A130836 -id:A130836 - cp's OEIS frontend

A130849 a(n) is half the sum of the terms in the n-th antidiagonal of the table A130836.

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, 55, 28, 59, 34, 53, 50, 59, 32, 75, 57, 64, 41, 84, 47, 88, 55, 66, 72, 97, 42, 97, 71, 90, 70, 113, 65, 104, 67, 104, 97, 128, 56, 133, 103, 102, 82, 129, 89, 150, 99, 130, 100

Offset: 1

Jacob Woolcutt (woolcutt(AT)gmail.com), Jul 21 2007

			d(3, 1) = 1
d(2, 2) = 0
d(1, 3) = 1
So a(3) = 1/2 * (1 + 0 + 1) = 1

Equals half the antidiagonal sums of A130836.

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}]

multDist(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]))))};
a(n)={sum(i=0, (n-1)/2, multDist(n-i, i+1))}; \\ edited by Michel Marcus, Sep 20 2018

a(n) = 1/2 * Sum_{i=0..n-1} d(n-i, i+1) where, if m = Sum_{k} p_k^e^k, and n = Sum_{k} p_k^f^k, then d(m, n) = Sum_{i=1..k} abs(e_i - f_i), the multiplicative distance between m and n.

Program and corrections by Charles R Greathouse IV, Sep 02 2009

Edited by Michel Marcus, Sep 20 2018

A089913 Table T(n,k) = lcm(n,k)/gcd(n,k) = n*k/gcd(n,k)^2 read by antidiagonals (n >= 1, k >= 1).

Original entry on oeis.org

1, 2, 2, 3, 1, 3, 4, 6, 6, 4, 5, 2, 1, 2, 5, 6, 10, 12, 12, 10, 6, 7, 3, 15, 1, 15, 3, 7, 8, 14, 2, 20, 20, 2, 14, 8, 9, 4, 21, 6, 1, 6, 21, 4, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 5, 3, 2, 35, 1, 35, 2, 3, 5, 11, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12, 13, 6, 33, 10, 45

Offset: 1

Marc LeBrun, Nov 14 2003

A multiplicative analog of absolute difference A049581. Exponents in prime factorization of T(n,k) are absolute differences of those of n and k. Commutative non-associative operator with identity 1. T(nx,kx)=T(n,k), T(n^x,k^x)=T(n,k)^x, etc.
The bivariate function log(T(., .)) is a distance (or metric) function. It is a weighted analog of A130836, in the sense that if e_i (resp. f_i) denotes the exponent of prime p_i in the factorization of m (resp. of n), then both log(T(m, n)) and A130836(m, n) are writable as Sum_{i} w_i * abs(e_i - f_i). For A130836, w_i = 1 for all i, whereas for log(T(., .)), w_i = log(p_i). - Luc Rousseau, Sep 17 2018
If the analog of absolute difference, as described in the first comment, is determined by factorization into distinct terms of A050376 instead of by prime factorization, the equivalent operation is defined by A059897 and is associative. The positive integers form a group under A059897. The two factorization methods give the same factorization for squarefree numbers (A005117), so that T(.,.) restricted to A005117 is associative. Thus the squarefree numbers likewise form a group under the operation defined by this sequence. - Peter Munn, Apr 04 2019

			T(6,10) = lcm(6,10)/gcd(6,10) = 30/2 = 15.
  1,  2,  3,  4,  5, ...
  2,  1,  6,  2, 10, ...
  3,  6,  1, 12, 15, ...
  4,  2, 12,  1, 20, ...
  5, 10, 15, 20,  1, ...
  ...

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A049581, A003990, A003991, A130836, A001222, A005117, A050376, A059897.

T:=Flat(List([1..13],n->List([1..n-1],k->Lcm(k,n-k)/Gcd(k,n-k)))); # Muniru A Asiru, Oct 24 2018

Flatten[Table[LCM[i, m - i]/GCD[i, m - i], {m, 15}, {i, m - 1}]] (* Ivan Neretin, Apr 27 2015 *)

A089913(n,k)=n*k/gcd(n,k)^2 \\ M. F. Hasler, Dec 06 2019

A130836(n, k) = A001222(T(n, k)). - Luc Rousseau, Sep 17 2018

A127185 Triangle of distances between n>=1 and n>=m>=1 measured by the number of non-common prime factors.

Original entry on oeis.org

0, 1, 0, 1, 2, 0, 2, 1, 3, 0, 1, 2, 2, 3, 0, 2, 1, 1, 2, 3, 0, 1, 2, 2, 3, 2, 3, 0, 3, 2, 4, 1, 4, 3, 4, 0, 2, 3, 1, 4, 3, 2, 3, 5, 0, 2, 1, 3, 2, 1, 2, 3, 3, 4, 0, 1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 0, 3, 2, 2, 1, 4, 1, 4, 2, 3, 3, 4, 0, 1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 0, 2, 1, 3, 2, 3, 2, 1, 3, 4, 2, 3, 3, 3, 0

Offset: 1

R. J. Mathar, Mar 25 2007

Consider the non-directed graph where each integer n >= 1 is a unique node labeled by n and where nodes n and m are connected if their list of exponents in their prime number decompositions n=p_1^n_1*p_2^n_2*... and m=p_1^m_1*p_2^m_2*... differs at one place p_i by 1. [So connectedness means n/m or m/n is a prime.] The distance between two nodes is defined by the number of hops on the shortest path between them. [Actually, the shortest path is not unique if the graph is not pruned to a tree by an additional convention like connecting only numbers that differ in the exponent of the largest prime factors; this does not change the distance here.] The formula says this can be computed by passing by the node of the greatest common divisor.

			T(8,10)=T(2^3,2*5)=3 as one must lower the power of p_1=2 two times and rise the power of p_3=5 once to move from 8 to 10. A shortest path is 8<->4<->2<->10 obtained by division through 2, division through 2 and multiplication by 5.
Triangle is read by rows and starts
   n\m 1 2 3 4 5 6 7 8 9 10
   ------------------------
    1| 0
    2| 1 0
    3| 1 2 0
    4| 2 1 3 0
    5| 1 2 2 3 0
    6| 2 1 1 2 3 0
    7| 1 2 2 3 2 3 0
    8| 3 2 4 1 4 3 4 0
    9| 2 3 1 4 3 2 3 5 0
   10| 2 1 3 2 1 2 3 3 4 0

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened)
Diego Dominici, An Arithmetic Metric, arXiv:0906.0632 [math.NT], 2009.

Cf. A130836.

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], {n, 1, 14}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jan 08 2014 *)

T(n, k) = my(g=gcd(n,k)); bigomega(n/g) + bigomega(k/g);
tabl(nn) = for(n=1, nn, for (k=1, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Dec 26 2018

A127185(m,n)=vecsum(abs(factor(m/n)[, 2])) \\ M. F. Hasler, Dec 07 2019

T(n,m) = A001222(n/g)+A001222(m/g) where g=gcd(n,m)=A050873(n,m).
Special cases: T(n,n)=0. T(n,1)=A001222(n).
T(m,n) = A130836(m,n) = Sum |e_k| if m/n = Product p_k^e_k. - M. F. Hasler, Dec 08 2019

cp's OEIS Frontend

A130849 a(n) is half the sum of the terms in the n-th antidiagonal of the table A130836.

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A089913 Table T(n,k) = lcm(n,k)/gcd(n,k) = n*k/gcd(n,k)^2 read by antidiagonals (n >= 1, k >= 1).

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A127185 Triangle of distances between n>=1 and n>=m>=1 measured by the number of non-common prime factors.

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