A130836 Square array d(m,n) = multiplicative distance between m>=1 and n>=1, read by antidiagonals.

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

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