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A063754 Dirichlet convolution of totient and cototient. 1
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
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
Sequence in context: A140435 A194181 A277934 * A163117 A099749 A210442
KEYWORD
easy,nonn,look
AUTHOR
Labos Elemer, Aug 14 2001
EXTENSIONS
Offset corrected by Robert Israel, Feb 04 2018
STATUS
approved

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Last modified March 19 01:22 EDT 2024. Contains 370952 sequences. (Running on oeis4.)