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 A309378 a(n) is the smallest squarefree number m with n prime factors such that Sum_{prime q|m} 1/q - 1/m = P/Q, where P <> Q are primes, for n > 1, or a(n) = 1 if no such m. 2
 1, 6, 105, 1330, 331230, 4081530, 127357230 (list; graph; refs; listen; history; text; internal format)
 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 Table of n, a(n) for n=1..7. 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 Cf. A000040, A120944, A190273, A190275, A326689, A326690. Sequence in context: A033589 A077289 A279520 * A221933 A110342 A227207 Adjacent sequences: A309375 A309376 A309377 * A309379 A309380 A309381 KEYWORD nonn,hard AUTHOR Amiram Eldar and Thomas Ordowski, Jul 26 2019 STATUS approved

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Last modified December 4 18:39 EST 2023. Contains 367563 sequences. (Running on oeis4.)