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 A130836 Square array d(m,n) = multiplicative distance between m>=1 and n>=1, read by antidiagonals. 4
 0, 1, 1, 1, 0, 1, 2, 2, 2, 2, 1, 1, 0, 1, 1, 2, 2, 3, 3, 2, 2, 1, 1, 2, 0, 2, 1, 1, 3, 2, 1, 3, 3, 1, 2, 3, 2, 2, 2, 2, 0, 2, 2, 2, 2, 2, 3, 4, 3, 3, 3, 3, 4, 3, 2, 1, 1, 1, 1, 2, 0, 2, 1, 1, 1, 1, 3, 2, 3, 4, 4, 3, 3, 4, 4, 3, 2, 3, 1, 2, 2, 2, 3, 3, 0, 3, 3, 2, 2, 2, 1, 2, 2, 2, 3, 1, 2, 4, 4, 2, 1, 3, 2, 2, 2 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS If m = p_1^e_1 * p_2^e_2 * ... * p_k^e^k and n = p_1^f_1 * p_2^f_2 * ... * p_k^f^k, we define d(m, n) = Sum_{i = 1..k} |e_i - f_i| to be the multiplicative distance between m and n (see A130849). Equivalently, if m/n = Product p_k^e_k, then d(m,n) = Sum |e_k|. - M. F. Hasler, Dec 08 2019 LINKS Robert Israel, Table of n, a(n) for n = 1..10011 (first 141 antidiagonals, flattened) D. Dominici, An Arithmetic Metric, arXiv:0906.0632 [math.NT], 2009. FORMULA a(n,m) = A127185(n,m). - R. J. Mathar, Oct 17 2007 EXAMPLE Array begins: 0 1 1 2 1 2 1 3 ... 1 0 2 1 2 1 2 2 ... 1 2 0 3 2 1 2 4 ... 2 1 3 0 3 2 3 1 ... ... MAPLE A001222 := proc(n) numtheory[bigomega](n) ; end: A127185 := proc(n, m) local g ; g := gcd(n, m) ; RETURN(A001222(n/g)+A001222(m/g)) ; end: A130836 := proc(n, m) A127185(n, m) ; end: for d from 1 to 17 do for n from 1 to d do printf("%d, ", A130836(n, d-n+1)) ; od: od: # R. J. Mathar, Oct 17 2007 g:= proc(s) local t;   if s = 1 then 0   elif type(s, function) then 1   elif type(s, `^`) then abs(op(2, s))   else add(procname(t), t=s)   fi end proc: f:= (m, n) -> g(ifactor(m)/ifactor(n)): seq(seq(f(m, n-m), m=1..n-1), n=1..20); # Robert Israel, Sep 17 2018 MATHEMATICA t[n_, n_] = 0; t[n_, 1] := PrimeOmega[n]; t[n_, m_] := With[{g = GCD[n, m]}, PrimeOmega[n/g] + PrimeOmega[m/g]];  Table[t[n-m+1, m], {n, 1, 14}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jan 17 2014, after R. J. Mathar *) PROG (PARI) T(m, n) = {if (m==n, 0, my(f=vecsort(concat(factor(m)[, 1], factor(n)[, 1]), , 8)); sum(i=1, #f, abs(valuation(m, f[i])-valuation(n, f[i]))))}; \\ Michel Marcus, Sep 20 2018 (PARI) A130836(m, n)=vecsum(abs(factor(m/n)[, 2])) \\ M. F. Hasler, Dec 07 2019 CROSSREFS Half of antidiagonal sums gives A130849. First row is A001222. Sequence in context: A334235 A230415 A101080 * A279185 A161385 A152907 Adjacent sequences:  A130833 A130834 A130835 * A130837 A130838 A130839 KEYWORD nonn,tabl,easy AUTHOR N. J. A. Sloane, Sep 28 2007 EXTENSIONS More terms from R. J. Mathar, Oct 17 2007 STATUS approved

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Last modified May 18 05:10 EDT 2021. Contains 343994 sequences. (Running on oeis4.)