A141399 Positive integers k such that all the distinct primes that divide k or k+1 are members of a set of consecutive primes. In other words, k is included if and only if k*(k+1) is contained in sequence A073491.

1, 2, 3, 5, 8, 9, 14, 15, 20, 24, 35, 80, 125, 224, 384, 440, 539, 714, 1715, 2079, 2400, 3024, 4374, 9800, 12375, 123200, 194480, 633555

Offset: 1

Author

Leroy Quet, Aug 03 2008

Keywords

• more
• nonn

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

Examples

			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)
		

Crossrefs

Cf. A007947, A055932, A073491, A138180.

Programs

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 *)

Extensions

More terms from Emeric Deutsch, Aug 12 2008