A355919 Let n = p_1*p_2*...*p_k be the prime factorization of n, with the primes sorted in descending order; let b(n) = 7^(p_1 - 1)*13^(p_2 - 1)*19^(p_3 - 1)*...*A002476(k)^(p_k - 1). Sequence lists m such that b(m) > A343771(m).

128, 256, 512, 1024, 2048, 3072, 4096, 6144, 8192, 12288, 16384, 24576, 32768, 49152, 65536, 73728, 98304, 131072, 147456, 196608, 262144, 294912

Offset: 1

Author

Jianing Song, Jul 20 2022

Keywords

• nonn
• more

Comments

{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.

Examples

			128 is a term since b(128) = 7 * 13 * 19 * 31 * 37 * 43 * 61 > A343771(128) = 7^3 * 13 * 19 * 31 * 37 * 43.
		

Crossrefs

Analog of A072066 and A340624.

Cf. A343771, A002476, A037019, A340388, A002324.

Programs

PARI

b(n) = my(f=factor(n), w=omega(n), p=1, product=1); forstep(i=w, 1, -1, for(j=1, f[i, 2], p=nextprime(p+1); while(!(p%6==1), p=nextprime(p+1)); product *= p^(f[i, 1]-1))); product
isA355919(n) = (b(n) > A343771(n)) \\ See A343771 for its program

Extensions

a(20)-a(22) from Jinyuan Wang, Aug 10 2022