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

Author

N. J. A. Sloane, Sep 28 2007

Keywords

• nonn
• tabl
• easy

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

Examples

			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 ...
...
		

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.

Crossrefs

Half of antidiagonal sums gives A130849. First row is A001222.

Programs

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 *)

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

Formula

a(n,m) = A127185(n,m). - R. J. Mathar, Oct 17 2007

Extensions

More terms from R. J. Mathar, Oct 17 2007