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A093641 Numbers of form 2^i * prime(j), i>=0, j>0, together with 1. 72
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
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. A000265 (odd part), A008578 (noncomposite).
Sequence in context: A207674 A162722 A123345 * A209638 A191844 A096157
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, Apr 07 2004
STATUS
approved

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Last modified March 28 04:13 EDT 2024. Contains 371235 sequences. (Running on oeis4.)