A322988 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(1) = 0 if n is a prime power > 2, f(2) = -1, and f(n) = A322990(n) for all other numbers.

1, 2, 3, 3, 3, 4, 3, 3, 3, 5, 3, 6, 3, 7, 8, 3, 3, 9, 3, 10, 11, 12, 3, 13, 3, 14, 3, 15, 3, 6, 3, 3, 16, 17, 18, 19, 3, 20, 21, 22, 3, 8, 3, 23, 24, 25, 3, 26, 3, 27, 28, 29, 3, 30, 31, 32, 33, 34, 3, 35, 3, 36, 37, 3, 38, 11, 3, 39, 40, 10, 3, 41, 3, 42, 43, 44, 45, 13, 3, 46, 3, 47, 3, 48, 49, 50, 51, 52, 3, 15, 53, 54, 55, 56, 57, 58, 3, 59, 60, 61, 3, 16, 3

Offset: 1

Author

Antti Karttunen, Jan 02 2019

Keywords

• nonn

Comments

For all i, j: a(i) = a(j) => A322989(i) = A322989(j).

For all i, j > 2: A305976(i) = A305976(j) => a(i) = a(j).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A289271, A289272, A305976, A322989, A322990.

Cf. A322805, A322822 for analogous constructions for filter sequences.

Programs

PARI

up_to = 8192;
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; };
A289271(n) = { my(v=0,i=0,x=1); for(d=2,oo,if(n==1, return(v)); if(1==gcd(x,d)&&1==omega(d), if(!(n%d)&&1==gcd(d,n/d), v += 2^i; n /= d; x *= d); i++)); }; \\ After Rémy Sigrist's program for A289271.
A289272(n) = { my(m=1, pp=1); while(n>0, pp++; while(!isprimepower(pp)||(gcd(pp,m)>1), pp++); if(n%2, m *= pp); n >>=1); (m); }; \\ Antti Karttunen, Jan 02 2019
A322990(n) = A289272(A289271(n)>>1);
A322988aux(n) = if(2==n,-1,if(isprimepower(n),0,A322990(n)));
v322988 = rgs_transform(vector(up_to,n,A322988aux(n)));
A322988(n) = v322988[n];