A354106 -id:A354106 - 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).

A354189 Numbers k for which phi(A267099(k)) is equal to phi(k), where A267099 is fully multiplicative involution swapping the positions of 4k+1 and 4k+3 primes, and phi is Euler totient function.

Original entry on oeis.org

1, 2, 4, 8, 15, 16, 30, 32, 35, 39, 60, 64, 70, 78, 91, 120, 128, 140, 156, 182, 187, 225, 240, 256, 280, 312, 364, 374, 450, 480, 512, 551, 560, 624, 728, 748, 851, 900, 960, 1024, 1102, 1120, 1248, 1271, 1365, 1456, 1496, 1702, 1800, 1920, 2048, 2204, 2240, 2279, 2496, 2542, 2730, 2747, 2759, 2805, 2867, 2912, 2992

Offset: 1

Antti Karttunen, May 19 2022

nonn

Not a subsequence of A072202. The first term that is included here, but not in that sequence is 69037, as A000010(69037) = A354102(69037) = 62400, although 69037 = 17*31*131. See A354194.

Antti Karttunen, Table of n, a(n) for n = 1..5200

Positions of zeros in A354101.
Subsequence of A354109.
Cf. A000079, A354192, A354194 (subsequences), A354188 (characteristic function).
Cf. A000010, A072202, A267099, A354102, A354106.

A354188(n) = (eulerphi(A267099(n)) == eulerphi(n)); \\ Uses the program given in A267099.
isA354189(n) = A354188(n);

{k | A354102(k) == A000010(k)}.

A354107 a(n) = A354102(n) mod 4.

Original entry on oeis.org

1, 1, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0

Offset: 1

Antti Karttunen, May 18 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A010873, A267099, A353768, A354102.

PARI
```
A354106(n) = (A354102(n)%4);
```

a(n) = A010873(A354102(n)).

a(n) = A353768(A267099(n)).

A354362 Intersection of A228058 and A260021.

Original entry on oeis.org

45, 49005, 597861, 715473, 1538757, 1891593, 1893213, 2714877, 3067713, 3890997, 4126221, 4479057, 5302341, 5465313, 5793525, 5890437, 6008013, 6478461, 6596073, 6882525, 7184133, 7419357, 8595477, 9771597, 10712493, 11300553, 11771001, 11888613, 12123837, 12947121, 13535181, 14240853, 15887421, 16240257, 17181153

Offset: 1

Antti Karttunen, May 24 2022

nonn

Cf. A228058, A260021.
Subsequence of A353679.
Cf. also A354106.

isA354362(n) = (isA228058(n) && isA260021(n)); \\ Uses the programs from A228058 and A260021.

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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A354189 Numbers k for which phi(A267099(k)) is equal to phi(k), 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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A354107 a(n) = A354102(n) mod 4.

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A354362 Intersection of A228058 and A260021.

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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.

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