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Search: id:A130849
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| A130849 |
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If m = p_1^e_1 * p_2^e_2 * ... * p_k^e^k, n = p_1^f_1 * p_2^f_2 * ... * p_k^f^k we define d(m, n) = Sum[ Abs[e_i - f_i], {i, 1, k}] to be the multiplicative distance between m and n. The numbers in the sequence are half the sum of the antidiagonals of the table (A130836) of distances between integers using this metric. |
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+0
2
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| 0, 1, 1, 4, 2, 7, 4, 9, 8, 15, 6, 19, 13, 16, 13, 28, 15, 32, 17, 28, 27, 40, 16, 41, 34, 39, 30, 55, 28, 59, 34, 53, 50, 59, 32, 75, 57, 64, 41, 84, 47, 88, 55, 66, 72, 97, 42, 97, 71, 90, 70, 113, 65, 104, 67, 104, 97, 128, 56, 133, 103, 102, 82, 129, 89, 150, 99, 130, 100
(list; graph; listen)
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OFFSET
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1,4
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FORMULA
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a(n) = 1/2 * Sum[d(n-i, i+1), {i, 0, n-1}]
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EXAMPLE
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d(3, 1) = 1
d(2, 2) = 0
d(1, 3) = 1
So a(3) = 1/2 * (1 + 0 + 1) = 1
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MATHEMATICA
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MultDistance[m_, n_] := Module[{ mfac = FactorInteger[m], nfac = FactorInteger[ n]}, Plus @@ Map[(If[Length[ # ] == 1, #[[1, 2]], Abs[ #[[1, 2]] - #[[2, 2]]]]) &, Split[ Sort[Flatten[{mfac, nfac}, 1]], (#1[[1]] == #2[[1]]) &]]] DiagSum[n_] := 1/2 Sum[MultDistance[n - i, i + 1], {i, 0, n - 1}] Table[DiagSum[j], {j, 1, 1000}]
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PROGRAM
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(PARI) multDist(m, n)={my(f=vecsort(concat(factor(m)[, 1], factor(n)[, 1]), , 8)); sum(i=1, #f, abs(valuation(m, f[i])-valuation(n, f[i])))}; a(n)={sum(i=0, (n/2, multDist(n-i, i+1))};
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CROSSREFS
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Equals half the antidiagonal sums of A130836.
Sequence in context: A026213 A123684 A002949 this_sequence A138754 A021963 A131914
Adjacent sequences: A130846 A130847 A130848 this_sequence A130850 A130851 A130852
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KEYWORD
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nonn,easy
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AUTHOR
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Jacob Woolcutt (woolcutt(AT)gmail.com), Jul 21 2007
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EXTENSIONS
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Program and corrections by Charles R Greathouse IV (charles.greathouse(AT)case.edu), Sep 02 2009
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