A391970 Numbers m such that the product m*(m+1) has a set of prime divisors, from greatest down to 2, that is missing exactly two prime divisors.

7, 10, 11, 32, 39, 84, 119, 143, 153, 168, 209, 220, 242, 272, 285, 324, 351, 374, 441, 455, 494, 624, 675, 728, 1224, 1539, 1700, 2057, 2184, 2499, 2600, 3059, 3135, 4913, 5082, 5775, 5928, 6174, 6655, 7865, 10625, 11270, 12635, 13376, 14079, 19227, 19550, 21504

Offset

1,1

Comments

The number of distinct prime divisors, omega(m*(m+1)) is exactly two less than the index of the greatest prime divisor, pi(gpf(m*(m+1))): A252489(m) - A059957(m) = 2.

Although 1109496723125 may be the last term in this sequence, if not, subsequent terms are greater than 2*10^14 by computational verification.

The sequence must terminate because the growth rate of A252489 exceeds A059957 enough to produce a continuing difference of more than two eventually, so thereafter the factorization of the product will have more than two gaps.

Links

Ken Clements, Table of n, a(n) for n = 1..90

Example

a(1) = 7 because 7*8 = 56 with gpf 7 and pi(7) = 4 while omega(56) = 2 because it has only 2 and 7 for prime divisors, missing prime divisors 3 and 5.
a(2) = 10 because 10*11 = 110 with gpf 11 and pi(11) = 5 while omega(110) = 3 because it has only 2, 5 and 11 for prime divisors, missing prime divisors 3 and 7.
a(3) = 11 because 11*12 = 132 with gpf 11 and pi(11) = 5 while omega(132) = 3 because it has only 2, 3, and 11 for prime divisors, missing prime divisors 5 and 7.

Mathematica

q[m_] := Module[{p = Union[Flatten[FactorInteger[#][[;; , 1]] & /@ {m, m + 1}]]}, PrimePi[p[[-1]]] == Length[p] + 2]; Select[Range[22000], q] (* Amiram Eldar, Dec 25 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 == 2
print([m for m in range(2, 10**6) if ok(m)])
(PARI) isok(m) = my(f=factor(m*(m+1))); omega(f) == primepi(vecmax(f)) - 2; \\ Michel Marcus, Dec 26 2025

Crossrefs

Cf. A391602, A391885.

Sequence in context: A105140 A107018 A247868 * A048792 A024585 A190482

Adjacent sequences: A391967 A391968 A391969 * A391971 A391972 A391973

Keyword

nonn

Author

Ken Clements, Dec 24 2025

Status

approved