%I #19 Jul 23 2022 10:12:17

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

%T 20,3,21,22,23,10,24,3,25,26,27,3,28,3,29,8,30,3,31,4,32,33,34,3,35,

%U 14,36,37,38,3,39,3,40,41,42,43,44,3,45,46,47,3,48,3,49,50,51,10,52,3,53,11,54,3,55,26,56,57,58,3,59,14,60,61,62,33,63,3,64,65,66

%N Lexicographically earliest infinite sequence such that a(i) = a(j) => A342671(i) = A342671(j) and A348717(i) = A348717(j) for all i, j >= 1.

%C Restricted growth sequence transform of the ordered pair [A342671(n), A348717(n)].

%C Terms that occur in positions given by A349166 may occur only a finite number of times in this sequence. See also the array A355924.

%H Antti Karttunen, <a href="/A355833/b355833.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%e a(100) = a(3025) [= 66 as allotted by the rgs-transform] because 3025 = A003961(A003961(100)), therefore it is in the same column of the prime shift array A246278 as 100 is], and as A342671(100) = A342671(3025) = 7.

%e a(300) = a(21175) [= 200 as allotted by the rgs-transform], as 21175 = A003961(A003961(300)) and as A342671(300) = A342671(21175) = 7.

%e a(1215) = a(21875) [= 831 as allotted by the rgs-transform] because 21875 = A003961(1215), therefore it is in the same column of the prime shift array A246278 as 1215 is, and as A342671(1215) = A342671(21875) = 7.

%e a(2835) = a(48125) [= 1953 as allotted by the rgs-transform] because 48125 = A003961(2835) and as A342671(2835) = A342671(48125) = 11.

%o (PARI)

%o up_to = 65537;

%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 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

%o A342671(n) = gcd(sigma(n), A003961(n));

%o A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));

%o Aux355833(n) = [A342671(n), A348717(n)];

%o v355833 = rgs_transform(vector(up_to,n,Aux355833(n)));

%o A355833(n) = v355833[n];

%Y Cf. A000203, A003961, A246278, A342671, A348717, A349165, A349166, A355924.

%Y Cf. also A305801, A305897, A355834, A355835.

%K nonn

%O 1,2

%A _Antti Karttunen_, Jul 20 2022