cp's OEIS Frontend

A328243 Numbers whose arithmetic derivative (A003415) is larger than 1 and one of the terms of A143293 (partial sums of primorials).

%I A328243 #28 Jan 22 2024 08:59:11
%S A328243 14,45,74,198,5114,10295,65174,1086194,20485574,40354813,465779078,
%T A328243 12101385979,15237604243,18046312939,29501083259,52467636437,
%U A328243 65794608773,86725630997,87741700037,131833085077,168380217557,176203950283,177332276971,226152989747,292546582253
%N A328243 Numbers whose arithmetic derivative (A003415) is larger than 1 and one of the terms of A143293 (partial sums of primorials).
%C A328243 From _David A. Corneth_, Oct 12 2019: (Start)
%C A328243 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.
%C A328243 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)
%C A328243 See sequences A369239 and A369240 for more observations and insights about the terms of this sequence. - _Antti Karttunen_, Jan 22 2024
%H A328243 Giovanni Resta, <a href="/A328243/b328243.txt">Table of n, a(n) for n = 1..39</a> (terms < 10^13)
%F A328243 A327969(a(n)) <= 5 for all n.
%o A328243 (PARI)
%o A328243 A002620(n) = ((n^2)>>2);
%o A328243 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A328243 A143293(n) = if(n==0, 1, my(P=1, s=1); forprime(p=2, prime(n), s+=P*=p); (s)); \\ From A143293.
%o A328243 A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
%o A328243 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; };
%o A328243 isA328243(n) = { my(u=A003415(n)); ((u>1)&&(1==A276150(A276086(u)))); }; \\ This is very slow program!
%o A328243 k=0; for(n=1,A002620(A143293(6)),if(isA328243(n), k++; print1(n,", ")));
%Y A328243 Sequence A369240 sorted into ascending order.
%Y A328243 Cf. A003415, A143293, A327969, A327978, A328313, A369239.
%K A328243 nonn
%O A328243 1,1
%A A328243 _Antti Karttunen_, Oct 10 2019
%E A328243 a(12)-a(25) from _David A. Corneth_ and _Giovanni Resta_, Oct 12 2019