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Lexicographically earliest infinite sequence such that a(i) = a(j) => A344696(i) = A344696(j) and A344697(i) = A344697(j), for all i, j >= 1.
1

%I #17 Jan 20 2023 01:31:14

%S 1,1,1,2,1,1,1,3,4,1,1,2,1,1,1,5,1,4,1,2,1,1,1,3,6,1,7,2,1,1,1,8,1,1,

%T 1,9,1,1,1,3,1,1,1,2,4,1,1,5,10,6,1,2,1,7,1,3,1,1,1,2,1,1,4,11,1,1,1,

%U 2,1,1,1,12,1,1,6,2,1,1,1,5,13,1,1,2,1,1,1,3,1,4,1,2,1,1,1,8,1,10,4,14,1,1,1,3,1

%N Lexicographically earliest infinite sequence such that a(i) = a(j) => A344696(i) = A344696(j) and A344697(i) = A344697(j), for all i, j >= 1.

%C Restricted growth sequence transform of the ordered pair [A344696(n), A344697(n)].

%C For all i, j, A003557(i) = A003557(j) => a(i) = a(j); in other words, this sequence is a function of A003557. This follows because A344696(n) = A344696(A057521(n)), A344697(n) = A344696(A057521(n)), and A057521(n) = A064549(A003557(n)).

%C Apparently, A081770 gives the positions of 2's, which occur on those n where A344696(n) = 7 and A344697(n) = 6.

%H Antti Karttunen, <a href="/A349574/b349574.txt">Table of n, a(n) for n = 1..65537</a>

%F For all n >= 1, a(n) = a(A057521(n)). [See comments]

%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 A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615

%o Aux349574(n) = { my(s=sigma(n),u=A001615(n),g=gcd(u,s)); [s/g, u/g]; };

%o v349574 = rgs_transform(vector(up_to, n, Aux349574(n)));

%o A349574(n) = v349574[n];

%Y Cf. A000203, A001615, A003557, A005117 (positions of 1's), A057521, A064549, A081770, A280292, A344695, A344696, A344697.

%K nonn

%O 1,4

%A _Antti Karttunen_, Nov 22 2021