm-001:1960:15#70">Aufgaben 339</a>, Elemente der Mathematik 15 (1960), pp. 39-40.
%F a(n) = A000203(n) + A000010(n).
%F a(n) = A051709(n) + 2n. - _N. J. A. Sloane_, Jun 12 2004
%F G.f.: Sum_{k>=1} (mu(k) + 1)*x^k/(1 - x^k)^2. - _Ilya Gutkovskiy_, Sep 29 2017
%e a(10) = 22 because there are 4 coprimes to 10 below 10, the divisors of 10 add up to 18, and 4 + 18 = 22.
%p with(numtheory); A065387:=n->phi(n) + sigma(n); seq(A065387(n), n=1..100); # _Wesley Ivan Hurt_, Apr 08 2014
%t Table[EulerPhi[n] + DivisorSigma[1,n], {n, 65}] (* _Alonso del Arte_ *)
%t a[n_] := SeriesCoefficient[Sum[(1+MoebiusMu[k])*x^k/(1-x^k)^2, {k, 1, n}], {x, 0, n}]; Array[a, 63] (* _Jean-François Alcover_, Sep 29 2017, after _Ilya Gutkovskiy_ *)
%o (PARI) for (n=1, 1000, write("b065387.txt", n, " ", sigma(n) + eulerphi(n)) ) \\ _Harry J. Smith_, Oct 17 2009
%o (Magma) [DivisorSigma(1,k)+EulerPhi(k):k in [1..65]]; // _Marius A. Burtea_, Feb 09 2020
%Y Cf. A000010, A000203, A065388, A015702, A051709, A011774.
%Y See A292768 for partial sums, A051612 for sigma - phi.
%K nonn,easy
%O 1,1
%A _Labos Elemer_, Nov 05 2001
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