This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A140777 #52 Sep 30 2024 02:13:31
%S A140777 0,2,6,10,18,22,30,34,42,54,58,70,78,82,90,102,114,118,130,138,142,
%T A140777 154,162,174,190,198,202,210,214,222,250,258,270,274,294,298,310,322,
%U A140777 330,342,354,358,378,382,390,394,418,442,450,454,462,474,478,498,510,522
%N A140777 a(n) = 2*prime(n) - 4.
%C A140777 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.
%C A140777 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
%C A140777 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
%C A140777 Subset of the union of A017641 & A017593. - _Michel Marcus_, Sep 01 2020
%F A140777 a(n) = 2*A040976(n). - _Michel Marcus_, Jan 09 2015
%e A140777 The smallest prime dividing 42 is 2. Since 2 + 42/2 = 23 is prime, 42 is included in this sequence.
%p A140777 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
%p A140777 seq(2*ithprime(i)-4, i=1..1000); # _Robert Israel_, Jan 09 2015
%t A140777 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 *)
%t A140777 Table[2 Prime[n] - 4, {n, 60}] (* _Vincenzo Librandi_, Feb 19 2015 *)
%o A140777 (PARI) vector(100, n, 2*prime(n) - 4) \\ _Michel Marcus_, Jan 09 2015
%o A140777 (Magma) [2*NthPrime(n)-4: n in [1..80]]; // _Vincenzo Librandi_, Feb 19 2015
%Y A140777 Cf. A000040, A020639, A040976, A111234, A140775, A140776.
%Y A140777 Cf. A060819, A145979, A165355.
%Y A140777 Cf. A017641, A017593.
%K A140777 nonn,easy
%O A140777 1,2
%A A140777 _Leroy Quet_, May 29 2008, May 31 2008
%E A140777 More terms from _Robert G. Wilson v_ and _R. J. Mathar_, May 30 2008