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A292585
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Restricted growth sequence transform of A278222(A292385(n)).
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6
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1, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 2, 4, 3, 2, 2, 5, 3, 5, 3, 4, 3, 5, 2, 6, 4, 2, 3, 7, 2, 7, 2, 4, 5, 2, 3, 8, 5, 3, 3, 9, 4, 9, 3, 3, 5, 9, 2, 10, 6, 4, 4, 11, 2, 4, 3, 7, 7, 11, 2, 12, 7, 3, 2, 5, 4, 12, 5, 7, 2, 12, 3, 13, 8, 3, 5, 3, 3, 13, 3, 3, 9, 13, 4, 4, 9, 5, 3, 14, 3, 4, 5, 8, 9, 4, 2, 15, 10, 3, 6, 16, 4, 16, 4, 3
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OFFSET
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1,2
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COMMENTS
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Term a(n) essentially records the run lengths of numbers of form 4k+1 encountered when starting from that node in binary tree A005940 which contains n, and by then traversing towards the root by iterating the map n -> A252463(n). The actual run lengths can be read from the exponents of primes in the prime factorization of A278222(A292385(m)), where m = min_{k=1..n} for which a(k) = a(n). In compound filter A292584 this is combined with similar information about the run lengths of the numbers of the form 4k+3 (A292583).
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LINKS
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EXAMPLE
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When traversing from the root of binary tree A005940 from the node which contains 5, one obtains path 5 -> 3 -> 2 -> 1. Of these numbers, 5 and 1 are of the form 4k+1, while others are not, thus there are two separate runs of length 1: [1, 1]. On the other hand, when traversing from 9 as 9 -> 4 -> 2 -> 1, again only two terms are of the form 4k+1: 9 and 1 and they are not next to each other, so we have the same two runs of one each: [1, 1]. Similarly for n = 7, and n = 10 as neither in path 7 -> 5 -> 3 -> 2 -> 1 nor in path 10 -> 5 -> 3 -> 2 -> 1 are any more 4k+1 terms (compared to the path beginning from 5). Thus a(5), a(7), a(9) and a(10) are all allotted the same value by the restricted growth sequence transform, which in this case is 3. Note that 3 occurs in this sequence for the first time at n=5, with A292385(5) = 5 and A278222(5) = 6 = 2^1 * 3^1, where those run lengths 1 and 1 are the prime exponents of 6.
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PROG
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(PARI)
allocatemem(2^30);
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset, vec, bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ This function from Charles R Greathouse IV, Aug 17 2011
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f)};
write_to_bfile(1, rgs_transform(vector(16384, n, A278222(A292385(n)))), "b292585_upto16384.txt");
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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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