

A093641


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


55



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
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OFFSET

1,2


COMMENTS

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


LINKS

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


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


EXAMPLE

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


MATHEMATICA

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


PROG

(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
(Haskell)
a093641 n = a093641_list !! (n1)
a093641_list = filter ((<= 2) . a001227) [1..]
 Reinhard Zumkeller, May 01 2012


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.
Sequence in context: A207674 A162722 A123345 * A209638 A191844 A096157
Adjacent sequences: A093638 A093639 A093640 * A093642 A093643 A093644


KEYWORD

nonn,easy


AUTHOR

Reinhard Zumkeller, Apr 07 2004


STATUS

approved



