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

4, 12, 60, 72, 240, 192, 2112, 1152, 14592, 26112, 15360, 139968, 138240, 675840, 2101248, 737280, 4866048, 786432, 22118400, 36175872, 194641920, 63700992, 138412032, 169869312, 1321205760, 11123294208, 16357785600, 25669140480

Offset: 2

Author

Rick L. Shepherd, Mar 21 2002

Keywords

• nonn

Comments

Because it is unknown whether an infinite number of twin primes exist, it is unknown whether this sequence is infinite.

Examples

			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.
		

Links

• Donovan Johnson, Table of n, a(n) for n = 2..431
• Eric Weisstein's World of Mathematics, Almost Prime
• Eric Weisstein's World of Mathematics, Twin Primes

Crossrefs

Cf. A001358 (semiprimes, with links to other almost primes), A001359 (lesser of twin primes), A014574, A075590.

Programs

Mathematica

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 *)

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
\\ 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).

Extensions

a(27) - a(29) from Robert G. Wilson v, Aug 02 2010