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A093641
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Numbers of form 2^i * prime(j), i>=0, j>0, together with 1.
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72
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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
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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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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EXAMPLE
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55 is not a member, as 5*11 is not of the form 2^i * prime.
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MATHEMATICA
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hookQ[n_]:=MatchQ[DeleteCases[FactorInteger[n], {2, _}], {}|{{_, 1}}];
Select[Range[100], hookQ] (* Gus Wiseman, Sep 15 2018 *)
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PROG
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(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..]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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