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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.

%I A327977 #30 Mar 24 2020 15:54:15
%S A327977 7,10,21,25,18,38,46,65,77,217,361,129,205,493,529,98,426,718,170,254,
%T A327977 462,982,1501,2077,2257,2105,2933,6953,11513,14393,16469,17813,19769,
%U A327977 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
%N 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.
%C A327977 Permutation of A328117.
%C A327977 The branching degree of vertex v is given by A099302(v).
%C A327977 Leaves form a subsequence of A098700.
%C A327977 For any number k at level n (where 7 is at level 2), we have A256750(k) = A327966(k) = n.
%C A327977 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.
%H A327977 Antti Karttunen, <a href="/A327977/b327977.txt">Table of n, a(n) for n = 1..204</a>
%H A327977 Victor Ufnarovski and Bo Ahlander, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.html">How to Differentiate a Number</a>, J. Integer Seqs., Vol. 6, 2003.
%e A327977 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:
%e A327977    (0)
%e A327977     |
%e A327977    (1)
%e A327977     |
%e A327977     7
%e A327977     |
%e A327977     10______________________________
%e A327977     |                               |
%e A327977     21________                     25
%e A327977     |         |                     |
%e A327977     18___    38_____               46_________________________________
%e A327977     |    |    |     |               |            |      |             |
%e A327977     65   77  217   361____         129____      205    493_____      529
%e A327977          |          |     |         |     |             |      |
%e A327977          98        426   718       170   254           462    982
%e A327977          |          |     |         |     |             |      |
%e A327977         [3]       [21]   [15]      [9]   [9]           [28]   [17]
%e A327977 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.
%o A327977 (PARI)
%o A327977 A002620(n) = ((n^2)>>2);
%o A327977 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
%o A327977 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); };
%o A327977 (PARI)
%o A327977 \\ With precomputed large A328117, use this:
%o A327977 v328117 = readvec("a328117.txt");
%o A327977 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); };
%o A327977 v327977 = A327977list(114);
%o A327977 A327977(n) = v327977[n];
%o A327977 for(n=1,#v327977,write("b327977.txt", n, " ", A327977(n)));
%o A327977 (Sage) # uses[A003415]
%o A327977 def A327977():
%o 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.'''
%o A327977   yield 7
%o A327977   for x in A327977():
%o A327977     for k in [1 .. 1+floor((x*x)/2)]:
%o A327977       if(A003415(k) == x): yield k
%o A327977 def take(n, g):
%o A327977   '''Returns a list composed of the next n elements returned by generator g.'''
%o A327977   z = []
%o A327977   if 0 == n: return(z)
%o A327977   for x in g:
%o A327977     z.append(x)
%o A327977     if n > 1: n = n-1
%o A327977     else: return(z)
%o A327977 take(52, A327977())
%Y A327977 Cf. A003415, A098699, A098700, A099302, A099303, A099307, A099308, A189760, A256750, A327966, A327968, A328117.
%Y A327977 Cf. A327975 for the subtree starting from 5, and also A263267 for another similar tree.
%K A327977 nonn,tabf
%O A327977 1,1
%A A327977 _Antti Karttunen_, Oct 02 2019