OFFSET
1,2
COMMENTS
The smallest prime in the set of consecutive primes is always 2, since k*(k+1) is even.
No further terms thru 5*10^8. - Ray Chandler, Jun 24 2009
a(29) > 2.29*10^25, if it exists. - Giovanni Resta, Nov 30 2019
This sequence contains k such that rad(k*(k+1)) is in A055932, where rad = A007947. - Michael De Vlieger, Jul 13 2024
EXAMPLE
20 is factored as 2^2 * 5^1. 21 is factored as 3^1 * 7^1. Since the distinct primes that divide 20 and 21 (which are 2,3,5,7) form a set of consecutive primes, then 20 is in the sequence.
From Michael De Vlieger, Jul 13 2024: (Start)
Table showing terms a(n) = k such that rad(k*(k+1)) = P(i), where P = A002110.
i P(i) { k : rad(k*(k+1)) = P(i) }
--------------------------------------------------
1 2 {1}
2 6 {2, 3, 8}
3 30 {5, 9, 15, 24, 80}
4 210 {14, 20, 35, 125, 224, 2400, 4374}
5 2310 {384, 440, 539, 3024, 9800}
6 30030 {1715, 2079, 123200}
7 510510 {714, 12375, 194480}
8 9699690 {633555}
9 223092870 {} (End)
MAPLE
with(numtheory): a:=proc(n) local F, m: F:=`union`(factorset(n), factorset(n+1)): m:=nops(F): if ithprime(m)=F[m] then n else end if end proc: seq(a(n), n=1..1000000); # Emeric Deutsch, Aug 12 2008
MATHEMATICA
Select[Range[2^16], Or[IntegerQ@ Log2[#], And[EvenQ[#], Union@ Differences@ PrimePi@ FactorInteger[#][[All, 1]] == {1}]] &[#*(# + 1)] &] (* Michael De Vlieger, Jul 13 2024 *)
CROSSREFS
KEYWORD
more,nonn
AUTHOR
Leroy Quet, Aug 03 2008
EXTENSIONS
More terms from Emeric Deutsch, Aug 12 2008
STATUS
approved