A051709 - OEIS

%I #58 Dec 03 2023 05:03:18

%S 0,0,0,1,0,2,0,3,1,2,0,8,0,2,2,7,0,9,0,10,2,2,0,20,1,2,4,12,0,20,0,15,

%T 2,2,2,31,0,2,2,26,0,24,0,16,12,2,0,44,1,13,2,18,0,30,2,32,2,2,0,64,0,

%U 2,14,31,2,32,0,22,2,28,0,75,0,2,14,24,2,36,0

%N a(n) = sigma(n) + phi(n) - 2n.

%C Sigma is the sum of divisors (A000203), and phi is the Euler totient function (A000010). - _Michael B. Porter_, Jul 05 2013

%C Because sigma and phi are multiplicative functions, it is easy to show that (1) if a(n)=0, then n is prime or 1 and (2) if a(n)=2, then n is the product of two distinct prime numbers. Note that a(n) is the n-th term of the Dirichlet series whose generating function is given below. Using the generating function, it is theoretically possible to compute a(n). Hence a(n)=0 could be used as a primality test and a(n)=2 could be used as a test for membership in P2 (A006881). - _T. D. Noe_, Aug 01 2002

%C It appears that a(n) - A002033(n) = zeta(s-1) * (zeta(s) - 2 + 1/zeta(s)) + 1/(zeta(s)-2). - _Eric Desbiaux_, Jul 04 2013

%C a(n) = 1 if and only if n = prime(k)^2 (n is in A001248). It seems that a(n) = k has only finitely many solutions for k >= 3. - _Jianing Song_, Jun 27 2021

%H Antti Karttunen, <a href="/A051709/b051709.txt">Table of n, a(n) for n = 1..65537</a> (First 1000 terms from T. D. Noe.)

%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_076.htm">Puzzle 76. z(n)=sigma(n) + phi(n) - 2n</a>, The Prime Puzzles and Problems Connection.

%F Dirichlet g.f.: zeta(s-1) * (zeta(s) - 2 + 1/zeta(s)). - _T. D. Noe_, Aug 01 2002

%F From _Antti Karttunen_, Mar 02 2018: (Start)

%F a(n) = A001065(n) - A051953(n).  [Difference between the sum of proper divisors of n and their Moebius-transform.]

%F a(n) = -Sum_{d|n, d<n} A008683(n/d)*A001065(d). (End)

%F Sum_{k=1..n} a(k) = (3/(Pi^2) + Pi^2/12 - 1) * n^2 + O(n*log(n)). - _Amiram Eldar_, Dec 03 2023

%e a(5) = sigma(5) + phi(5) - 2*5 = 6 + 4 - 10 = 0.

%t Table[DivisorSigma[1,n]+EulerPhi[n]-2n,{n,80}] (* _Harvey P. Dale_, Apr 08 2015 *)

%o (PARI) a(n)=sigma(n)+eulerphi(n)-2*n \\ _Charles R Greathouse IV_, Jul 05 2013

%o (PARI) A051709(n) = -sumdiv(n,d,(d<n)*moebius(n/d)*(sigma(d)-d)); \\ _Antti Karttunen_, Mar 02 2018

%Y Cf. A000010, A000203, A001065, A001248, A005843, A006881, A051612, A051953, A065387, A072780, A228498 (= a(n^2)), A297159, A324048, A344994, A344995, A344996, A345048, A345054.

%Y Cf. A278373 (range of this sequence), A056996 (numbers not present).

%Y Cf. also A344753, A345001 (analogous sequences).

%K nonn

%O 1,6

%A _Jud McCranie_ and _Carlos Rivera_