A321503 Numbers m such that m and m+1 both have at least 3 distinct prime factors.

230, 285, 429, 434, 455, 494, 560, 594, 609, 615, 644, 645, 650, 665, 714, 740, 741, 759, 804, 805, 819, 825, 854, 860, 884, 902, 935, 945, 969, 986, 987, 1001, 1014, 1022, 1034, 1035, 1044, 1064, 1065, 1070, 1085, 1104, 1105, 1130, 1196, 1209, 1220, 1221, 1235, 1239, 1245, 1265

Offset: 1

Author

M. F. Hasler, Nov 13 2018

Keywords

• nonn

Comments

Disjoint union of A140077 (omega({m, m+1}) = {3}) and A321493 (not both have exactly 3 prime divisors). The latter contains terms with indices {15, 60, 82, 98, 99, 104, ...} of this sequence.

Numbers m and m+1 can never have a common prime factor (consider them mod p), therefore the terms are > sqrt(A002110(3+3)), A002110 = primorial.

Links

• M. F. Hasler, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A000977.

Cf. A255346, A321504 .. A321506, A321489 (analog for k = 2, ..., 7 prime divisors).

Cf. A321493, A321494 .. A321497 (subsequences of the above: m or m+1 has more than k prime divisors).

Cf. A074851, A140077, A140078, A140079 (complementary subsequences: m and m+1 have exactly k = 2, 3, 4, 5 prime divisors).

Programs

Mathematica

aQ[n_]:=Module[{v={PrimeNu[n], PrimeNu[n+1]}}, Min[v]>2]; Select[Range[1300], aQ] (* Amiram Eldar, Nov 12 2018 *)

PARI

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

Formula

a(n) ~ n. - Charles R Greathouse IV, Jan 25 2025