|
Search: id:A164643
|
|
|
| A164643 |
|
Semiprimes pq with pq - 1 divisible by p + q. |
|
+0
2
|
|
| 6, 21, 301, 697, 1333, 1909, 2041, 3901, 24601, 26977, 96361, 130153, 163201, 250321, 275833, 296341, 389593, 486877, 495529, 542413, 808861, 1005421, 1005649, 1055833, 1063141, 1232053, 1284121, 1403221, 1618597, 1787917
(list; graph; listen)
|
|
|
OFFSET
|
1,1
|
|
|
COMMENT
|
The first three terms are Syl(0)*Syl(1), Syl(1)*Syl(2) and Syl(2)*Syl(3). Syl means Sylvester's sequence, see A000058.
Products of two consecutive numbers p and q in Sylvester's sequence with primes p and q are in the sequence.
Let p and q be consecutive prime Sylvester numbers. Then: pq - 1 = p*(p^2 - p + 1) - 1 = p^3 - p^2 + p - 1 = (p^2 + 1)*(p - 1) = (p + p^2 - p + 1)*(p - 1) = (p + q)*(p - 1) it means that: (pq - 1) is divisible by (p + q). [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Aug 21 2009]
(p-k)*(q-k) = k^2 + 1 for some integer k, providing a fast way for finding appropriate p,q [From Max Alekseyev (maxale(AT)gmail.com), Aug 26 2009]
|
|
MAPLE
|
isA001358 := proc(n) RETURN ( numtheory[bigomega](n) =2 ) ; end:
isA164643 := proc(n) if isA001358(n) then p := op(1, op(1, ifactors(n)[2]) ) ; q := n/p ; if (p*q-1) mod (p+q) =0 then true; else false; fi; else false; fi; end:
for n from 4 to 3000000 do if isA164643(n) then print(n) ; fi; od: # R. J. Mathar, Aug 24 2009
|
|
CROSSREFS
|
Cf. A001358, A000058.
Sequence in context: A058821 A054366 A143049 this_sequence A007594 A012858 A084539
Adjacent sequences: A164640 A164641 A164642 this_sequence A164644 A164645 A164646
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Aug 19 2009
|
|
EXTENSIONS
|
Extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 24 2009
More terms from Max Alekseyev (maxale(AT)gmail.com), Aug 26 2009
|
|
|
Search completed in 0.002 seconds
|
