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

1,1

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

Formula

A321503 \ A140077.

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 *)

Prog

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

Crossrefs

Cf. A140077, A321503.

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

Sequence in context: A251031 A256509 A259674 * A260282 A360358 A166841

Adjacent sequences: A321490 A321491 A321492 * A321494 A321495 A321496

Keyword

nonn

Author

M. F. Hasler, Nov 13 2018

Status

approved