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A068525 Smallest k-almost prime between twin primes (for k >= 2).

%I A068525 #21 Feb 16 2025 08:32:45
%S A068525 4,12,60,72,240,192,2112,1152,14592,26112,15360,139968,138240,675840,
%T A068525 2101248,737280,4866048,786432,22118400,36175872,194641920,63700992,
%U A068525 138412032,169869312,1321205760,11123294208,16357785600,25669140480
%N A068525 Smallest k-almost prime between twin primes (for k >= 2).
%C A068525 Because it is unknown whether an infinite number of twin primes exist, it is unknown whether this sequence is infinite.
%H A068525 Donovan Johnson, <a href="/A068525/b068525.txt">Table of n, a(n) for n = 2..431</a>
%H A068525 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime</a>
%H A068525 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TwinPrimes.html">Twin Primes</a>
%e A068525 a(6)=240 because 240=2^4*3*5 is a 6-almost prime, 239 and 241 are twin primes and there is no 6-almost prime smaller than 240 which is between a pair of twin primes.
%t A068525 f[n_] := Plus @@ Last /@ FactorInteger@n; p = 3; t = Table[0, {30}]; While[p < 26*10^9, If[ PrimeQ[p + 2], a = f[p + 1]; If[ t[[a]] == 0, t[[a]] = p + 1; Print[{a, p + 1}]]]; p = NextPrime@p]; t (* _Robert G. Wilson v_, Aug 02 2010 *)
%o A068525 (PARI) v=vector(32) for(n=3,2250000000, if(n%1000000==0,print(n)); if(isprime(n) && isprime(n+2),k=bigomega(n+1); if(v[k]==0,v[k]=n+1; print(v[k],", ",k)))); v
%o A068525 \\ The PARI program prints a progress mark per million integers examined. v[k] is loaded with the first k-almost prime encountered between primes and is printed upon discovery. The entire vector is printed at program completion (or can be printed after interrupting the PARI program with CTRL-C).
%Y A068525 Cf. A001358 (semiprimes, with links to other almost primes), A001359 (lesser of twin primes), A014574, A075590.
%K A068525 nonn
%O A068525 2,1
%A A068525 _Rick L. Shepherd_, Mar 21 2002
%E A068525 a(27) - a(29) from _Robert G. Wilson v_, Aug 02 2010