A335422 -id:A335422 - cp's OEIS frontend

A335421 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(A335422(i)) = A046523(A335422(j)) for all i, j >= 0.

1, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 6, 3, 7, 1, 8, 4, 9, 3, 10, 5, 5, 2, 10, 6, 5, 3, 8, 7, 5, 1, 11, 8, 12, 4, 13, 9, 14, 3, 15, 10, 16, 5, 10, 5, 17, 2, 15, 10, 16, 6, 10, 5, 10, 3, 18, 8, 19, 7, 3, 5, 5, 1, 20, 11, 20, 8, 21, 12, 22, 4, 22, 13, 23, 9, 8, 14, 20, 3, 22, 15, 22, 10, 24, 16, 25, 5, 26, 10, 14, 5, 20, 17, 27, 2, 22, 15, 23, 10, 10, 16, 28, 6, 26

Offset: 0

Antti Karttunen, Jun 09 2020

For all i, j: A324400(i) = A324400(j) => a(i) = a(j) => A335420(i) = A335420(j).

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A046523, A335420, A335422.

Cf. also A286622, A324400.

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; };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A054429(n) = ((3<<#binary(n\2))-n-1); \\ From A054429
A163511(n) = if(!n,1,A005940(1+A054429(n)))
A335422(n) = A005940(1+A163511(n));
v335421 = rgs_transform(vector(1+up_to,n,A046523(A335422(n-1))));
A335421(n) = v335421[1+n];

For all n >= 0, a(2^n) = 1.

A335420 a(n) = A000120(A163511(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 2, 3, 2, 4, 2, 3, 1, 3, 4, 6, 2, 4, 3, 3, 2, 4, 4, 3, 2, 3, 3, 3, 1, 6, 3, 5, 4, 7, 6, 6, 2, 4, 4, 6, 3, 4, 3, 5, 2, 4, 4, 6, 4, 4, 3, 4, 2, 6, 3, 5, 3, 2, 3, 3, 1, 6, 6, 6, 3, 7, 5, 5, 4, 5, 7, 7, 6, 3, 6, 6, 2, 5, 4, 5, 4, 8, 6, 7, 3, 6, 4, 6, 3, 6, 5, 4, 2, 5, 4, 7, 4, 4, 6, 5, 4, 6

Offset: 0

Antti Karttunen, Jun 09 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A000079 (positions of ones), A000120, A001222, A163511, A335421, A335422.

Cf. also A323901, A334204.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A054429(n) = ((3<<#binary(n\2))-n-1);
A163511(n) = if(!n,1,A005940(1+A054429(n)));
A335420(n) = hammingweight(A163511(n));

from sympy import nextprime
def A335420(n):
    c, p, k = 1, 1, n
    while k:
        c *= (p:=nextprime(p))**(s:=(~k&k-1).bit_length())
        k >>= s+1
    return (c*p).bit_count() # Chai Wah Wu, Jul 25 2023

a(n) = A000120(A163511(n)).
a(n) = A001222(A335422(n)).
a(n) = a(2n) = a(A000265(n)).
For all n >= 0, a(2^n) = 1.

cp's OEIS Frontend

A335421 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(A335422(i)) = A046523(A335422(j)) for all i, j >= 0.

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