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Smallest odd prime q such that pq+2 is has exactly 3 prime factors (with multiplicity), where p is the n-th odd prime.
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%I #7 Jul 11 2015 16:58:11

%S 41,5,37,23,13,19,17,5,5,11,7,3,11,17,17,11,7,13,5,11,7,11,11,23,5,13,

%T 11,13,11,5,5,47,17,5,3,11,7,19,5,3,13,3,7,5,19,7,31,29,7,29,7,3,11,

%U 11,11,7,7,3,3,11,5,7,3,7,5,31,13,5,7,5,11,13,7,13,11,5,19,5,7,5,3,3,7,5,3

%N Smallest odd prime q such that pq+2 is has exactly 3 prime factors (with multiplicity), where p is the n-th odd prime.

%C Trieneprime (3-prime-factor) analog of A126610 (semiprimes) and A126608-126609 (primes). This is the second row of the infinite array: A(k,n) = Smallest odd prime q such that pq+k has exactly 3 prime factors (with multiplicity), where p is the n-th odd prime. The odd trieneprimes arising begin: 125, 27, 261, 255, 171, 325. The above A(k,n) is the 3rd plane of the infinite 3-D array: A(j,k,n) = Smallest odd prime q such that pq+k has exactly j prime factors, where p is the n-th odd prime.

%F a(n) = Min {q in A065091: q*A000040(n)+2 is in A014612}.

%e a(1) = 41 because oddprime(1)*1+2 = 3*41+2 = 125 = 5*5*5.

%e a(2) = 5 because oddprime(2)*5+2 = 5*5+2 = 27 = 3*3*3.

%e a(3) = 37 because oddprime(3)*37+2 = 7*37+2 = 261 = 3*3*29.

%e a(4) = 23 because oddprime(4)*3+2 = 11*23+2 = 255 = 3*5*17.

%e a(5) = 13 because oddprime(5)*13+2 = 13*13+2 = 171 = 3*3*19.

%e a(6) = 19 because oddprime(6)*19+2 = 17*19+2 = 325 = 5*5*13.

%p A001222 := proc(n) local ifcts,i ; ifcts := ifactors(n)[2] ; add(op(2,op(i,ifcts)),i=1..nops(ifcts)) ; end: isA014612 := proc(n) if A001222(n) = 3 then true ; else false ; fi ; end: A126636 := proc(n) local p,q; p := ithprime(n+1) ; q := 3 ; while not isA014612(p*q+2) do q := nextprime(q) ; od ; RETURN(q) ; end : for n from 1 to 100 do printf("%d, ",A126636(n)) ; od ; # _R. J. Mathar_, Feb 13 2007

%Y Cf. A000040, A001358, A014612, A065091, A126608-126609, A126610.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Feb 08 2007

%E More terms from _R. J. Mathar_, Feb 13 2007