A063754 - OEIS

A063754

Dirichlet convolution of totient and cototient.

0, 1, 1, 3, 1, 7, 1, 8, 5, 11, 1, 20, 1, 15, 13, 20, 1, 31, 1, 32, 17, 23, 1, 52, 9, 27, 21, 44, 1, 71, 1, 48, 25, 35, 21, 88, 1, 39, 29, 84, 1, 99, 1, 68, 61, 47, 1, 128, 13, 83, 37, 80, 1, 123, 29, 116, 41, 59, 1, 200, 1, 63, 81, 112, 33, 155, 1, 104, 49, 159, 1, 228, 1, 75, 101

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

a(n) = 1 if and only if n is prime. - Robert Israel, Feb 04 2018

a(n) = n+1 if and only if n = 2*p with p an odd prime (A100484 \ {4}). - Bernard Schott, Jun 19 2023

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = Sum_{d|n} A000010(d)*A051953(n/d).

From Richard L. Ollerton, May 06 2021: (Start)

a(n) = Sum_{k=1..n} A051953(gcd(n,k)).

a(n) = Sum_{k=1..n} A051953(n/gcd(n,k))*A000010(gcd(n,k))/A000010(n/gcd(n,k)).

a(n) = A018804(n) - A029935(n). (End)

Sum_{k=1..n} a(k) ~ (1/(2*zeta(2)))*(1 - 1/zeta(2)) * n^2 * (log(n) + 2*gamma - 1/2 - ((zeta(2)-2)/(zeta(2)-1))*(zeta'(2)/zeta(2))), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 11 2024

EXAMPLE

n = 24: divisors = {1, 2, 3, 4, 6, 8, 12, 24}, d-phi(d) = {0, 1, 1, 2, 4, 4, 8, 16}, phi(n/d) = {8, 4, 4, 2, 2, 2, 1, 1}, products = {0, 4, 4, 4, 8, 8, 8, 16}, a(24) = 52.

MAPLE

f:= n -> add(numtheory:-phi(d)*(n/d - numtheory:-phi(n/d)), d=numtheory:-divisors(n)):

map(f, [$1..100]); # Robert Israel, Feb 04 2018

MATHEMATICA

f1[p_, e_] := (e*(p - 1)/p + 1)*p^e; f2[p_, e_] := (e+1)*(p^e - p^(e-1)) - (e-1)*(p^(e-1) - p^(e-2)); a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; a[1] = 0; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

PROG

(PARI) a(n) = sumdiv(n, d, eulerphi(d)*(n/d - eulerphi(n/d))); \\ Michel Marcus, Feb 05 2018

CROSSREFS

Cf. A000010, A018804, A029935, A051953, A100484.

Cf. A001620, A013661, A306016.

Sequence in context: A140435 A194181 A277934 * A163117 A099749 A210442

Adjacent sequences: A063751 A063752 A063753 * A063755 A063756 A063757

KEYWORD

easy,nonn,look

AUTHOR

Labos Elemer, Aug 14 2001

EXTENSIONS

Offset corrected by Robert Israel, Feb 04 2018

STATUS

approved