A349128 a(n) = phi(A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and phi is Euler totient function.

1, 1, 1, 1, 2, 1, 4, 1, 2, 2, 6, 1, 10, 4, 2, 1, 12, 2, 16, 2, 4, 6, 18, 1, 6, 10, 4, 4, 22, 2, 28, 1, 6, 12, 8, 2, 30, 16, 10, 2, 36, 4, 40, 6, 4, 18, 42, 1, 20, 6, 12, 10, 46, 4, 12, 4, 16, 22, 52, 2, 58, 28, 8, 1, 20, 6, 60, 12, 18, 8, 66, 2, 70, 30, 6, 16, 24, 10, 72, 2, 8, 36, 78, 4, 24, 40, 22, 6, 82, 4, 40

Offset

1,5

Comments

See comments in A349127.

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Index entries for sequences computed from indices in prime factorization.

Formula

Multiplicative with a(2^e) = 1, and for odd primes p, a(p^e) = (q - 1)*q^(e-1), where q = prevprime(p), where prevprime is A151799.

For odd n, a(n) = A349127(n), for even n, a(n) = a(n/2).

For all n >= 1, a(n) = a(2*n) = a(A000265(n)).

For all n >= 1, a(A000040(1+n)) = A006093(n) = A000040(n)-1.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (64/(3*Pi^4)) / Product_{p prime > 2} (1+1/p-q(p)/p^2-q(p)/p^3) = 0.17889586..., where q(p) = prevprime(p) = A151799(p). - Amiram Eldar, Dec 24 2022

Mathematica

f[p_, e_] := If[p == 2, 1, Module[{q = NextPrime[p, -1]}, (q - 1)*q^(e - 1)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 24 2022 *)

Prog

(PARI) A349128(n) = { my(f = factor(n), q); prod(i=1, #f~, if(2==f[i, 1], 1, q = precprime(f[i, 1]-1); (q-1)*(q^(f[i, 2]-1)))); };

Crossrefs

Agrees with A347115, A348045 and A349127 on odd numbers.

Cf. A285702 (odd bisection).

Cf. A000010, A064989, A151799, A349122 (inverse Möbius transform).

Cf. also A003972.

Sequence in context: A181982 A070194 A323300 * A366450 A105584 A072064

Adjacent sequences: A349125 A349126 A349127 * A349129 A349130 A349131

Keyword

nonn,mult

Author

Antti Karttunen, Nov 13 2021

Status

approved