|
Search: id:A158870
|
|
|
| A158870 |
|
Sums of the form (twin primes + 1) which are also an upper twin prime. |
|
+0
2
|
|
| 13, 61, 1321, 1621, 4261, 5101, 6661, 6781, 11701, 12541, 21061, 66361, 83221, 88261, 107101, 110881, 114661, 127681, 130201, 140761, 141961, 144541, 148201, 149521, 157561, 161341, 163861, 175081, 186481, 204601, 230941, 249541, 267961
(list; graph; listen)
|
|
|
OFFSET
|
1,1
|
|
|
COMMENT
|
If the sum is a member of a twin prime pair, it always is the upper member, shown in A158866.
Moreover, except the first term, these numbers are of the form 10k+1. [We prove this by
exhausting the possibilities when calculating the upper, summing and inspecting
the lower of the sum. Here are the possible outcomes.
p1(k), p2(k) p2(m) = p1(k)+p2(k)+1
----------- ---------------------------------
10k+1 10k+3 20k+4+1 not prime
10k+3 10k+5 p2(k) not prime
10k+5 10k+7 p1(k) not prime
10k+7 10k+9 20k+16+1 upper => p1(m) not prime
10k+9 10k+11 20k+20+1 = 10(2k+2)+1
So the only form that was not eliminated, is 10k+1. 13 defies this scheme
because 10k+5 is prime for k=0, q.e.d.]
|
|
FORMULA
|
{A054735(k)+1: A054735(k)+1 = A006512(j), any j,k} - R. J. Mathar, Apr 06 2009
|
|
EXAMPLE
|
The 30th lower twin prime is 659. 659+661+1 = 1321, prime and 1319 is too.
Then 1319 is the lower member of the twin prime pair (1319,1321). So 1321 is
in the sequence.
|
|
PROGRAM
|
(PARI) gp > g(n)=for(x=1, n, y=2*twinl(x)+3; if(isprime(y)&&isprime(y-2), print1(y", ")))
|
|
CROSSREFS
|
Cf. A158866.
Sequence in context: A147185 A122885 A135535 this_sequence A145044 A044151 A044532
Adjacent sequences: A158867 A158868 A158869 this_sequence A158871 A158872 A158873
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Cino Hilliard (hillcino368(AT)hotmail.com), Mar 28 2009
|
|
EXTENSIONS
|
Edited by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 06 2009
|
|
|
Search completed in 0.002 seconds
|
