%I A130849
%S A130849 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,
%T A130849 55,28,59,34,53,50,59,32,75,57,64,41,84,47,88,55,66,72,97,42,97,71,90,
%U A130849 70,113,65,104,67,104,97,128,56,133,103,102,82,129,89,150,99,130,100
%N A130849 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.
%F A130849 a(n) = 1/2 * Sum[d(n-i, i+1), {i, 0, n-1}]
%e A130849 d(3, 1) = 1
%e A130849 d(2, 2) = 0
%e A130849 d(1, 3) = 1
%e A130849 So a(3) = 1/2 * (1 + 0 + 1) = 1
%t A130849 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}]
%o A130849 (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))};
%Y A130849 Equals half the antidiagonal sums of A130836.
%Y A130849 Sequence in context: A026213 A123684 A002949 this_sequence A138754 A021963 
               A131914
%Y A130849 Adjacent sequences: A130846 A130847 A130848 this_sequence A130850 A130851 
               A130852
%K A130849 nonn,easy
%O A130849 1,4
%A A130849 Jacob Woolcutt (woolcutt(AT)gmail.com), Jul 21 2007
%E A130849 Program and corrections by Charles R Greathouse IV (charles.greathouse(AT)case.edu), 
               Sep 02 2009