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A328243 Numbers whose arithmetic derivative (A003415) is larger than 1 and one of the terms of A143293 (partial sums of primorials). 5
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 (list; graph; refs; listen; history; text; internal format)
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)

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

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

Sequence in context: A216258 A064348 A206215 * A123295 A092350 A090197

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

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Last modified April 7 10:12 EDT 2020. Contains 333300 sequences. (Running on oeis4.)