A305975 Filter sequence: All prime powers p^k, k >= 1, are allotted to distinct equivalence classes according to their exponent k, while all other numbers occur in singular equivalence classes of their own.

1, 2, 2, 3, 2, 4, 2, 5, 3, 6, 2, 7, 2, 8, 9, 10, 2, 11, 2, 12, 13, 14, 2, 15, 3, 16, 5, 17, 2, 18, 2, 19, 20, 21, 22, 23, 2, 24, 25, 26, 2, 27, 2, 28, 29, 30, 2, 31, 3, 32, 33, 34, 2, 35, 36, 37, 38, 39, 2, 40, 2, 41, 42, 43, 44, 45, 2, 46, 47, 48, 2, 49, 2, 50, 51, 52, 53, 54, 2, 55, 10, 56, 2, 57, 58, 59, 60, 61, 2, 62, 63, 64, 65, 66, 67, 68, 2, 69, 70, 71

Offset: 1

Author

Antti Karttunen, Jul 02 2018

Keywords

• nonn

Comments

Restricted growth sequence transform of A305974.

For all i, j: A305800(i) = A305800(j) => a(i) = a(j) => A305976(i) = A305976(j).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..100000

Crossrefs

Cf. A000961 (A246655), A085970, A305800, A305974, A305976, A305979.

Programs

PARI

up_to = 100000;
partialsums(f,up_to) = { my(v = vector(up_to), s=0); for(i=1,up_to,s += f(i); v[i] = s); (v); }
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; };
v065515 = partialsums(n -> (omega(n)<=1), up_to);
A065515(n) = v065515[n];
A085970(n) = (n - A065515(n));
A305974(n) = if(1==n,n,my(e = isprimepower(n)); if(e,-e,1+A085970(n)));
v305975 = rgs_transform(vector(up_to,n,A305974(n)));
A305975(n) = v305975[n];

Formula

a(prime) = 2, a(prime^2) = 3, a(prime^3) = 5, a(prime^4) = 10, a(prime^5) = 19.