

A065387


a(n) = sigma(n) + phi(n).


38



2, 4, 6, 9, 10, 14, 14, 19, 19, 22, 22, 32, 26, 30, 32, 39, 34, 45, 38, 50, 44, 46, 46, 68, 51, 54, 58, 68, 58, 80, 62, 79, 68, 70, 72, 103, 74, 78, 80, 106, 82, 108, 86, 104, 102, 94, 94, 140, 99, 113, 104, 122, 106, 138, 112, 144, 116, 118, 118, 184, 122, 126, 140
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OFFSET

1,1


COMMENTS

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
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
If n is semiprime, a(n) = 2n+1+ceiling(sqrt(n))floor(sqrt(n)).  Wesley Ivan Hurt, May 05 2015
Atanassov proves that a(n) >= n + A001414(n).  Charles R Greathouse IV, Dec 06 2016


REFERENCES

K. Atanassov, New integer functions, related to ψ and σ functions. IV., Bull. Number Theory Related Topics 12 (1988), pp. 3135.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 149.


LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..10000 (First 1000 terms from T. D. Noe.)
A. Makowski, Aufgaben 339, Elemente der Mathematik 15 (1960), pp. 3940.


FORMULA

a(n) = A000203(n) + A000010(n).
a(n) = A051709(n) + 2n.  N. J. A. Sloane, Jun 12 2004
G.f.: Sum_{k>=1} (mu(k) + 1)*x^k/(1  x^k)^2.  Ilya Gutkovskiy, Sep 29 2017


EXAMPLE

a(10) = 22 because there are 4 coprimes to 10 below 10, the divisors of 10 add up to 18, and 4 + 18 = 22.


MAPLE

with(numtheory); A065387:=n>phi(n) + sigma(n); seq(A065387(n), n=1..100); # Wesley Ivan Hurt, Apr 08 2014


MATHEMATICA

Table[EulerPhi[n] + DivisorSigma[1, n], {n, 65}] (* Alonso del Arte *)
a[n_] := SeriesCoefficient[Sum[(1+MoebiusMu[k])*x^k/(1x^k)^2, {k, 1, n}], {x, 0, n}]; Array[a, 63] (* JeanFrançois Alcover, Sep 29 2017, after Ilya Gutkovskiy *)


PROG

(PARI) for (n=1, 1000, write("b065387.txt", n, " ", sigma(n) + eulerphi(n)) ) \\ Harry J. Smith, Oct 17 2009


CROSSREFS

Cf. A000010, A000203, A065388, A015702, A051709, A011774.
See A292768 for partial sums, A051612 for sigma  phi.
Sequence in context: A184416 A187225 A003661 * A219787 A065388 A157936
Adjacent sequences: A065384 A065385 A065386 * A065388 A065389 A065390


KEYWORD

nonn,easy


AUTHOR

Labos Elemer, Nov 05 2001


STATUS

approved



