

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
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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 nth 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(s1) * (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(s1) * (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 Moebiustransform.]
a(n) = Sum_{dn, d<n} A008683(n/d)*A001065(d).
(End)


EXAMPLE

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


MATHEMATICA

Table[DivisorSigma[1, n]+EulerPhi[n]2n, {n, 80}] (* Harvey P. Dale, Apr 08 2015 *)


PROG

(PARI) a(n)=sigma(n)+eulerphi(n)2*n \\ Charles R Greathouse IV, Jul 05 2013
(PARI) A051709(n) = sumdiv(n, d, (d<n)*moebius(n/d)*(sigma(d)d)); \\ Antti Karttunen, Mar 02 2018


CROSSREFS

Cf. A000010, A000203, A001065, A001248, A005843, A006881, A051612, A051953, A065387, A072780, A228498 (= a(n^2)), A297159, A324048, A344994, A344995, A344996, A345048, A345054.
Cf. A278373 (range of this sequence), A056996 (numbers not present).
Cf. also A344753, A345001 (analogous sequences).
Sequence in context: A197117 A343879 A275387 * A318326 A329646 A293813
Adjacent sequences: A051706 A051707 A051708 * A051710 A051711 A051712


KEYWORD

nonn


AUTHOR

Jud McCranie and Carlos Rivera


STATUS

approved



