A394554 Smallest number m such that m*(m+1)*(m+2)*(m+3) has exactly n distinct prime factors.

1, 2, 4, 10, 19, 55, 130, 284, 712, 1768, 5795, 17017, 68263, 195545, 545258, 1698179, 11990404, 28003819, 99712613, 452419979, 1990586012, 5313193818, 22323360772

Offset

2,2

Links

Table of n, a(n) for n=2..24.

Example

For n = 9, a(9)*(a(9) + 1)*(a(9) + 2)*(a(9) + 3) = 284*285*286*287 = (2*2*71)*(3*5*19)*(2*11*13)*(7*41) with 9 distinct prime factors.

Mathematica

a[n_] :=(k=1; While[PrimeNu[Times @@ {k, k + 1, k + 2, k + 3}] != 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(4)], dict(), 2
    for m in count(1):
        v = len(wlst[0] | wlst[1] | wlst[2] | wlst[3])
        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+4))]
print(list(islice(agen(), 15))) # Michael S. Branicky, May 05 2026

Crossrefs

Cf. A002110, A059958, A392672.

Sequence in context: A011963 A083844 A026554 * A259132 A369391 A099413

Adjacent sequences: A394551 A394552 A394553 * A394555 A394556 A394557

Keyword

nonn,more

Author

Zhining Yang, May 01 2026

Extensions

a(21) and a(23) corrected by Chai Wah Wu, May 05 2026

Status

approved