cp's OEIS Frontend

A093641 Numbers of form 2^i * prime(j), i>=0, j>0, together with 1.

%I A093641 #42 Feb 02 2025 20:13:45
%S A093641 1,2,3,4,5,6,7,8,10,11,12,13,14,16,17,19,20,22,23,24,26,28,29,31,32,
%T A093641 34,37,38,40,41,43,44,46,47,48,52,53,56,58,59,61,62,64,67,68,71,73,74,
%U A093641 76,79,80,82,83,86,88,89,92,94,96,97,101,103,104,106,107,109,112
%N A093641 Numbers of form 2^i * prime(j), i>=0, j>0, together with 1.
%C A093641 a(n) is either 1, prime, or of form 2a(m), m<n.
%C A093641 1 and Heinz numbers of hook integer partitions. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). A hook is a partition of the form (n,1,1,...,1). - _Gus Wiseman_, Sep 15 2018
%C A093641 Numbers whose odd part is noncomposite. - _Peter Munn_, Aug 06 2020
%H A093641 Charles R Greathouse IV, <a href="/A093641/b093641.txt">Table of n, a(n) for n = 1..10000</a>
%F A093641 A001227(a(n)) <= 2. - _Reinhard Zumkeller_, May 01 2012
%F A093641 Number A(x) of a(n) not exceeding x equals 1 + pi(x) + pi(x/2) + pi(x/4) + ..., where pi(x) is the number of primes <= x. If x goes to infinity, A(x)~2*x/log(x) and a(n)~n*log(n)/2 (n-->infinity). - _Vladimir Shevelev_, Feb 06 2014
%e A093641 55 is not a member, as 5*11 is not of the form 2^i * prime.
%t A093641 hookQ[n_]:=MatchQ[DeleteCases[FactorInteger[n],{2,_}],{}|{{_,1}}];
%t A093641 Select[Range[100],hookQ] (* _Gus Wiseman_, Sep 15 2018 *)
%o A093641 (PARI) upTo(lim)=my(v=List([1])); for(e=0, log(lim)\log(2), forprime(p=2, lim>>e, listput(v,p<<e))); vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Aug 21 2011
%o A093641 (PARI) isok(m) = my(k=m/2^valuation(m,2)); (k == 1) || isprime(k); \\ _Michel Marcus_, Mar 16 2023
%o A093641 (Haskell)
%o A093641 a093641 n = a093641_list !! (n-1)
%o A093641 a093641_list = filter ((<= 2) . a001227) [1..]
%o A093641 -- _Reinhard Zumkeller_, May 01 2012
%o A093641 (Python)
%o A093641 from sympy import primepi
%o A093641 def A093641(n):
%o A093641     def bisection(f,kmin=0,kmax=1):
%o A093641         while f(kmax) > kmax: kmax <<= 1
%o A093641         kmin = kmax >> 1
%o A093641         while kmax-kmin > 1:
%o A093641             kmid = kmax+kmin>>1
%o A093641             if f(kmid) <= kmid:
%o A093641                 kmax = kmid
%o A093641             else:
%o A093641                 kmin = kmid
%o A093641         return kmax
%o A093641     def f(x): return n-1+x-sum(primepi(x>>i) for i in range(x.bit_length()))
%o A093641     return bisection(f,n,n) # _Chai Wah Wu_, Feb 02 2025
%Y A093641 A093640(a(n)) = A000005(a(n)); A000040 and A000079 are subsequences.
%Y A093641 A105440 is a subsequence, see also A105442. - _Reinhard Zumkeller_, Apr 09 2005
%Y A093641 Cf. A078822, A007088.
%Y A093641 Complement of A105441; A001221(a(n))<=2; A005087(a(n))<=1; A087436(a(n))<=1.
%Y A093641 See also A105442.
%Y A093641 Union of A038550 and A000079, see also A008578.
%Y A093641 Cf. A082733, A153452, A296188, A296561, A300121, A304438, A305940, A317554.
%Y A093641 Cf. A000265 (odd part), A008578 (noncomposite).
%K A093641 nonn,easy
%O A093641 1,2
%A A093641 _Reinhard Zumkeller_, Apr 07 2004