

A309378


a(n) is the smallest squarefree number m with n prime factors such that Sum_{prime qm} 1/q  1/m = P/Q, where P <> Q are primes, for n > 1, or a(n) = 1 if no such m.


2




OFFSET

1,2


COMMENTS

Associated fractions P/Q for n > 1 are 2/3, 2/3, 17/19, 191/181, 19/17, 5701/4241, .... Note that Q  m.
a(n) is the least m with Omega(m) = omega(m) = n such that A326689(m) is a prime P and A326690(m) is a prime Q, or a(n) = 1 if no such m.


LINKS



EXAMPLE

1/2 + 1/3  1/6 = 2/3,
1/3 + 1/5 + 1/7  1/105 = 2/3,
1/2 + 1/5 + 1/7 + 1/19  1/1330 = 17/19,
....
6 = 2*3, 105 = 3*5*7, 1330 = 2*5*7*19, 331230 = 2*3*5*61*181, 127357230 = 2*3*5*17*53*151, ...  Jonathan Sondow, Jul 27 2019


MATHEMATICA

m=2; s={}; Do[f = FactorInteger[n]; p = f[[;; , 1]]; e = f[[;; , 2]]; If[Max[e] > 1  Length[e] < m, Continue[]]; frac = Total@(1/p)  1/n; num = Numerator[frac]; den = Denominator[frac]; If[den != num && PrimeQ[num] && PrimeQ[den], AppendTo[s, n]; m++], {n, 1, 5*10^6}]; s


PROG

(PARI) a(n) = {for(i = 2, oo, if(is(i, n), return(i)))}
is(m, qp) = {my(f = factor(m)); if(#f~ != qp, return(0)); if(Set(f[, 2]) != Set([1]), return(0)); s = sum(i = 1, qp, 1/f[i, 1])  1/m; isprime(denominator(s)) && isprime(numerator(s))} \\ David A. Corneth, Jul 27 2019


CROSSREFS



KEYWORD

nonn,hard


AUTHOR



STATUS

approved



