A140777 a(n) = 2*prime(n) - 4.

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

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

Table of n, a(n) for n=1..56.

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, A040976, A111234, A140775, A140776.

Cf. A060819, A145979, A165355.

Cf. A017641, A017593.

Sequence in context: A229218 A255174 A375637 * 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