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A382110 Smallest number k such that k-n and k+n are consecutive primes and k has exactly n distinct prime factors.

%I A382110 #27 Mar 25 2025 16:39:08
%S A382110 4,15,154,3045,22386,2467465,3015870,368961285,6326289970,
%T A382110 2313524242029,1568018377380,5808562826801735,1575649493651310,
%U A382110 6177821212870783905,171718219950879367766,2039004035049368722335,13156579658122684173390,112733682549950000276753015
%N A382110 Smallest number k such that k-n and k+n are consecutive primes and k has exactly n distinct prime factors.
%C A382110 a(10) > 5*10^11, if it exists. - _Amiram Eldar_, Mar 18 2025
%C A382110 a(10) <= 2313524242029. a(11) <= 1811331573870. - _Giorgos Kalogeropoulos_, Mar 21 2025
%H A382110 Daniel Suteu, <a href="/A382110/b382110.txt">Table of n, a(n) for n = 1..20</a>
%e A382110 a(1) = 4, because 4 - 1 = 3 and 4 + 1 = 5 are two consecutive primes and omega(4) = 1.
%e A382110 a(2) = 15, because 15 - 2 = 13 and 15 + 2 = 17 are two consecutive primes and omega(15) = 2.
%t A382110 Do[k=0;Until[PrimeQ[k-n]&&NextPrime[k-n]==k+n&&PrimeNu[k]==n,k++];a[n]=k,{n,7}];Array[a,7] (* _James C. McMahon_, Mar 20 2025 *)
%o A382110 (PARI) list(len) = {my(v = vector(len), prv = 3, c = 0, d); forprime(p = 5, , d = (p-prv)/2; if(d <= len && v[d] == 0 && omega(prv+d) == d, c++; v[d] = prv + d; if(c == len, break)); prv = p); v;} \\ _Amiram Eldar_, Mar 18 2025
%o A382110 (PARI)
%o A382110 generate(A, B, n) = A=max(A, vecprod(primes(n))); (f(m, p, j) = my(list=List()); forprime(q=p, sqrtnint(B\m, j), my(v=m*q); while(v <= B, if(j==1, if(v>=A && (nextprime(v) - v == n) && (v - precprime(v) == n), listput(list, v)), if(v*(q+1) <= B, list=concat(list, f(v, q+1, j-1)))); v *= q)); list); vecsort(Vec(f(1, if(n%2 == 0, 3, 2), n)));
%o A382110 a(n) = my(x=vecprod(primes(n)), y=2*x); while(1, my(v=generate(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ _Daniel Suteu_, Mar 25 2025
%Y A382110 Cf. A001221, A087378.
%K A382110 nonn
%O A382110 1,1
%A A382110 _Jean-Marc Rebert_, Mar 16 2025
%E A382110 a(10)-a(18) from _Daniel Suteu_, Mar 25 2025