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A113474
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a(n) = a(floor(n/2)) + floor(n/2) with a(1) = 1.
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3
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1, 2, 2, 4, 4, 5, 5, 8, 8, 9, 9, 11, 11, 12, 12, 16, 16, 17, 17, 19, 19, 20, 20, 23, 23, 24, 24, 26, 26, 27, 27, 32, 32, 33, 33, 35, 35, 36, 36, 39, 39, 40, 40, 42, 42, 43, 43, 47, 47, 48, 48, 50, 50, 51, 51, 54, 54, 55, 55, 57, 57, 58, 58, 64, 64, 65, 65, 67, 67, 68, 68, 71, 71
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OFFSET
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1,2
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COMMENTS
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a(2^n) = 2^n, in other cases a(n) < n. Except for the initial 1 all entries are repeated. Apparently no simple formula is known for a(n).
1/a(n) is the probability that a randomly chosen divisor of n! is odd. This is because the product n! contains the prime factor 2 a total of a(n) - 1 times (cf. A011371) and thus the prime factor 2 can occur in a divisor of n! a total of a(n) times, ranging between 0 and a(n) - 1 times. - Martin Renner, Dec 28 2022
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LINKS
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FORMULA
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a(n) = ( Sum_{k=0..n} floor(n/2^k) ) - n + 1.
a(n) = 2 + Sum_{k=0..n} ( floor(n/2^k)-1 ).
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MATHEMATICA
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a[1]=1; a[n_]:=a[n]=a[Floor[n/2]]+Floor[n/2]; Table[a[n], {n, 100}]
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PROG
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(PARI) for(n=1, 75, print1(1 - n + sum(k=0, n, n\2^k), ", ")) \\ G. C. Greubel, Mar 11 2017
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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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