OFFSET
1,1
COMMENTS
From David A. Corneth, Oct 12 2019: (Start)
Let k' be the arithmetic derivative of k. Then to find terms of the form k = p * q where p, q are prime, we could see that k' = p + q. Then as one of them needs to be two, say p, needs to be 2, we have q = A143293(m) - 2 a prime. This would give terms 2 * q.
If terms are of the form k = p * q * r where p, q, r are distinct primes then k' = p*q + p*r + q*r. For m we like, we could solve p*q + p*r + q*r = A143293(m). checking p * q below some bound, we can solve for r and get r = (A143293(m) - p*q) / (p + q). With some extra constraints and searching different prime signatures, one might confirm terms found are all below some chosen upper bound. (End)
See sequences A369239 and A369240 for more observations and insights about the terms of this sequence. - Antti Karttunen, Jan 22 2024
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..39 (terms < 10^13)
FORMULA
A327969(a(n)) <= 5 for all n.
PROG
(PARI)
A002620(n) = ((n^2)>>2);
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A276086(n) = { my(i=0, m=1, pr=1, nextpr); while((n>0), i=i+1; nextpr = prime(i)*pr; if((n%nextpr), m*=(prime(i)^((n%nextpr)/pr)); n-=(n%nextpr)); pr=nextpr); m; };
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, Oct 10 2019
EXTENSIONS
a(12)-a(25) from David A. Corneth and Giovanni Resta, Oct 12 2019
STATUS
approved