cp's OEIS Frontend

Numbers k such that A340388(k) is not the smallest number whose prime factors are all congruent to 1 modulo 4 and with exactly k divisors.

Despite being an analog of A072066, this sequence seems to be considerably sparser than A072066. What's the reason for that?

All powers of 2 that are greater than or equal to 16 are here. All numbers of the form 3 * 2^e with e >= 8 are here.

All powers of 3 that are greater than or equal to 3^15 = 14348907 are here. For example, we have A340388(3^15) = (5 * 13 * 17 * 29 * ... * 113 * 137)^2, while a(3^15) <= (5^4 * 13 * 17 * 29 * .. * 113)^2, so 3^15 is a term. Apparently 3^15 is the smallest odd term in this sequence.

Similarly, let q be a prime, then all powers of q that are greater than or equal to q^(N+1) are here, where N is the number of primes congruent to 1 modulo 4 below 5^q. It seems that q^(N+1) is the smallest q-rough term in this sequence.

{b(n)} is an analog of A037019 and of A340388: all prime factors of b(n) are all congruent to 1 modulo 6 and b(n) has exactly n divisors, so A002324(b(n)) = n. By definition we have A343771(n) <= b(n), and it seems that the equality holds for most n. This sequence lists the exceptions.

Since {b(n)} agrees with A343771(n) for most n, it cannot have its own entry.

Let q be a prime, then q^e is here if and only if e >= N+1, where N is the number of primes congruent to 1 modulo 6 below 7^q (N = 6, 32, 958, ... for q = 2, 3, 5, ...).

Proof: p_1 < p_2 < ... be the primes congruent to 1 modulo 6. Suppose that A343771(q^e) = (p_1)^(q^(m_1)-1) * (p_2)^(q^(m_2)-1) * ... * (p_r)^(q^(m_r)-1) with r <= e, m_1 >= m_2 >= ... >= m_r. If m_1 >= 2, then r < e, so we can substitute (p_1)^(q^(m_1)-1) with (p_1)^(q^(m_1-1)-1) * (p_{r+1})^(q-1), which a smaller number with exactly q^e divisors, a contradiction. So we have m_1 = 1, namely A343771(q^e) = b(q^e). On the other hand, if e >= N+1, then A343771(q^e) <= (p_1)^(q^2-1) * (p_2)^(q-1) * ... * (p_{e-1})^(q-1) < b(q^e).

It seems that q^(N+1) is the smallest q-rough term in this sequence.