A354190 -id:A354190 - cp's OEIS frontend

A354102 a(n) = phi(A267099(n)), where A267099 is fully multiplicative involution swapping the positions of 4k+1 and 4k+3 primes, and phi is Euler totient function.

1, 1, 4, 2, 2, 4, 12, 4, 20, 2, 16, 8, 6, 12, 8, 8, 10, 20, 28, 4, 48, 16, 36, 16, 6, 6, 100, 24, 18, 8, 40, 16, 64, 10, 24, 40, 22, 28, 24, 8, 30, 48, 52, 32, 40, 36, 60, 32, 156, 6, 40, 12, 42, 100, 32, 48, 112, 18, 72, 16, 46, 40, 240, 32, 12, 64, 88, 20, 144, 24, 96, 80, 58, 22, 24, 56, 192, 24, 100, 16, 500

Offset: 1

Antti Karttunen, May 18 2022

Antti Karttunen, Table of n, a(n) for n = 1..19683

Möbius transform of A267099.
Cf. A000720, A008683, A267101, A354101, A354103, A354104 (Dirichlet inverse), A354105 (sum with it), A354106, A354107 (a(n) mod 4), A354190, A354191.
Coincides with A000010 on A354189.

A354102(n) = eulerphi(A267099(n)); \\ Uses the program given in A267099.

Multiplicative with a(p^e) = (q-1) * q^(e-1), where q = A267101(A000720(p)).
a(n) = A000010(A267099(n)).
a(n) = Sum_{d|n} A008683(n/d) * A267099(d).
a(n) = A354101(n) + A000010(n) = A354190(n) - A354191(n).
For all n >= 0, a(4n+2) = a(2n+1).

A354191 a(n) = phi(A267099(sigma(n))) - phi(A267099(n)), where A267099 is fully multiplicative involution swapping the positions of 4k+1 and 4k+3 primes.

Original entry on oeis.org

0, 3, -2, 10, 2, 4, -8, 4, -14, 18, -8, 16, 6, 4, 8, 32, 10, 4, -24, 44, -32, 24, -20, 0, 34, 42, -92, 24, -10, 72, -24, 224, -32, 90, 8, 32, 6, -12, 24, 32, 18, 16, -20, 64, -16, 44, -28, 48, -44, 154, 40, 144, 58, -68, 48, -16, -96, 22, -56, 176, -6, 24, -216, 116, 84, 96, -68, 220, -80, 136, -16, -32, -36, 90

Offset: 1

Antti Karttunen, May 19 2022

sign

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A000010, A000203, A267099, A354102, A354106 (positions of 0's), A354190.

Cf. also A353636.

A354191(n) = (eulerphi(A267099(sigma(n))) - eulerphi(A267099(n))); \\ Uses the program given in A267099.

a(n) = A354190(n) - A354102(n).

cp's OEIS Frontend

A354102 a(n) = phi(A267099(n)), where A267099 is fully multiplicative involution swapping the positions of 4k+1 and 4k+3 primes, and phi is Euler totient function.

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