OFFSET
1,1
COMMENTS
Permutation of A328115.
The branching degree of vertex v is given by A099302(v).
Leaves form a subsequence of A098700.
Most terms of A189760 (apart from 0, 1, 2, 414, ...) seem to be located in this tree, in positions where they have no smaller siblings.
Question: Does this subtree contain infinitely long paths? How many? Cf. conjecture number 8 in Ufnarovski and Ahlander paper, and a similar tree starting from 7, A327977.
LINKS
Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.
EXAMPLE
Because we have A003415(5) = 1, A003415(6) = 5, A003415(9) = 6, A003415(14) = 9, A003415(33) = A003415(49) = 14, A003415(62) = 33, etc, this subtree is laid out as below. The terms of this sequence are obtained by scanning each successive level of the tree from left to right, from the node 5 onward:
(0)
|
(1)
|
5
|
6
|
9
|
14________________
| |
33 49
| |
62________ 94_____________________________
| | | | | | | |
| | | | | | | |
177 817 961 445 913 1633 2173 2209
| | | |
| | | |
1146 886 1822 4414
| | | |
| | | |
(19193, (1155, (19921, ..., 829921) (22045, ..., 4870849)
25829, 2649, [49 children for 4414]
..., ..., [27 children for 1822]
328313) 196249)
[19 children for 886]
[38 children
for 1146]
The row lengths thus start as: 1, 1, 1, 1, 2, 2, 8, 4, 133 (= 38+19+27+49), ...
PROG
(PARI)
A002620(n) = ((n^2)>>2);
A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A327975list(e) = { my(lista=List([5]), f); for(n=1, e, f = lista[n]; for(k=1, 1+A002620(f), if(A003415(k)==f, listput(lista, k)))); Vec(lista); };
v328975 = A327975list(21);
A327975(n) = v328975[n];
(Sage) # uses[A003415]
def A327975():
'''Breadth-first reading of irregular subtree rooted at 5, defined by the edge-relation A003415(child) = parent.'''
yield 5
for x in A327975():
for k in [1 .. 1+(x*x)//2]:
if A003415(k) == x: yield k
def take(n, g):
'''Returns a list composed of the next n elements returned by generator g.'''
z = []
if 0 == n: return z
for x in g:
z.append(x)
if n > 1: n = n-1
else: return(z)
take(60, A327975())
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Antti Karttunen, Oct 02 2019
STATUS
approved