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A127185 Triangle of distances between n>=1 and n>=m>=1 measured by the number of non-common prime factors. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..105.

D. Dominici, An Arithmetic Metric, arXiv:0906.0632 [math.NT], 2009.

FORMULA

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

EXAMPLE

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

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

PROG

(PARI) 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

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

CROSSREFS

Cf. A130836.

Sequence in context: A125942 A291440 A061986 * A159780 A324285 A138036

Adjacent sequences:  A127182 A127183 A127184 * A127186 A127187 A127188

KEYWORD

easy,nonn,tabl

AUTHOR

R. J. Mathar, Mar 25 2007

STATUS

approved

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Last modified February 20 22:47 EST 2020. Contains 332086 sequences. (Running on oeis4.)