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

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, 34, 37, 38, 40, 41, 43, 44, 46, 47, 48, 52, 53, 56, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 79, 80, 82, 83, 86, 88, 89, 92, 94, 96, 97, 101, 103, 104, 106, 107, 109, 112

Offset: 1

Author

Reinhard Zumkeller, Apr 07 2004

Keywords

• nonn
• easy

Comments

a(n) is either 1, prime, or of form 2a(m), m

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

Numbers whose odd part is noncomposite. - Peter Munn, Aug 06 2020

Examples

			55 is not a member, as 5*11 is not of the form 2^i * prime.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

A093640(a(n)) = A000005(a(n)); A000040 and A000079 are subsequences.

A105440 is a subsequence, see also A105442. - Reinhard Zumkeller, Apr 09 2005

Cf. A078822, A007088.

Complement of A105441; A001221(a(n))<=2; A005087(a(n))<=1; A087436(a(n))<=1.

See also A105442.

Union of A038550 and A000079, see also A008578.

Cf. A082733, A153452, A296188, A296561, A300121, A304438, A305940, A317554.

Cf. A000265 (odd part), A008578 (noncomposite).

Programs

Haskell

a093641 n = a093641_list !! (n-1)
a093641_list = filter ((<= 2) . a001227) [1..]
-- Reinhard Zumkeller, May 01 2012

Mathematica

hookQ[n_]:=MatchQ[DeleteCases[FactorInteger[n],{2,}],{}|{{,1}}];
Select[Range[100],hookQ] (* Gus Wiseman, Sep 15 2018 *)

PARI

upTo(lim)=my(v=List([1])); for(e=0, log(lim)\log(2), forprime(p=2, lim>>e, listput(v,p<Charles R Greathouse IV, Aug 21 2011

PARI

isok(m) = my(k=m/2^valuation(m,2)); (k == 1) || isprime(k); \\ Michel Marcus, Mar 16 2023

Python

from sympy import primepi
def A093641(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 f(x): return n-1+x-sum(primepi(x>>i) for i in range(x.bit_length()))
    return bisection(f,n,n) # Chai Wah Wu, Feb 02 2025

Formula

A001227(a(n)) <= 2. - Reinhard Zumkeller, May 01 2012

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

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