A369966 Numbers of the form 4m+2 that have an even number of prime factors with multiplicity.

6, 10, 14, 22, 26, 34, 38, 46, 54, 58, 62, 74, 82, 86, 90, 94, 106, 118, 122, 126, 134, 142, 146, 150, 158, 166, 178, 194, 198, 202, 206, 210, 214, 218, 226, 234, 250, 254, 262, 274, 278, 294, 298, 302, 306, 314, 326, 330, 334, 342, 346, 350, 358, 362, 382, 386, 390, 394, 398, 414, 422, 446, 454, 458, 462, 466

Offset

1,1

Comments

Numbers that have one even prime factor (2) and an odd number of odd prime factors with multiplicity.

Numbers that have an even number of prime factors with multiplicity and whose arithmetic derivative (A003415) is odd.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Formula

a(n) = 2*A067019(n).

Mathematica

Select[Range[2, 400, 4], EvenQ[PrimeOmega[#]] &] (* Paolo Xausa, Jan 21 2026 *)

Prog

(PARI) \\ See A369965
(Python)
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
from oeis_sequences.OEISsequences import bisection
def A369966(n):
    def g(x, a, b, c, m): yield from (((d, ) for d in enumerate(primerange(b, isqrt(x//c)+1), a)) if m==2 else (((a2, b2), )+d for a2, b2 in enumerate(primerange(b, integer_nthroot(x//c, m)[0]+1), a) for d in g(x, a2, b2, c*b2, m-1)))
    def f(x): return int(n+2+x-(y:=x>>1)+sum(sum(primepi(y//prod(c[1] for c in a))-a[-1][0] for a in g(y, 0, 1, 1, m)) for m in range(2, x.bit_length(), 2))+sum(sum(primepi((y>>1)//prod(c[1] for c in a))-a[-1][0] for a in g(y>>1, 0, 1, 1, m)) for m in range(2, x.bit_length()-1, 2)))
    return bisection(f, n, n) # Chai Wah Wu, Dec 19 2025

Crossrefs

Intersection of A016825 and A028260.

Intersection of A028260 and A235991.

Cf. A003415, A067019, A369668 (subsequence), A369965 (characteristic function).

Cf. A369661 (numbers whose arithmetic derivative is in this sequence).

Sequence in context: A080784 A072978 A381642 * A055163 A342192 A119431

Adjacent sequences: A369963 A369964 A369965 * A369967 A369968 A369969

Keyword

nonn

Author

Antti Karttunen, Feb 08 2024

Status

approved