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

Author

Jinyuan Wang, Mar 01 2023

Keywords

• nonn
• more

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)

Examples

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

Links

• Math Overflow, The range of the Euler totient function and multiplication by 28, 2018.
• Michel Marcus, Known results, Aug 11 2023.

Crossrefs

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

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

Programs

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

Formula

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

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