1,5

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(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).

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

---------------------

.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.

Sequence in context: A125939 A125942 A061986 this_sequence A159780 A055136 A074397

Adjacent sequences: A127182 A127183 A127184 this_sequence A127186 A127187 A127188

easy,nonn,tabl

R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 25 2007