A329872 -id:A329872 - cp's OEIS frontend

A361686 a(n) is the least totient divisor of A329872(n), where A329872 are nontotients (A005277) that are the product of two totients (A002202).

22, 22, 10, 46, 58, 46, 78, 82, 58, 46, 102, 22, 106, 82, 46, 138, 106, 82, 166, 172, 178, 190, 106, 226, 82, 166, 238, 172, 178, 22, 106, 262, 190, 282, 22, 106, 310, 316, 226, 82, 166, 238, 172, 346, 46, 178, 22, 358, 22, 10, 366, 106, 262, 382, 82, 388, 58, 22, 22, 46, 418

Offset: 1

Michel Marcus, Mar 29 2023

nonn

Let k be the least instance a(k) = m, then A329872(k) = m*A361058(m). For instance a(3)=10, and A329872(3) = 1100 = 10*110 = 10*A361058(10).

Can we get a(k)=30 or a(k)=52 (see A361058)?

			a(3)=10 because A329872(3)=1100 which can be expressed as 1*1100, 2*550, 4*275, 5*220, 10*110, ... where 10*110 is the first case where both factors are nontotients.

Michel Marcus, Table of n, a(n) for n = 1..7280

Cf. A002202, A005277, A329872, A361058.

is(n) = if(!istotient(n), my(v=divisors(n)); for(i=1, (1+#v)\2, if(istotient(v[i])&&istotient(n/v[i]), return(1))); 0); \\ A329872
lista(nn) = for (n=1, nn, if (is(n), my(d=divisors(n)); for (i=1, (1+#d)\2, if (istotient(d[i]) && istotient(n/d[i]), print1(d[i], ", "); break););););

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

Original entry on oeis.org

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

Jinyuan Wang, Mar 01 2023

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)

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

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

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

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

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

check(n, k) = istotient(k) && !istotient(n*k); \\ Michel Marcus, Apr 05 2023; just for checking

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

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

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A361686 a(n) is the least totient divisor of A329872(n), where A329872 are nontotients (A005277) that are the product of two totients (A002202).

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

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