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A125508
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Lowest number having a particular shape of factor decomposition binary tree.
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2
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1, 2, 4, 8, 16, 20, 32, 40, 64, 72, 88, 128, 160, 176, 200, 220, 256, 272, 288, 320, 336, 360, 400, 420, 460, 480, 512, 540, 544, 640, 704, 864, 880, 920, 1024, 1056, 1152, 1184, 1200, 1280, 1344, 1440, 1600, 1640, 1680, 1800, 1840, 1920, 2048, 2176, 2368
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OFFSET
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1,2
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COMMENTS
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For a natural number, n, make it the root node of a binary tree. The left child node (L) is the largest divisor of n which is greater than 1 but less than or equal to the square root of n, if this exists. The right child node is n/L, if the left node exists. Thus if n is a prime it is a leaf node; otherwise if it is composite then it is the product of its two children. If n = 1 then we have an empty tree. A term in this sequence a(n) is such that no number (x) exists 1<=x<a(n) with an isomorphic binary tree.
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LINKS
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EXAMPLE
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a(8) = 40, make this the root node.
40 is first decomposed into 5 and 8; 5 is a leaf node.
8 is decomposed into 2 and 4; 2 is a leaf node.
4 is decomposed into 2 and 2; both are leaf nodes.
There is no number x (where 1 <= x < 40) which has a factor decomposition binary tree isomorphic to this.
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PROG
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(PARI) fissr(n) = {if (isprime(n), n, my(d = divisors(n)); my(nddt = #d\2 + 1); if (#d % 2, [fissr(d[nddt]), n, fissr(d[nddt])], [fissr(d[nddt-1]), n, fissr(d[nddt])]); ); }
fiss(n) = if (n==1, [], if (type(fv = fissr(n))== "t_INT", [fv], fv));
empty(v) = {my(nv = vector(#v)); for (i=1, #v, if (type(v[i]) == "t_INT", nv[i] = 0, nv[i] = empty(v[i])); ); nv; }
eqvec(va, vb) = {if (type(va) != type(vb), return (0)); if (type(va) == "t_INT", return (va == vb)); if (#va != #vb, return (0)); for (i=1, #va, if (!eqvec(va[i], vb[i]), return (0)); ); return (1); }
isalready(erep, vrep) = {if (! #vrep, return (0)); for (j=1, #vrep, if (eqvec(erep, vrep[j]), return (1); ); ); return (0); }
addrep(erep, vrep) = {nvrep = vector(#vrep+1, i, if (i <= #vrep, vrep[i], 0)); nvrep[#vrep+1] = erep; nvrep; }
lista(nn) = {vrep = []; for (n=1, nn, rep = fiss(n); erep = empty(rep); if (! isalready(erep, vrep), print1(n, ", "); vrep = addrep(erep, vrep); ); ); } \\ Michel Marcus, May 25 2014
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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