A391885 Numbers m such that the product m*(m+1) has a set of prime divisors, from greatest down to 2, that is missing exactly one prime divisor.

4, 6, 21, 27, 44, 48, 49, 54, 55, 63, 65, 77, 90, 98, 99, 104, 120, 175, 195, 350, 363, 560, 594, 935, 1000, 1155, 1274, 2430, 2925, 4095, 4199, 4224, 5984, 10647, 10829, 14364, 14399, 21735, 27455, 28560, 37179, 43680, 52325, 89375, 136850, 336140, 601425, 709631

Offset

1,1

Comments

The number of distinct prime divisors, omega(m*(m+1)) is exactly one less than the index of the greatest prime divisor, pi(gpf(m*(m+1))): A252489(m) - A059957(m) = 1. The last term of the sequence is 80061344 because after this number, the growth rate of A252489 exceeds A059957 enough to produce a difference of more than one, so thereafter the factorization of the product will have more than one gap.

Links

Ken Clements, Table of n, a(n) for n = 1..58

Example

a(1) = 4 because 4*5 = 20 with gpf 5 and pi(5) = 3 while omega(20) = 2 because it has only 2 and 5 for prime divisors, missing prime divisor 3.
a(2) = 6 because 6*7 = 42 with gpf 7 and pi(7) = 4 while omega(42) = 3 because it has only 2, 3 and 7 for prime divisors, missing prime divisor 5.
a(3) = 21 because 21*22 = 462 with gpf 11 and pi(11) = 5 while omega(462) = 4 because it has only 2, 3, 7 and 11 for prime divisors, missing prime divisor 5.

Mathematica

q[m_] := Module[{p = Union[Flatten[FactorInteger[#][[;; , 1]] & /@ {m, m + 1}]]}, PrimePi[p[[-1]]] == Length[p] + 1]; Select[Range[720000], q] (* Amiram Eldar, Dec 23 2025 *)

Prog

(Python)
from sympy import primepi, primefactors
def ok(m):
    pf = primefactors(m*(m+1))
    omega, pidx = len(pf), primepi(pf[-1])
    return pidx - omega == 1
print([m for m in range(2, 10**4) if ok(m)])

Crossrefs

Cf. A391449, A252489, A059957, A391602, A391970.

Sequence in context: A048789 A038069 A274425 * A019147 A026722 A323845

Adjacent sequences: A391882 A391883 A391884 * A391886 A391887 A391888

Keyword

nonn,fini,full

Author

Ken Clements, Dec 22 2025

Status

approved