A392672 Smallest number m such that m*(m+1)*(m+2) has exactly n distinct prime factors.

1, 3, 5, 13, 33, 153, 285, 713, 5795, 18445, 90363, 304589, 1013725, 5617820, 23738713, 123702369, 556876529, 3760113565, 18723991845, 41704979953

Offset

2,2

Comments

Conjecture: this is an increasing sequence.

a(22) <= 1782994645795, a(23) <= 6127197154440, a(24) <= 34529471219670, a(25) <= 906874931149913. - David A. Corneth, May 14 2026

Links

Table of n, a(n) for n=2..21.

XiaoYang Zhang, What are the numbers that contain the most distinct prime factors among three consecutive natural numbers, within the ranges of 10^2, 10^3, 10^4, etc? (in Chinese).

Example

For n = 9, a(9)*(a(9) + 1)*(a(9) + 2) = 713*714*715 = (23*31)*(2*3*7*17)*(5*11*13) with 9 distinct prime factors.

Mathematica

a[n_]:=(k=1; While[PrimeNu[Times@@{k, k + 1, k + 2}]!=n, k++]; k);
Table[a[n], {n, 2, 10}]

Prog

(Python)
from itertools import count, islice
from sympy import factorint
def agen(): # generator of terms
    wlst, adict, n = [set(factorint(1+i)) for i in range(3)], dict(), 2
    for m in count(1):
        v = len(wlst[0] | wlst[1] | wlst[2])
        if v >= n and v not in adict:
            adict[v] = m
            while n in adict:
                yield adict[n]
                n += 1
        wlst = wlst[1:] + [set(factorint(m+3))]
print(list(islice(agen(), 13))) # Michael S. Branicky, May 05 2026

Crossrefs

Cf. A001221, A002110, A059958, A394554.

Sequence in context: A296023 A027039 A283844 * A324783 A032009 A032027

Adjacent sequences: A392669 A392670 A392671 * A392673 A392674 A392675

Keyword

nonn,more

Author

Zhining Yang, Apr 30 2026

Status

approved