A321493 Numbers m such that m and m+1 both have at least 3, but m or m+1 has at least 4 distinct prime factors.

714, 1364, 1595, 1770, 1785, 1869, 2001, 2090, 2145, 2184, 2210, 2261, 2345, 2379, 2414, 2639, 2805, 2820, 2849, 2870, 2925, 3002, 3009, 3059, 3080, 3219, 3255, 3289, 3354, 3366, 3444, 3450, 3485, 3534, 3654, 3689, 3705

Offset: 1

Author

M. F. Hasler, Nov 13 2018

Keywords

• nonn

Comments

A321503 lists numbers m such that m and m+1 both have at least 3 distinct prime factors, while A140077 lists numbers such that m and m+1 have exactly 3 distinct prime factors. This sequence is the complement of the latter in the former, it consists of terms with indices (15, 60, 82, 98, 99, 104, ...) of the former.

Since m and m+1 can't share a prime factor, we have a(n)*(a(n)+1) >= p(3+4)# = A002110(7). Remarkably enough, a(1) = A000196(A002110(3+4)) exactly!

Links

• M. F. Hasler, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A140077, A321503.

Cf. A321494, A321495, A321496, A321497 (analog for 4, 5, 6, 7 factors).

Programs

Mathematica

aQ[n_]:=Module[{v={PrimeNu[n],PrimeNu[n+1]}},Min[v]>2 && v!={3,3}]; Select[Range[120000], aQ] (* Amiram Eldar, Nov 12 2018 *)
dpfQ[{a_,b_}]:=a>2&&b>2&&(a>3||b>3); Position[Partition[PrimeNu[Range[4000]],2,1],?dpfQ]//Flatten (* _Harvey P. Dale, Apr 21 2025 *)

PARI

select( is(n)=omega(n)>2&&omega(n+1)>2&&(omega(n)>3||omega(n+1)>3), [1..1300])

Formula

A321503 \ A140077.