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 A140777 a(n) = 2*prime(n) - 4. 3
 0, 2, 6, 10, 18, 22, 30, 34, 42, 54, 58, 70, 78, 82, 90, 102, 114, 118, 130, 138, 142, 154, 162, 174, 190, 198, 202, 210, 214, 222, 250, 258, 270, 274, 294, 298, 310, 322, 330, 342, 354, 358, 378, 382, 390, 394, 418, 442, 450, 454, 462, 474, 478, 498, 510, 522 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS A number n is included if (p + n/p) is prime, where p is the smallest prime that divides n. Since all terms of this sequence are even (or otherwise p + n/p would be even and not a prime), p is always 2. So this sequence is the set of all even numbers n where (2 + n/2) is prime. The entries are also encountered via the bilinear transform approximation to the natural log (unit circle). Specifically, evaluating 2(x-1)/(x+1) at x = 2, 3, 4, ..., the terms of this sequence are seen ahead of each new prime encountered. Additionally, the position of those same primes will occur at the entry positions. For clarity, the evaluation output is 2, 3, 1, 1, 6, 5, 4, 3, 10, 7, 3, 2, 14, 9, 8, 5, 18, 11, ..., where the entries ahead of each new prime are 2, 6, 10, 18, ... . As an aside, the same mechanism links this sequence to A165355. - Bill McEachen, Jan 08 2015 As a follow-up to previous comment, it appears that the numerators and denominators of 2(x-1)/(x+1) are respectively given by A145979 and A060819, but with different offsets. - Michel Marcus, Jan 14 2015 Subset of the union of A017641 & A017593. - Michel Marcus, Sep 01 2020 LINKS FORMULA a(n) = 2*A040976(n). - Michel Marcus, Jan 09 2015 EXAMPLE The smallest prime dividing 42 is 2. Since 2 + 42/2 = 23 is prime, 42 is included in this sequence. MAPLE A020639 := proc(n) local dvs, p ; dvs := sort(convert(numtheory[divisors](n), list)) ; for p in dvs do if isprime(p) then RETURN(p) ; fi ; od: error("%d", n) ; end: A111234 := proc(n) local p ; p := A020639(n) ; p+n/p ; end: isA140777 := proc(n) RETURN(isprime(A111234(n))) ; end: for n from 2 to 1200 do if isA140777(n) then printf("%d, ", n) ; fi ; od: # R. J. Mathar, May 31 2008 seq(2*ithprime(i)-4, i=1..1000); # Robert Israel, Jan 09 2015 MATHEMATICA fQ[n_] := Block[{p = First@ First@ Transpose@ FactorInteger@ n}, PrimeQ[p + n/p] == True]; Select[ Range[2, 533], fQ@# &] (* Robert G. Wilson v, May 30 2008 *) Table[2 Prime[n] - 4, {n, 60}] (* Vincenzo Librandi, Feb 19 2015 PROG (PARI) vector(100, n, 2*prime(n) - 4) \\ Michel Marcus, Jan 09 2015 (MAGMA) [2*NthPrime(n)-4: n in [1..80]]; // Vincenzo Librandi, Feb 19 2015 CROSSREFS Cf. A000040, A020639, A111234, A140775, A140776. Sequence in context: A274679 A229218 A255174 * A309502 A281664 A330507 Adjacent sequences:  A140774 A140775 A140776 * A140778 A140779 A140780 KEYWORD nonn,easy AUTHOR Leroy Quet, May 29 2008, May 31 2008 EXTENSIONS More terms from Robert G. Wilson v and R. J. Mathar, May 30 2008 STATUS approved

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Last modified December 7 13:08 EST 2021. Contains 349581 sequences. (Running on oeis4.)