A350065 - OEIS

%I #13 Jan 29 2022 22:32:14

%S 1,2,2,3,2,3,2,3,3,4,2,3,2,3,3,5,2,3,2,3,4,5,2,3,3,5,3,5,2,5,2,3,3,5,

%T 3,6,2,3,5,5,2,3,2,3,3,7,2,3,3,4,5,3,2,3,4,3,5,8,2,3,2,5,3,8,3,5,2,5,

%U 3,3,2,5,2,5,3,5,3,5,2,3,5,5,2,8,5,9,7,7,2,8,4,5,8,10,5,5,2,4,5,5,2,8,2,3,5

%N Lexicographically earliest infinite sequence such that a(i) = a(j) => A350063(i) = A350063(j), for all i, j >= 1.

%C Restricted growth sequence transform of A350063.

%C For all i, j >= 1: A305897(i) = A305897(j) => a(i) = a(j) => A324117(i) = A324117(j).

%C For all i, j >= 2: a(i) = a(j) => A342656(i) = A342656(j).

%H Antti Karttunen, <a href="/A350065/b350065.txt">Table of n, a(n) for n = 1..10000</a> (based on Hans Havermann's factorization of A156552)

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

%o (PARI)

%o up_to = 3000;

%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 A000265(n) = (n>>valuation(n,2));

%o A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523

%o A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };

%o A350063(n) = if(1==n,0,A046523(A000265(A156552(n))));

%o v350065 = rgs_transform(vector(up_to, n, A350063(n)));

%o A350065(n) = v350065[n];

%o (PARI)

%o \\ Version using the factorization file available at https://oeis.org/A156552/a156552.txt

%o v156552sigs = readvec("a156552.txt");

%o up_to = #v156552sigs;

%o A350063(n) = if(n<=2,n-1,my(es=v156552sigs[n][2]); if(n%2, es = vector(#es-1,i,es[1+i])); my(f=vecsort(es, , 4), p=0); prod(i=1, #f, (p=nextprime(p+1))^f[i]));

%o v350065 = rgs_transform(vector(up_to, n, A350063(n)));

%o A350065(n) = v350065[n]; \\ _Antti Karttunen_, Jan 29 2022

%Y Cf. A000265, A046523, A156552, A322993, A350063, A350064, A350067, A350068.

%Y Cf. also A305897, A324117, A342656.

%K nonn

%O 1,2

%A _Antti Karttunen_, Jan 29 2022