A324735 -id:A324735 - cp's OEIS frontend

A324728 Binary length of A324712: a(n) = 0 if A324712(n) = 0, otherwise a(n) = 1+A000523(A324712(n)).

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

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. A000523, A323243, A324712, A324733, A324735, A324811.

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))); };

If A324712(n) = 0, then a(n) = 0, otherwise a(n) = 1+A000523(A324712(n)).

a(A000040(n)) = n.

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).

A324734 Number of 0-bits present between the most and the least significant 1-bits in A324712(n), and 0 if A324712(n) = 0; a(n) = A324733(n) - A324829(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 17 2019

nonn

Cf. A323243, A324712, A324733, A324829, 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)
A324829(n) = hammingweight(A324712(n));
A324733(n) = { my(k=A324712(n)); if(!k,k,1 + (A000523(k)-A007814(k))); };
A324734(n) = (A324733(n) - A324829(n));

a(n) = A324733(n) - A324829(n).

a(p) = 0 for all primes p.

cp's OEIS Frontend

A324728 Binary length of A324712: a(n) = 0 if A324712(n) = 0, otherwise a(n) = 1+A000523(A324712(n)).

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

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A324734 Number of 0-bits present between the most and the least significant 1-bits in A324712(n), and 0 if A324712(n) = 0; a(n) = A324733(n) - A324829(n).

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