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A304768
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Augmented integer conjugate of n. a(n) = (1/n) * A007947(n)^(1 + A051903(n)) where A007947 is squarefree kernel and A051903 is maximum prime exponent.
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2
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1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 18, 13, 14, 15, 2, 17, 12, 19, 50, 21, 22, 23, 54, 5, 26, 3, 98, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 250, 41, 42, 43, 242, 75, 46, 47, 162, 7, 20, 51, 338, 53, 24, 55, 686, 57, 58, 59, 450, 61, 62, 147, 2, 65, 66, 67
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OFFSET
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1,2
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COMMENTS
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Image is the weak numbers A052485, on which n -> a(n) is an involution whose fixed points are the squarefree numbers A005117.
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LINKS
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FORMULA
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If n = Product_{i = 1..k} prime(x_i)^y_i, then a(n) = Product_{i = 1..k} prime(x_i)^(max{y_1,...,y_k} - y_i + 1).
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MATHEMATICA
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acj[n_]:=Module[{f, m}, f=FactorInteger[n]; m=Max[Last/@f]; Times@@Table[p[[1]]^(m-p[[2]]+1), {p, f}]];
Array[acj, 100]
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PROG
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(PARI) a(n) = {if(n==1, 1, my(f = factor(n), e = vecmax(f[, 2]) + 1); prod(i = 1, #f~, f[i, 1]^e) / n); } \\ Amiram Eldar, Feb 12 2023
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CROSSREFS
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Cf. A001597, A001694, A005117, A007916, A007947, A013929, A051903, A052410, A062759, A066638, A072774, A087320, A303554.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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