login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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-Fran├žois 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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