A378641 -id:A378641 - cp's OEIS frontend

A378637 Largest m <= n such that phi(m) divides n, where phi is the Euler totient function (A000010).

1, 2, 2, 4, 2, 6, 2, 8, 2, 6, 2, 12, 2, 6, 2, 16, 2, 18, 2, 12, 2, 6, 2, 24, 2, 6, 2, 12, 2, 22, 2, 32, 2, 6, 2, 36, 2, 6, 2, 33, 2, 18, 2, 23, 2, 6, 2, 48, 2, 22, 2, 12, 2, 54, 2, 30, 2, 6, 2, 50, 2, 6, 2, 64, 2, 46, 2, 12, 2, 22, 2, 72, 2, 6, 2, 12, 2, 18, 2, 75

Offset: 1

Paolo Xausa, Dec 03 2024

Paolo Xausa, Table of n, a(n) for n = 1..10000

Right border of A378636.

Cf. A000010, A319048, A378641.

A378637[n_] := If[OddQ[n] && n > 2, 2, Module[{m = n}, While[!Divisible[n, EulerPhi[m]], m--]; m]];
Array[A378637, 100]

a(n) = my(m=n); while (n % eulerphi(m), m--); m; \\ Michel Marcus, Dec 05 2024

a(2*k+1) = 2, for k >= 1.

A378642 Number of numbers m <= n such that phi(m) does not divide n, where phi is the Euler totient function (A000010).

Original entry on oeis.org

0, 0, 1, 0, 3, 1, 5, 1, 7, 5, 9, 1, 11, 9, 13, 5, 15, 9, 17, 10, 19, 17, 21, 5, 23, 21, 25, 19, 27, 19, 29, 16, 31, 29, 33, 16, 35, 33, 37, 22, 39, 33, 41, 34, 43, 41, 45, 16, 47, 43, 49, 43, 51, 41, 53, 41, 55, 53, 57, 34, 59, 57, 61, 42, 63, 55, 65, 59, 67, 63

Offset: 1

Paolo Xausa, Dec 05 2024

Paolo Xausa, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A069932, A378637, A378641.

Table[Count[Divisible[n, #[[;;n]]], False], {n, Length[#]}] & [EulerPhi[Range[100]]]

a(n) = n - sumdiv(n, d, #select(x -> x<=n, invphi(d))); \\ Amiram Eldar, Dec 10 2024, using Max Alekseyev's invphi.gp

a(n) = n - A069932(n).

A378640 Smallest m such that phi(m) does not divide n, where phi is the Euler totient function (A000010).

Original entry on oeis.org

3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 11, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 11, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 11, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 11, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 15, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 11, 3, 5, 3, 7, 3, 5, 3, 7, 3, 5, 3, 11, 3, 5

Offset: 1

Paolo Xausa, Dec 05 2024

Up to n = 10^7 the distinct terms of the sequence (which are also the record values) are {3, 5, 7, 11, 15, 17, 19, 23, 29, 47, 51, 53}. Is this A076245 (for n >= 2)?
First differs from A095366 at n = 60.
It appears that a(n) = A095366(n) except when n = 60*(2*k + 1), with k >= 0, where a(n) = 15 while A095366(n) = 17.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000010, A076245, A095366, A378638, A378641, A378642.

A378640[n_] := If[OddQ[n], 3, Module[{m = 4}, While[Divisible[n, EulerPhi[++m]]]; m]];
Array[A378640, 100]

a(n) = 3 if n is odd.

cp's OEIS Frontend

A378637 Largest m <= n such that phi(m) divides n, where phi is the Euler totient function (A000010).

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A378642 Number of numbers m <= n such that phi(m) does not divide n, where phi is the Euler totient function (A000010).

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A378640 Smallest m such that phi(m) does not divide n, where phi is the Euler totient function (A000010).

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