A342011 - OEIS

%I #14 Mar 12 2021 23:11:50

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

%T 19,20,21,5,22,1,23,2,24,25,6,26,27,28,29,30,31,32,33,34,1,35,36,4,37,

%U 38,39,7,40,41,42,43,44,45,46,8,47,2,48,49,50,3,51,52,53,54,1,55,56,57,58,59,60,61,62,9,63,64

%N Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), for all i, j >= 1, with f(1) = 2 and f(n) = A004490(n)/A004490(n-1) when n > 1, where A004490(n) is the n-th colossally abundant number.

%C This is also the restricted growth sequence transform of A073751, provided that quotient A004490(1+n)/A004490(n) is always prime, which is implied by a conjecture mentioned in Lagarias' paper. Note that the b-file of A073751 is computed based on the knowledge that the conjecture holds at least for the first 10^7 quotients.

%H Antti Karttunen, <a href="/A342011/b342011.txt">Table of n, a(n) for n = 1..10000</a> (computed from the b-file of A073751 provided by T. D. Noe)

%H J. C. Lagarias, <a href="https://arxiv.org/abs/math/0008177">An elementary problem equivalent to the Riemann hypothesis</a>, arXiv:math/0008177 [math.NT], 2000-2001; Am. Math. Monthly 109 (#6, 2002), 534-543.

%F a(n) = A000720(A073751(n)), up to the first n where A004490(n)/A004490(n-1) is not a prime.

%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 v073751 = readvec("b073751_to.txt"); \\ Prepared with gawk '{ print $2 }' < b073751.txt > b073751_to.txt

%o v342011 = rgs_transform(v073751);

%o A342011(n) = v342011[n];

%o for(n=1,#v342011,write("b342011.txt", n, " ", A342011(n)));

%Y Cf. A000720, A004490, A073751, A342010, A342012, A342013.

%K nonn

%O 1,2

%A _Antti Karttunen_, Mar 08 2021