A206719 -id:A206719 - cp's OEIS frontend

A304751 Filter sequence: Restricted growth sequence transform of function that gives the least natural number with the same prime signature that (0,1)-polynomial encoded in the binary expansion of n has when it is factored over Q.

1, 2, 2, 3, 2, 4, 2, 5, 4, 4, 2, 6, 2, 4, 4, 7, 2, 8, 2, 6, 4, 4, 2, 9, 2, 4, 6, 6, 2, 8, 2, 10, 4, 4, 4, 11, 2, 4, 4, 9, 2, 8, 2, 6, 8, 4, 2, 12, 4, 4, 4, 6, 2, 11, 2, 9, 4, 4, 2, 11, 2, 4, 8, 13, 4, 8, 2, 6, 2, 8, 2, 14, 2, 4, 8, 6, 2, 8, 2, 12, 2, 4, 2, 11, 4, 4, 2, 9, 2, 15, 2, 6, 4, 4, 4, 16, 2, 8, 6, 6, 2, 8, 2, 9, 8

Offset: 1

Antti Karttunen, Jun 08 2018

nonn

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

Cf. A206719, A206074 (gives the positions of 2's), A257000.

Cf. also A046523 (A101296), A278233, A284010, A284011.

up_to = 65537;
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
Aux304751(n) = { my(p=0, f=vecsort((factor(Pol(binary(n)))[, 2]), , 4)); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }
v304751 = rgs_transform(vector(up_to,n,Aux304751(n)));
A304751(n) = v304751[n];

For all i, j: a(i) = a(j) => A206719(i) = A206719(j).

For all i, j: a(i) = a(j) => A257000(i) = A257000(j).

A305822 Number of irreducible factors (counted with multiplicity) of the (0,1)-polynomial encoded in the binary expansion of n has when it is factored over Q.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 3, 3, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 5, 2, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 4, 1, 2, 3, 6, 2, 3, 1, 3, 1, 3, 1, 5, 1, 2, 3, 3, 1, 3, 1, 5, 1, 2, 1, 4, 2, 2, 1, 4, 1, 4, 1, 3, 2, 2, 2, 6, 1, 3, 3, 3, 1, 3, 1, 4, 3

Offset: 1

Antti Karttunen, Jun 10 2018

nonn

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

Cf. A206074 (gives the positions of 1's), A206719, A257000, A304751.
Cf. also A001222, A091222.
Differs from A277013 for the first time at n=65, where a(65) = 2, while A277013(65) = 1.

A305822(n) = vecsum(factor(Pol(binary(n)))[, 2]);

For all n >= 1, a(n) >= A206719(n).

cp's OEIS Frontend

A304751 Filter sequence: Restricted growth sequence transform of function that gives the least natural number with the same prime signature that (0,1)-polynomial encoded in the binary expansion of n has when it is factored over Q.

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