1,2

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

The first comment implies that one could replace A073491 in the second definition by the more specific A055932. Equivalently, these are integers k such that the number of distinct prime factors of k*(k+1), A059957(k) = A001222(k^2+k), is equal to the index of the largest prime factor of k*(k+1), A252489(k) = A000720(A074399(k)). - M. F. Hasler, Mar 31 2026

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)

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

Select[Range[2^16], Or[IntegerQ@ Log2[#], And[EvenQ[#], Union@ Differences@ PrimePi@ FactorInteger[#][[All, 1]] == {1}]] &[#*(# + 1)] &] (* Michael De Vlieger, Jul 13 2024 *)

(PARI) select( {is_A141399(n)=prime(#n=factor(n^2+n)[, 1])==n[#n]}, [1..20000]) \\ M. F. Hasler, Mar 31 2026

(Python)

from itertools import islice

from heapq import heappop, heappush

from sympy import integer_nthroot, nextprime

def A141399_gen(): # generator of terms

h, hset = [(1, (1, ))], {1}

while True:

m, ps = heappop(h)

a, b = integer_nthroot(m+1, 2)

if b:

yield a-1>>1

for p in ps:

mp = m*p

if mp not in hset:

heappush(h, (mp, ps))

hset.add(mp)

q = nextprime(max(ps, default=1))

mp = m*q

if mp not in hset:

heappush(h, (mp, (ps+(q, ))))

hset.add(mp)

A141399_list = list(islice(A141399_gen(), 28)) # Chai Wah Wu, Apr 05 2026

Cf. A007947, A055932, A073491, A138180, A391449.

Sequence in context: A009388 A190803 A125871 * A104737 A286495 A295082

Adjacent sequences: A141396 A141397 A141398 * A141400 A141401 A141402

nonn,more,changed

Leroy Quet, Aug 03 2008

More terms from Emeric Deutsch, Aug 12 2008

Dubious claim of finiteness removed by Sean A. Irvine, Mar 27 2026

Definition corrected by M. F. Hasler, Mar 31 2026

approved