A324401 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j >= 1, where f(n) = -1 if n is an odd prime, f(n) = -2 if n = 2^k, with k > 1, and f(n) = n for all other numbers.

1, 2, 3, 4, 3, 5, 3, 4, 6, 7, 3, 8, 3, 9, 10, 4, 3, 11, 3, 12, 13, 14, 3, 15, 16, 17, 18, 19, 3, 20, 3, 4, 21, 22, 23, 24, 3, 25, 26, 27, 3, 28, 3, 29, 30, 31, 3, 32, 33, 34, 35, 36, 3, 37, 38, 39, 40, 41, 3, 42, 3, 43, 44, 4, 45, 46, 3, 47, 48, 49, 3, 50, 3, 51, 52, 53, 54, 55, 3, 56, 57, 58, 3, 59, 60, 61, 62, 63, 3, 64, 65, 66, 67, 68, 69, 70, 3, 71, 72, 73, 3, 74, 3, 75, 76

Offset: 1

Author

Antti Karttunen, Mar 01 2019

Keywords

• nonn

Comments

For all i, j:

A305801(i) = A305801(j) => a(i) = a(j),

a(i) = a(j) => A305976(i) = A305976(j) => A001221(i) = A001221(j),

a(i) = a(j) => A322591(i) = A322591(j),

a(i) = a(j) => A323235(i) = A323235(j),

a(i) = a(j) => A324399(i) = A324399(j),

a(i) = a(j) => A297159(i) = A297159(j).

Links

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

Crossrefs

Cf. A000523, A000720.

Cf. also A297159, A305801, A305976, A322591, A323235, A324399, A324400.

Programs

PARI

A000523(n) = if(n<1, 0, #binary(n)-1);
A324401(n) = if(n<4,n,if(isprime(n),3,if(!bitand(n,n-1),4,4+n-A000523(n)-primepi(n))));

PARI

up_to = 65537;
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; };
Aux324401(n) = if((n>2) && (isprime(n)||!bitand(n,n-1)),-(2-(n%2)),n);
\\ Equally: Aux324401(n) = if(n<=2,n,if(isprime(n),-1,if(!bitand(n,n-1),-2,n)));
v324401 = rgs_transform(vector(up_to, n, Aux324401(n)));
A324401(n) = v324401[n];

Formula

If n <= 2, a(n) = n, for n > 2, if n is an odd prime, a(n) = 3, if n = 2^k, with k >= 2, a(n) = 4, otherwise a(n) = 4 + n - A000523(n) - A000720(n).