%I #72 Dec 19 2018 15:13:47

%S 1,2,3,4,5,6,7,8,9,10,6,11,12,13,13,14,10,15,16,17,18,19,13,20,21,22,

%T 23,24,25,26,18,27,28,29,28,30,31,32,33,34,22,35,36,37,38,26,28,39,40,

%U 41,26,42,29,43,26,44,45,46,32,47,48,35,49,50,51,52,53,54,35,52,26,55,56,57,58,59,35,60,45,61,62,63,51,64,65,66,67,68,46,69,70,47,71

%N Lexicographically earliest such sequence a that a(i) = a(j) => A003557(i) = A003557(j) and A048250(i) = A048250(j), for all i, j.

%C Restricted growth sequence transform of A291750, which means that this is the lexicographically least sequence a, such that for all i, j: a(i) = a(j) <=> A291750(i) = A291750(j) <=> A003557(i) = A003557(j) and A048250(i) = A048250(j). That this is equal to the definition given in the title follows because any such lexicographically least sequence satisfying relation <=> is also the least sequence satisfying relation => with the same parameters.

%C Sigma (A000203) and psi (A001615) are functions of this sequence. See comments in A291750 for the reason. For example, to find the value of A001615(n) when we know just a(n), but without knowing n, let m be the least i for which a(i) = a(n); then A001615(n) = A003991(A291750(m)) = A003557(m) * A048250(m).

%H Antti Karttunen, <a href="/A291751/b291751.txt">Table of n, a(n) for n = 1..65537</a>

%o (PARI)

%o 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 A003557(n) = n/factorback(factor(n)[, 1]); \\ From A003557

%o A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));

%o A291750(n) = (1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n));

%o v291751 = rgs_transform(vector(65537,n,A291750(n)));

%o A291751(n) = v291751[n];

%Y Cf. A001615, A003557, A048250, A291750, A291752, A294877, A295300, A295886, A295887, A295888, A319698, A322021.

%Y Differs from A286603 for the first time at n = 25, where a(25) = 21, while A286603(25) = 14.

%K nonn

%O 1,2

%A _Antti Karttunen_, Sep 06 2017

%E Name changed and comments added by _Antti Karttunen_, Nov 24 2018