cp's OEIS Frontend

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.

%I A141399 #25 Jul 13 2024 23:25:32
%S A141399 1,2,3,5,8,9,14,15,20,24,35,80,125,224,384,440,539,714,1715,2079,2400,
%T A141399 3024,4374,9800,12375,123200,194480,633555
%N 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.
%C A141399 The smallest prime in the set of consecutive primes is always 2, since k*(k+1) is even.
%C A141399 No further terms thru 5*10^8. - _Ray Chandler_, Jun 24 2009
%C A141399 a(29) > 2.29*10^25, if it exists. - _Giovanni Resta_, Nov 30 2019
%C A141399 This sequence contains k such that rad(k*(k+1)) is in A055932, where rad = A007947. - _Michael De Vlieger_, Jul 13 2024
%e A141399 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.
%e A141399 From _Michael De Vlieger_, Jul 13 2024: (Start)
%e A141399 Table showing terms a(n) = k such that rad(k*(k+1)) = P(i), where P = A002110.
%e A141399 i         P(i)  { k : rad(k*(k+1)) = P(i) }
%e A141399 --------------------------------------------------
%e A141399 1           2   {1}
%e A141399 2           6   {2, 3, 8}
%e A141399 3          30   {5, 9, 15, 24, 80}
%e A141399 4         210   {14, 20, 35, 125, 224, 2400, 4374}
%e A141399 5        2310   {384, 440, 539, 3024, 9800}
%e A141399 6       30030   {1715, 2079, 123200}
%e A141399 7      510510   {714, 12375, 194480}
%e A141399 8     9699690   {633555}
%e A141399 9   223092870   {}          (End)
%p A141399 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
%t A141399 Select[Range[2^16], Or[IntegerQ@ Log2[#], And[EvenQ[#], Union@ Differences@ PrimePi@ FactorInteger[#][[All, 1]] == {1}]] &[#*(# + 1)] &] (* _Michael De Vlieger_, Jul 13 2024 *)
%Y A141399 Cf. A007947, A055932, A073491, A138180.
%K A141399 more,nonn
%O A141399 1,2
%A A141399 _Leroy Quet_, Aug 03 2008
%E A141399 More terms from _Emeric Deutsch_, Aug 12 2008