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A241916
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a(2^k) = 2^k, and for other numbers, if n = 2^e1 * 3^e2 * 5^e3 * ... p_k^e_k, then a(n) = 2^(e_k - 1) * 3^(e_{k-1}) * ... * p_{k-1}^e2 * p_k^(e1+1). Here p_k is the greatest prime factor of n (A006530), and e_k is its exponent (A071178), and the exponents e1, ..., e_{k-1} >= 0.
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30
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1, 2, 3, 4, 5, 9, 7, 8, 6, 25, 11, 27, 13, 49, 15, 16, 17, 18, 19, 125, 35, 121, 23, 81, 10, 169, 12, 343, 29, 75, 31, 32, 77, 289, 21, 54, 37, 361, 143, 625, 41, 245, 43, 1331, 45, 529, 47, 243, 14, 50, 221, 2197, 53, 36, 55, 2401, 323, 841, 59, 375, 61, 961, 175, 64
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OFFSET
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1,2
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COMMENTS
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For other numbers than the powers of 2 (that are fixed), this permutation reverses the sequence of exponents in the prime factorization of n from the exponent of 2 to that of the largest prime factor, except that the exponents of 2 and the greatest prime factor present are adjusted by one. Note that some of the exponents might be zeros.
From the above it follows, that this fixes both primes (A000040) and powers of two (A000079), among other numbers.
Even positions from n=4 onward contain only terms of A070003, and the odd positions only the terms of A102750, apart from 1 which is at a(1), and 2 which is at a(2).
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LINKS
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FORMULA
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If 2n has prime factorization Product_{i=1..k} prime(x_i), then a(n) = Product_{i=1..k-1} prime(x_k-x_i+1). The opposite version is A000027, even bisection of A246277. - Gus Wiseman, Dec 28 2022
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MATHEMATICA
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nn = 65; f[n_] := If[n == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ n]; g[w_List] := Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, w]; Table[If[IntegerQ@ #, n/4, g@ Reverse@(# - Join[{1}, ConstantArray[0, Length@ # - 2], {1}] &@ f@ n)] &@ Log2@ n, {n, 4, 4 nn, 4}] (* Michael De Vlieger, Aug 27 2016 *)
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PROG
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(PARI)
A209229(n) = (n && !bitand(n, n-1));
A241916(n) = if(1==A209229(n), n, my(f = factor(2*n), nbf = #f~, igp = primepi(f[nbf, 1]), g = f); for(i=1, nbf, g[i, 1] = prime(1+igp-primepi(f[i, 1]))); factorback(g)/2); \\ Antti Karttunen, Jul 02 2018
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CROSSREFS
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The sum of prime indices of a(n) is A243503(n).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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