Hello Math-Fun & SeqFan,

consider this (partial) graph:

+------2281..

|

+---69..

| .70..

| / ..

|/ ..

+----13.. .

| +--14.. .

|/ .

6----15.. .

/|\ .

/ | +--16.. .

/ +----17.. .

/ .. .

+--4----7.. . .

| \ . .

1--2--3 \__8.. . .

| . .

| +----9.. . .

| | __10.. . .

| |/ . .

+--5 . .

|\__11.. . .

| . .

+---12.. . .

|\ . .

| \ .. .

| +-67.. .

+----68.. .

|\ ....

| \ ....

| +-2279..

+----2280..

... each integer indicates the number of paths linking the said integer to the rest of the graph.

The graph is very simple to build:

- start with 1;

- next node must be the smallest integer not used so far. [The yellow sequence is (almost) A108225].

Question:

- starting from 1 and always moving away, is there an “only primes” path one could follow? If yes, which one? Is it unique?

This path would obviously start with 1-2-3-5-11...

Best,

É.