A088986 Numbers k such that each of k through k+4 are divisible by exactly two primes.

54, 91, 92, 115, 141, 142, 143, 144, 158, 205, 212, 213, 214, 215, 295, 301, 323, 324, 325, 391, 535, 685, 721, 799, 1135, 1345, 1465, 1535, 1711, 1941, 1981, 2101, 2215, 2302, 2303, 2304, 2425, 2641, 3865, 4411, 5461, 6505, 6625, 6925, 7165, 7231, 7261

Offset: 1

Author

Labos Elemer, Sep 30 2003

Keywords

• nonn

Comments

Identical with A045933 from first-to 38th terms, but deviates later because A045933 includes start of chains with more than 2 prime-factors.

Contrary to longer chains (6, 7, 8, ...) of omega = 2 this sequence seems to be either infinite or very long. See comments in A088983 [especially Eggleton via Kimberley, 2017].

Primes counted without multiplicity. - Harvey P. Dale, Oct 20 2011

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..750 from Harvey P. Dale)

Crossrefs

Cf. A001221, A045933, A088983.

Programs

Mathematica

Transpose[Select[Partition[Transpose[Select[Table[{n,PrimeNu[n]},{n,10000}],Last[#]==2&]][[1]],5,1],Last[#]-First[#]==4&]][[1]] (* Harvey P. Dale, Oct 20 2011 *)

PARI

lista(kmax) = {my(q = vector(5)); for(k = 6, kmax, q = concat(vecextract(q, "^1"), omega(k) == 2); if(vecmin(q) == 1, print1(k-4, ", ")));} \\ Amiram Eldar, Jul 11 2024

Python

from sympy import primefactors
def ok(n):
    return all(len(primefactors(n + i))==2 for i in range(5))
print([n for n in range(1, 8001) if ok(n)]) # Indranil Ghosh, Jul 17 2017