A324728 -id:A324728 - cp's OEIS frontend

A324811 a(n) = A324728(n) - A061395(n).

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

Offset: 1

Antti Karttunen, Mar 17 2019

sign

The first negative term is a(182) = -6, as A324712(182) = 0 and 182 = 2*7*13 = prime(1) * prime(4) * prime(6).
The next negative term after that is a(198) = -4, as A324712(198) = 1, and 198 = 2 * 3^2 * 11 = prime(1) * prime(2)^2 * prime(5).
There are only 161 negative terms among the first 10000 terms.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000523, A061395, A323243, A324712, A324728.

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A324712(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A323243(d)))); (v); }; \\ Needs also code from A323243.
A000523(n) = if( n<1, 0, #binary(n) - 1); \\ From A000523
A324728(n) = { my(k=A324712(n)); if(!k,k,(1+A000523(k))); };
A324811(n) = (A324728(n) - A061395(n));

a(n) = A324728(n) - A061395(n).

a(p) = 0 for all primes p.

A324733 a(n) = 0 if A324712(n) = 0, otherwise a(n) = 1 + A000523(A324712(n)) - A007814(A324712(n)).

Original entry on oeis.org

0, 1, 2, 3, 3, 1, 4, 2, 4, 4, 5, 4, 6, 3, 4, 1, 7, 3, 8, 3, 6, 5, 9, 2, 5, 6, 3, 6, 10, 4, 11, 3, 5, 7, 5, 4, 12, 7, 6, 1, 13, 2, 14, 7, 6, 9, 15, 3, 6, 4, 1, 8, 16, 1, 7, 5, 8, 8, 17, 4, 18, 11, 3, 4, 6, 7, 19, 9, 9, 4, 20, 5, 21, 12, 6, 10, 6, 7, 22, 4, 3, 13, 23, 7, 6, 14, 10, 5, 24, 7, 8, 11, 11, 15, 8, 3, 25, 2, 8, 6, 26, 9, 27, 2, 5

Offset: 1

Antti Karttunen, Mar 17 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A323243, A324712, A324724, A324728, A324734, A324735.

A324712(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A323243(d)))); (v); }; \\ Needs also code from A323243.
A000523(n) = if( n<1, 0, #binary(n) - 1); \\ From A000523
A007814(n) = valuation(n,2)
A324733(n) = { my(k=A324712(n)); if(!k,k,1 + (A000523(k)-A007814(k))); };

If A324712(n) = 0, then a(n) = 0, otherwise a(n) = 1 + A324728(n) - A324724(n).

A324735 Difference between the binary length and the binary weight of A324712. a(n) = 0 if A324712(n) = 0.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 2, 0, 1, 0, 0, 0, 2, 2, 4, 0, 0, 0, 2, 2, 2, 0, 3, 1, 3, 3, 2, 0, 2, 0, 3, 3, 3, 4, 3, 0, 2, 4, 5, 0, 3, 0, 3, 2, 3, 0, 3, 2, 2, 7, 3, 0, 2, 4, 3, 4, 5, 0, 2, 0, 5, 3, 5, 4, 1, 0, 5, 3, 2, 0, 3, 0, 3, 3, 2, 4, 1, 0, 3, 5, 6, 0, 1, 6, 4, 6, 4, 0, 1, 2, 5, 5, 6, 5, 4, 0, 3, 4, 3, 0, 4, 0, 7, 3

Offset: 1

Antti Karttunen, Mar 17 2019

nonn

Number of nonleading 0-bits in A324712(n), with a(n) = 0 if A324712(n) = 0.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A323243, A324712, A324728, A324733, A324829, A324724, A324734.

A324712(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A323243(d)))); (v); }; \\ Needs also code from A323243.
A000523(n) = if( n<1, 0, #binary(n) - 1); \\ From A000523
A324735(n) = { my(k=A324712(n)); if(!k,k,1 + (A000523(k)-hammingweight(k))); };

a(n) = A324728(n) - A324829(n).
If A324712(n) > 0, then a(n) = -1 + A324734(n) + A324724(n).
a(p) = 0 for all primes p.

cp's OEIS Frontend

A324811 a(n) = A324728(n) - A061395(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A324733 a(n) = 0 if A324712(n) = 0, otherwise a(n) = 1 + A000523(A324712(n)) - A007814(A324712(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A324735 Difference between the binary length and the binary weight of A324712. a(n) = 0 if A324712(n) = 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula