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A327977
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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.
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7
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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)
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(1)
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7
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10______________________________
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21________ 25
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18___ 38_____ 46_________________________________
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65 77 217 361____ 129____ 205 493_____ 529
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98 426 718 170 254 462 982
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[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.
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PROG
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(PARI)
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]<u, if(A003415(v328117[i])==f, listput(lista, v328117[i])); i++)); Vec(lista); };
v327977 = A327977list(114);
for(n=1, #v327977, write("b327977.txt", n, " ", A327977(n)));
'''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 k in [1 .. 1+floor((x*x)/2)]:
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)
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CROSSREFS
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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.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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