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A300827 Lexicographically earliest sequence such that a(i) = a(j) => A324193(i) = A324193(j) for all i, j >= 1.

%I A300827 #38 Jan 03 2020 12:18:05
%S A300827 1,2,2,3,2,4,2,4,5,6,2,7,2,8,9,10,2,11,2,12,13,14,2,15,16,17,9,18,2,
%T A300827 19,2,20,21,22,23,24,2,25,26,27,2,28,2,29,30,31,2,32,33,34,35,36,2,37,
%U A300827 38,39,40,41,2,42,2,43,44,45,46,47,2,48,49,50,2,51,2,52,53,54,55,56,2,57,58,59,2,60,61,62,63,64,2,65,66,67
%N A300827 Lexicographically earliest sequence such that a(i) = a(j) => A324193(i) = A324193(j) for all i, j >= 1.
%C A300827 Apart from primes, the sequence contains duplicate values at points p*q and p^3, where p*q are the product of two successive primes, with p < q (sequences A006094, A030078). Question: are there any other cases where a(x) = a(y), with x < y ?
%C A300827 The reason why this is not equal to A297169: Even though A297112 contains only powers of two after the initial zero, as A297112(n) = 2^A033265(A156552(d)) for n > 1, and A297168(n) is computed as Sum_{d|n, d<n} A297112(d), still a single 1-bit in binary expansion of A297168(n) might be formed as a sum of several terms of A297112(d), i.e., could be born of carries.
%C A300827 From _Antti Karttunen_, Feb 28 2019: (Start)
%C A300827   A297168(n) = Sum_{d|n, d<n} A297112(d) will not produce any carries (in base-2) if and only if n is a power of prime. Only in that case the number of summands (A000005(n)-1) is equal to the number of prime factors counted with multiplicity, A001222(n) = A000120(A156552(n)). (A notable subset of such numbers is A324201, numbers that are mapped to even perfect numbers by A156552). Precisely because there are so few points with duplicate values (apart from primes), this sequence is not particularly good for filtering other sequences, because the number of false positives is high. Any of the related sequences like A324203, A324196, A324197 or A324181 might work better in that respect. In any case, the following implications hold (see formula section of A324193 for the latter): (End)
%C A300827 For all i, j:
%C A300827   a(i) = a(j) => A297168(i) = A297168(j). (The same holds for A297169).
%C A300827   a(i) = a(j) => A324181(i) = A324181(j) => A324120(i) = A324120(j).
%H A300827 Antti Karttunen, <a href="/A300827/b300827.txt">Table of n, a(n) for n = 1..65537</a>
%H A300827 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A300827 Restricted growth sequence transform of sequence f, defined as f(1) = 0, f(2) = 1, and for n > 2, f(n) = Product_{d|n, 1<d<n} prime(1+A297167(d)).
%F A300827 a(p) = 2 for all primes p.
%F A300827 a(A006094(n)) = a(A030078(n)), for all n >= 1.
%e A300827 For n = 15, with proper divisors 3 and 5, we have f(n) = prime(1+A297167(3)) * prime(1+A297167(5)) = prime(2)*prime(3) = 3*5 = 15.
%e A300827 For n = 27, with proper divisors 3 and 9, we have f(n) = prime(1+A297167(3)) * prime(1+A297167(9)) = prime(2)*prime(3) = 3*5 = 15.
%e A300827 Because f(15) = f(27), the restricted growth sequence transform allots the same number (in this case 9) for both, so a(15) = a(27) = 9.
%o A300827 (PARI)
%o A300827 up_to = 65537;
%o A300827 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; };
%o A300827 write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
%o A300827 A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1]))); \\ After _M. F. Hasler_'s code for A006530.
%o A300827 A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) -1));
%o A300827 Aux300827(n) = { my(m=1); if(n<=2, n-1, fordiv(n,d,if((d>1)&(d<n),m *= prime(1+A297167(d)))); (m)); };
%o A300827 write_to_bfile(1,rgs_transform(vector(up_to,n,Aux300827(n))),"b300827.txt");
%Y A300827 Cf. A000120, A000961, A006094, A030078, A046660, A061395, A101296, A156552, A297112, A297167, A297168, A297169, A324193, A324120, A324201.
%Y A300827 Cf. also A324181, A323914, A324196, A324197, A324203.
%K A300827 nonn
%O A300827 1,2
%A A300827 _Antti Karttunen_, Mar 13 2018
%E A300827 Name changed by _Antti Karttunen_, Feb 21 2019