Antti Karttunen, <a href="/A351228/b351228_1.txt">Table of n, a(n) for n = 1..11643 (terms up to 3*A002110(9))</a>

proposed

editing

Cf. A003415, A024451, A048103, A060735, A235991, A276085, A276086, A327862, A351088, A351089, A351232, A369656, A369663, A369664.

From Antti Karttunen, Feb 14 2024: (Start)

Question 1: Is there an upper bound sequence u(n) for each subrange [A060735(n), A060735(n+1)], so that the terms of this sequence are guaranteed to stay confined to subranges [A060735(n), u(n)], or, put in other words, that no term of this sequence may ever occur in subrange [u(n)+1, A060735(n+1)]? That is, when prime(i_1)*prime(i_2)*...*prime(i_k) [product of k primes, not necessarily all distinct] >= k * (A002110(i_1-1)+A002110(i_2-1)+...+A002110(i_k-1))^((k-1)/k)? [Ufnarovski and Åhlander's Theorem 9, point (4).]

Question 2: What can be said about the numbers k and x, for which d(k) = e(k) = x? Here we use a letter d for arithmetic derivative, A003415, e for primorial base exponent function, A276086, and e^-1 for the inverse of the latter, the primorial base log-function, A276085.

First of all, such numbers must be in the intersection of A369970 (k such that e(k)|d(k)) and A358222 (composite k such that d(k)|e(k), thus k shoud be also in A358215). Also k should be in A369666, k such that e^-1(d(k)) == k (mod 4).

Then, if x is a prime, then k must be a primorial, with k = A002110(i), and x = prime(1+i) for some i, the only possible solution is k = 6 and x = 5, because A024451(i) = d(A002110(i)) >= A000040(1+i) for i >= 2, with equality obtained only at i=2.

If x were an odd semiprime, then k = e^-1(x) were a multiple of 4, in which case also d(k) were a multiple of 4, so this certainly is not possible. More generally, this applies to all terms in A046337. If x is an even semiprime = 2*prime(i), then k = e^1(x) = 1+A002110(i-1). In cases where i-1 is in A014545, 0, 1, 2, 3, 4, 5, 11, 75, ..., we would have A003415(k) = 1, which is a contradiction, but then, in other cases, by Ufnarovski and Åhlander's Theorem 9, point (3), A003415(k) >= 2*sqrt(k), so 2*prime(i) >= 2*sqrt(1+A002110(i-1)), which also cannot be true for i > 4, so therefore x cannot be a semiprime.

Either in case (1): d(k) = x is an odd number with an odd number of prime factors, and k = e-1(x) == 2 (mod 4), i.e., both k and x are in A235991, (i.e., k is in A369656), or case (2): k is in A327862 (i.e., x is of the form 4m+2), and by a simple consideration of modulo 4 values, k in this case must be either in A369663 or in A369664.

(End)

Question 1: Is there an upper bound sequence u(n) for each subrange [A060735(n), A060735(n+1)], so that the terms of this sequence are guaranteed to stay confined to subranges [A060735(n), u(n)], or, put in other words, that no term of this sequence may ever occur in subrange [u(n)+1, A060735(n+1)]? That is, when prime(i)*prime(j)*...*prime(k) [not necessarily all distinct] >= sqrt(A002110(i-1)+A002110(j-1)+...+A002110(k-1))?

Question 1: Is there an upper bound sequence u(n) for each subrange [A060735(n), A060735(n+1)], so that the terms of this sequence are guaranteed to stay confined to subranges [A060735(n), u(n)], or, put in other words, that no term of this sequence may ever occur in subrange [u(n)+1, A060735(n+1)]?

Question 2: What can be said about the numbers k and x, for which d(k) = e(k) = x? Here we use a letter d for arithmetic derivative, A003415, e for primorial base exponent function, A276086, and e^-1 for the inverse of the latter, the primorial base log-functon, A276085.

First of all, such numbers must be in the intersection of A369970 (k such that e(k)|d(k)) and A358222 (composite k such that d(k)|e(k), thus k shoud be also in A358215). Also k should be in A369666, k such that e-1(d(k)) == k (mod 4).

Antti Karttunen, <a href="/A351228/b351228_1.txt">Table of n, a(n) for n = 1..11643 (terms up to 3*A002110(9))</a>Victor Ufnarovski and Bo Åhlander, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.html">How to Differentiate a Number</a>, J. Integer Seqs., Vol. 6, 2003, #03.3.4.

If x were an odd semiprime, then k = e^-1(x) were a multiple of 4, in which case also d(k) were a multiple of 4, so this certainly is not possible. More generally, this applies to all terms in A046337. If x is an even semiprime = 2*prime(i), then k = e^1(x) = 1+A002110(i-1). In cases where i-1 is in A014545, 0, 1, 2, 3, 4, 5, 11, 75, ..., we would have A003415(k) = 1, which is a contradiction, but then, by Ufnarovski and Åhlander's Theorem 9, point (3), A003415(k) >= 2*sqrt(k), so 2*prime(i) >= 2*sqrt(1+A002110(i-1)), which also cannot be true for i > 4, so therefore x cannot be a semiprime.

Antti Karttunen, <a href="/A351228/b351228_1.txt">Table of n, a(n) for n = 1..11643 (terms up to 3*A002110(9))</a>