A361058 Least totient number k > 1 such that n*k is a nontotient number, or 0 if no such number exists.

0, 0, 30, 0, 10, 0, 2, 0, 10, 110, 22, 0, 2, 22, 6, 0, 2, 0, 2, 0, 54, 22, 10, 0, 2, 22, 22, 212983792, 6

Offset

1,3

Comments

After a(30) which is unknown, the sequence continues: 2, 0, 18, 2, 10, 0, 2, 2, 6, 0, 6, 0, 2, 22, 2, 46, 2, 0, 2, 22, 10, 146068, 6, 0, 10, and a(56) is unknown. - Michel Marcus, Mar 11 2023

When n is in A002202, then n*a(n) is a term of A329872; in other words a(n) is the value k, such that k*a(n) is the least term of A329872 that is divisible by n. - Michel Marcus, Mar 26 2023

a(30) > 2.5*10^10, if it is not 0. - Amiram Eldar, May 07 2023

a(568) <= 2^17*71^13 where 568 = 2^3*71 (so similar to a(652) = 2^4*163^3 where 652 = 2^2*163). - Michel Marcus, May 14 2023

From Michel Marcus, Jun 08 2023: (Start)

Experimentally there are 2 cases: n is a totient value or is a nontotient.

If n is a nontotient, then it is relatively easy to find the titular k.

If n is a totient value, then we see that there are 4 cases:

there are no such k and a(n)=0,

k is known, and by definition k is a totient value.

k is not known but we know a large totient value K for which n*K is nontotient,

k is currently unknown.

For several k or K, n*k are squares of terms of A281187. (End)

Links

Table of n, a(n) for n=1..29.

Math Overflow, The range of the Euler totient function and multiplication by 28, 2018.

Michel Marcus, Known results, Aug 11 2023.

Formula

a(n) = 0 if n is in A301587.

a(A007617(n)) = A350085(n). - Michel Marcus, Apr 08 2023

Example

a(3) = 30 because 30 is in A002202 and 3*30 = 90 is in A007617.

Prog

(PARI) a(n) = if (vecsearch([1, 2, 4, 6, 8, 12, 16, 18, 20, 24], n), return(0)); my(k=2); while (istotient(n*k), k++; while (!istotient(k), k++)); k; \\ Michel Marcus, Mar 08 2023
(PARI) check(n, k) = istotient(k) && !istotient(n*k); \\ Michel Marcus, Apr 05 2023; just for checking

Crossrefs

Cf. A002202 (totient numbers), A007617 (nontotient numbers).

Cf. A281187, A301587, A316665, A329872, A350085, A350086.

Sequence in context: A023926 A022068 A277043 * A198805 A030128 A137348

Adjacent sequences: A361055 A361056 A361057 * A361059 A361060 A361061

Keyword

nonn,more

Author

Jinyuan Wang, Mar 01 2023

Status

approved