A324724 -id:A324724 - cp's OEIS frontend

A324725 a(n) = sign(A324543(n)) * A001511(A324543(n)), with a(n) = 0 if A324543(n) = 0.

0, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 4, 5, 1, -1, 1, 3, 1, 5, 1, 4, 1, 3, 3, 1, 1, 2, 1, 4, 3, 5, 3, 3, 1, 2, 2, 6, 1, -2, 1, 1, 1, 6, 1, 4, 1, -1, 2, 1, 1, 3, 1, 3, 2, 2, 1, 2, 1, 5, 3, 4, 4, 1, 1, 1, 3, -4, 1, 3, 1, 3, 1, 1, 3, -1, 1, 4, 5, 5, 1, 1, 2, 4, 2, 4, 1, 1, 1, 1, 7, 5, 2, 7, 1, -2, 1, 2, 1, 1, 1, 8, 2

Offset: 1

Antti Karttunen, Mar 17 2019

sign

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. A001511, A324343, A324543, A324724, A324817, A324828.

A324543(n) = sumdiv(n,d,moebius(n/d)*A323243(d));
A001511ext(n) = if(!n,n,sign(n)*(1+valuation(n,2))); \\ Like A001511 but gives 0 for 0 and -A001511(-n) for negative numbers.
A324725(n) = A001511ext(A324543(n));

If A324543(n) = 0, then a(n) = 0, otherwise a(n) = sign(A324543(n)) * A001511(A324543(n)).
a(p) = 1 for all primes p.
A324828(n) = [1 == abs(a(n))], where [ ] is the Iverson bracket.

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.

A324817 a(n) = sign(A323244(n))*A001511(A323244(n)), with a(n) = 0 if A323244(n) = 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 2, 0, 1, 1, 2, 1, 5, 2, 2, 1, 3, 1, 2, -1, 2, 1, 2, -3, 2, 3, 2, 1, 2, 1, 2, 2, 2, -2, 2, 1, 9, -3, 2, 1, 3, 1, 2, 4, 2, 1, 2, -3, 1, 4, 2, 1, 3, -1, 2, -3, 3, 1, 2, 1, 2, 5, 2, 2, 2, 1, 2, 2, 4, 1, 2, 1, 2, 2, 2, -2, 2, 1, 2, -3, 2, 1, 2, -4, 2, -3, 2, 1, 3, -1, 2, 2, 2, -3, 2, 1, 1, -3, 2, 1, 3, 1, 2, -3

Offset: 1

Antti Karttunen, Mar 17 2019

sign

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. A001511, A156552, A323243, A323244, A324724, A324725, A324731, A324732.

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552 by David A. Corneth
A323244(n) = ((2*A156552(n))-A323243(n)); \\ Needs also code from A323243.
A001511ext(n) = if(!n,n,sign(n)*(1+valuation(n,2))); \\ Like A001511 but gives 0 for 0 and -A001511(-n) for negative numbers.
A324817(n) = A001511ext(A323244(n));

If A323244(n) = 0, then a(n) = 0, otherwise a(n) = sign(A323244(n)) * A001511(A323244(n)).

a(p) = 1 for all primes p.

cp's OEIS Frontend

A324725 a(n) = sign(A324543(n)) * A001511(A324543(n)), with a(n) = 0 if A324543(n) = 0.

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

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A324817 a(n) = sign(A323244(n))*A001511(A323244(n)), with a(n) = 0 if A323244(n) = 0.

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