cp's OEIS Frontend

A105418 Smallest prime p such that the sum of it and the following prime has n prime factors including multiplicity, or 0 if no such prime exists.

%I A105418 #24 Aug 07 2024 00:54:03
%S A105418 2,0,3,11,53,71,61,191,953,1151,3833,7159,4093,30713,36857,110587,
%T A105418 360439,663547,2064379,786431,3932153,5242877,9437179,63700991,
%U A105418 138412031,169869311,436207613,3875536883,1358954453,1879048183,10066329587,8053063661,14495514619
%N A105418 Smallest prime p such that the sum of it and the following prime has n prime factors including multiplicity, or 0 if no such prime exists.
%C A105418 a(2) = 0 since it is impossible.
%H A105418 Daniel Suteu, <a href="/A105418/b105418.txt">Table of n, a(n) for n = 1..500</a> (terms 1..38 from Amiram Eldar)
%e A105418 a(5) = 53 because (53 + 59) = 112 = 2^4*7.
%e A105418 a(24) = 63700991 because (63700991 + 63700993) = 127401984 = 2^19*3^5.
%e A105418 a(28) = 3875536883 because (3875536883 + 3875536909) = 7751073792 = 2^25*3*7*11.
%e A105418 a(29) = 1358954453 because (1358954453 + 1358954539) = 2717908992 = 2^25*3^4.
%e A105418 a(30) = 1879048183 because (1879048183 + 1879048201) = 3758096384 = 2^29*7.
%t A105418 f[n_] := Plus @@ Flatten[ Table[ #[[2]], {1}] & /@ FactorInteger[n]]; t = Table[0, {40}]; Do[a = f[Prime[n] + Prime[n + 1]]; If[a < 41 && t[[a]] == 0, t[[a]] = Prime[n]; Print[{a, Prime[n]}]], {n, 111500000}]; t
%o A105418 (PARI)
%o A105418 almost_primes(A, B, n) = A=max(A, 2^n); (f(m, p, n) = my(list=List()); if(n==1, forprime(q=max(p, ceil(A/m)), B\m, listput(list, m*q)), forprime(q=p, sqrtnint(B\m, n), list=concat(list, f(m*q, q, n-1)))); list); vecsort(Vec(f(1, 2, n)));
%o A105418 a(n) = if(n==2, return(0)); my(x=2^n, y=2*x); while(1, my(v=almost_primes(x, y, n)); for(k=1, #v, my(p=precprime(max(v[k]>>1, 2)), q=nextprime(p+1)); if(p+q == v[k], return(p))); x=y+1; y=2*x); \\ _Daniel Suteu_, Aug 06 2024
%Y A105418 Cf. A001043, A001222, A071215, A098037, A098048.
%K A105418 nonn
%O A105418 1,1
%A A105418 _Giovanni Teofilatto_ and _Robert G. Wilson v_, Apr 06 2005
%E A105418 a(28)=3875536883 from _Ray Chandler_ and _Robert G. Wilson v_, Apr 10 2005
%E A105418 Edited by _Ray Chandler_, Apr 10 2005
%E A105418 a(31)-a(33) from _Daniel Suteu_, Nov 18 2018
%E A105418 Definition slightly modified by _Harvey P. Dale_, Jul 17 2024