cp's OEIS Frontend

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.

%I A072978 #39 May 15 2025 00:49:55
%S A072978 1,6,10,14,22,26,34,36,38,46,58,60,62,74,82,84,86,94,100,106,118,122,
%T A072978 132,134,140,142,146,156,158,166,178,194,196,202,204,206,214,216,218,
%U A072978 220,226,228,254,260,262,274,276,278,298,302,308,314,326,334,340,346
%N 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.
%C A072978 (number of odd prime factors) = (number of even prime factors).
%C A072978 A000400, A011557, A001023, A001024, A009965, A009966 and A009975 are subsequences. - _Reinhard Zumkeller_, Jan 06 2008
%C A072978 Subsequence of A028260. - _Reinhard Zumkeller_, Sep 20 2008
%H A072978 Reinhard Zumkeller, <a href="/A072978/b072978.txt">Table of n, a(n) for n = 1..10000</a>
%F A072978 A007814(a(n)) = A087436(a(n)). - _Reinhard Zumkeller_, Jan 06 2008
%t A072978 Join[{1}, Select[Range[2, 500, 2], First[#] == Total[Rest[#]] & [FactorInteger[#][[All, 2]]] &]] (* _Paolo Xausa_, Feb 19 2025 *)
%o A072978 (Python)
%o A072978 from math import prod, isqrt
%o A072978 from sympy import primerange, integer_nthroot, primepi
%o A072978 def A072978(n):
%o A072978     def bisection(f,kmin=0,kmax=1):
%o A072978         while f(kmax) > kmax: kmax <<= 1
%o A072978         kmin = kmax >> 1
%o A072978         while kmax-kmin > 1:
%o A072978             kmid = kmax+kmin>>1
%o A072978             if f(kmid) <= kmid:
%o A072978                 kmax = kmid
%o A072978             else:
%o A072978                 kmin = kmid
%o A072978         return kmax
%o A072978     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)))
%o A072978     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))
%o A072978     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
%o A072978     return bisection(f,n,n) # _Chai Wah Wu_, Apr 10 2025
%o A072978 (PARI) isok(k) = {my(v = valuation(k, 2)); bigomega(k >> v) == v;} \\ _Amiram Eldar_, May 15 2025
%Y A072978 Cf. A001222, A007814, A087436.
%Y A072978 Cf. A000400, A011557, A001023, A001024, A009965, A009966, A009975, A028260.
%K A072978 nonn,nice
%O A072978 1,2
%A A072978 _Reinhard Zumkeller_, Aug 20 2002