A374131 - OEIS

%I #11 Jun 30 2024 16:13:05

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

%T 19,4,20,21,22,23,24,4,25,26,27,4,28,4,29,30,31,4,32,16,10,33,34,4,35,

%U 36,37,38,39,4,40,4,41,42,43,44,45,4,46,47,48,4,14,4,49,50,33,51,52,4,50,53,54,4,55,56,57,58,59,4,60,61,62,63,64,65,66,4,15,67,68

%N Lexicographically earliest infinite sequence such that for all i, j >= 1, a(i) = a(j) => f(i) = f(j), where f(1) = 1, and for n > 1, f(n) = [A083345(n), A374132(n), A374133(n)], where A083345 is the numerator of the fully additive function with a(p) = 1/p, and A374132 and A374133 are the 2- and 3-adic valuations of A276085, which is fully additive with a(p) = p#/p.

%C Restricted growth sequence transform of the function f given in the definition.

%C For all i, j >= 1:

%C   A305900(i) = A305900(j) => a(i) = a(j),

%C   a(i) = a(j) => A035263(i) = A035263(j),

%C   a(i) = a(j) => A369001(i) = A369001(j),

%C   a(i) = a(j) => A369004(i) = A369004(j),

%C   a(i) = a(j) => A372573(i) = A372573(j),

%C   a(i) = a(j) => A373137(i) = A373137(j),

%C   a(i) = a(j) => A373258(i) = A373258(j),

%C   a(i) = a(j) => A373483(i) = A373483(j).

%H Antti Karttunen, <a href="/A374131/b374131.txt">Table of n, a(n) for n = 1..100000</a>

%o (PARI)

%o up_to = 100000;

%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 A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };

%o A276085(n) = { my(f=factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1,primepi(f[k, 1]-1),prime(i))); };

%o Aux374131(n) = if(1==n, n, my(u=A276085(n)); [A083345(n), valuation(u, 2), valuation(u, 3)]);

%o v374131 = rgs_transform(vector(up_to, n, Aux374131(n)));

%o A374131(n) = v374131[n];

%Y Cf. A083345, A374132, A374133.

%Y Cf. A035263, A305900, A369001, A369004, A372573, A373137, A373258, A373483.

%Y Cf. also A373151.

%K nonn

%O 1,2

%A _Antti Karttunen_, Jun 30 2024