A327977 Breadth-first reading of the subtree rooted at 7 of the tree where each parent node is the arithmetic derivative (A003415) of all its children.

7, 10, 21, 25, 18, 38, 46, 65, 77, 217, 361, 129, 205, 493, 529, 98, 426, 718, 170, 254, 462, 982, 1501, 2077, 2257, 2105, 2933, 6953, 11513, 14393, 16469, 17813, 19769, 21653, 24053, 25769, 27413, 29993, 34553, 35369, 41273, 42233, 42869, 44969, 45113, 45173, 11917, 27757, 38881, 45937, 62317, 76897, 84781, 102637, 111457, 114481, 117217, 118477, 120781, 127117, 128881, 501, 1141

Offset: 1

Author

Antti Karttunen, Oct 02 2019

Keywords

• nonn
• tabf

Comments

Permutation of A328117.

The branching degree of vertex v is given by A099302(v).

Leaves form a subsequence of A098700.

For any number k at level n (where 7 is at level 2), we have A256750(k) = A327966(k) = n.

Question: Does this subtree contain infinitely long paths? How many? Cf. conjecture number 8 in Ufnarovski and Ahlander paper. As an example of possible beginning of such a sequence they give: 1 ← 7 ← 10 ← 25 ← 46 ← 129 ← 170 ← 501 ← 414 ← 2045.

Examples

			The 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 7 onward:
   (0)
    |
   (1)
    |
    7
    |
    10______________________________
    |                               |
    21________                     25
    |         |                     |
    18___    38_____               46_________________________________
    |    |    |     |               |            |      |             |
    65   77  217   361____         129____      205    493_____      529
         |          |     |         |     |             |      |
         98        426   718       170   254           462    982
         |          |     |         |     |             |      |
        [3]       [21]   [15]      [9]   [9]           [28]   [17]
On the last level illustrated above, the numbers in brackets [ ] tell how many children the node has. E.g, there are three for 98: 1501, 2077, 2257, as A003415(1501) = A003415(2077) = A003415(2257) = 98, and nine for 170: 501, 1141, 2041, 2869, 4309, 5461, 6649, 6901, 7081.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..204
• Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.

Crossrefs

Cf. A003415, A098699, A098700, A099302, A099303, A099307, A099308, A189760, A256750, A327966, A327968, A328117.

Cf. A327975 for the subtree starting from 5, and also A263267 for another similar tree.

Programs

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
A327977list(e) = { my(lista=List([7]), f); for(i=1, e, f = lista[i]; for(k=1,1+A002620(f),if(A003415(k)==f, listput(lista,k)))); Vec(lista); };

PARI

\\ With precomputed large A328117, use this:
v328117 = readvec("a328117.txt");
A327977list(e) = { my(lista=List([7]), f, i); for(n=1, e, f = lista[n]; print("n=",n," #lista=", #lista, " A002620(",f,")=",A002620(f)); my(u=1+A002620(f)); if(u>=v328117[#v328117],print("Not enough precomputed terms of A328117 as search upper limit ", u, " > ", v328117[#v328117], " (the last item in v328117). Number of expansions so far=", n); return(1/0)); i=1; while(v328117[i]A003415(v328117[i])==f, listput(lista,v328117[i])); i++)); Vec(lista); };
v327977 = A327977list(114);
A327977(n) = v327977[n];
for(n=1,#v327977,write("b327977.txt", n, " ", A327977(n)));

Sage

# uses[A003415]
def A327977():
  '''Breadth-first reading of irregular subtree rooted at 7, defined by the edge-relation A003415(child) = parent. Starts giving terms from 7 onward, after a(0) = 0 and a(1) = 1.'''
  yield 7
  for x in A327977():
    for k in [1 .. 1+floor((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(52, A327977())