A072978 Numbers of the form m*2^Omega(m), where m>1 is odd and Omega(m)=A001222(m), the number of prime factors of m.

1, 6, 10, 14, 22, 26, 34, 36, 38, 46, 58, 60, 62, 74, 82, 84, 86, 94, 100, 106, 118, 122, 132, 134, 140, 142, 146, 156, 158, 166, 178, 194, 196, 202, 204, 206, 214, 216, 218, 220, 226, 228, 254, 260, 262, 274, 276, 278, 298, 302, 308, 314, 326, 334, 340, 346

Offset: 1

Author

Reinhard Zumkeller, Aug 20 2002

Keywords

• nonn
• nice

Comments

(number of odd prime factors) = (number of even prime factors).

A000400, A011557, A001023, A001024, A009965, A009966 and A009975 are subsequences. - Reinhard Zumkeller, Jan 06 2008

Subsequence of A028260. - Reinhard Zumkeller, Sep 20 2008

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001222, A007814, A087436.

Cf. A000400, A011557, A001023, A001024, A009965, A009966, A009975, A028260.

Programs

Mathematica

Join[{1}, Select[Range[2, 500, 2], First[#] == Total[Rest[#]] & [FactorInteger[#][[All, 2]]] &]] (* Paolo Xausa, Feb 19 2025 *)

PARI

isok(k) = {my(v = valuation(k, 2)); bigomega(k >> v) == v;} \\ Amiram Eldar, May 15 2025

Python

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A072978(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    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 h(x,n): return sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,1,3,1,n))
    def f(x): return int(n+x-primepi(x>>1)-sum(h(x>>m,m) for m in range(2,x.bit_length()+1))) if x>1 else 1
    return bisection(f,n,n) # Chai Wah Wu, Apr 10 2025

Formula

A007814(a(n)) = A087436(a(n)). - Reinhard Zumkeller, Jan 06 2008