

A127185


Triangle of distances between n>=1 and n>=m>=1 measured by the number of noncommon 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
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OFFSET

1,5


COMMENTS

Consider the nondirected 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 (* JeanFrançois 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



