A093641 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

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

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

(Haskell)

a093641 n = a093641_list !! (n-1)

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.