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

14, 45, 74, 198, 5114, 10295, 65174, 1086194, 20485574, 40354813, 465779078, 12101385979, 15237604243, 18046312939, 29501083259, 52467636437, 65794608773, 86725630997, 87741700037, 131833085077, 168380217557, 176203950283, 177332276971, 226152989747, 292546582253

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]));
A143293(n) = if(n==0, 1, my(P=1, s=1); forprime(p=2, prime(n), s+=P*=p); (s)); \\ From A143293.
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; };
isA328243(n) = { my(u=A003415(n)); ((u>1)&&(1==A276150(A276086(u)))); }; \\ This is very slow program!
k=0; for(n=1, A002620(A143293(6)), if(isA328243(n), k++; print1(n, ", ")));

Crossrefs

Sequence A369240 sorted into ascending order.

Cf. A003415, A143293, A327969, A327978, A328313, A369239.

Sequence in context: A216258 A064348 A206215 * A369240 A123295 A092350

Adjacent sequences: A328240 A328241 A328242 * A328244 A328245 A328246

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