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 A051709 a(n) = sigma(n) + phi(n) - 2n. 17
 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, 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, 2, 14, 31, 2, 32, 0, 22, 2, 28, 0, 75, 0, 2, 14, 24, 2, 36, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Sigma is the sum of divisors (A000203), and phi is the Euler totient function (A000010). - Michael B. Porter, Jul 05 2013 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 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 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 LINKS Antti Karttunen, Table of n, a(n) for n = 1..65537 (First 1000 terms from T. D. Noe.) Carlos Rivera, Puzzle 76. z(n)=sigma(n) + phi(n) - 2n, The Prime Puzzles and Problems Connection. FORMULA Dirichlet g.f.: zeta(s-1) * (zeta(s) - 2 + 1/zeta(s)). - T. D. Noe, Aug 01 2002 From Antti Karttunen, Mar 02 2018: (Start) a(n) = A001065(n) - A051953(n).  [Difference between the sum of proper divisors of n and their Moebius-transform.] a(n) = -Sum_{d|n, d

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Last modified May 25 10:05 EDT 2022. Contains 354066 sequences. (Running on oeis4.)