This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A355919 #13 Aug 10 2022 06:08:17
%S A355919 128,256,512,1024,2048,3072,4096,6144,8192,12288,16384,24576,32768,
%T A355919 49152,65536,73728,98304,131072,147456,196608,262144,294912
%N 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).
%C A355919 {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.
%C A355919 Since {b(n)} agrees with A343771(n) for most n, it cannot have its own entry.
%C A355919 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, ...).
%C A355919 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).
%C A355919 It seems that q^(N+1) is the smallest q-rough term in this sequence.
%e A355919 128 is a term since b(128) = 7 * 13 * 19 * 31 * 37 * 43 * 61 > A343771(128) = 7^3 * 13 * 19 * 31 * 37 * 43.
%o A355919 (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
%o A355919 isA355919(n) = (b(n) > A343771(n)) \\ See A343771 for its program
%Y A355919 Analog of A072066 and A340624.
%Y A355919 Cf. A343771, A002476, A037019, A340388, A002324.
%K A355919 nonn,more
%O A355919 1,1
%A A355919 _Jianing Song_, Jul 20 2022
%E A355919 a(20)-a(22) from _Jinyuan Wang_, Aug 10 2022