A324377 -id:A324377 - cp's OEIS frontend

A324287 a(n) = A002487(A005187(n)).

0, 1, 2, 1, 3, 1, 3, 5, 4, 1, 4, 7, 5, 7, 7, 5, 5, 1, 5, 9, 7, 10, 11, 8, 7, 9, 9, 7, 13, 8, 3, 10, 6, 1, 6, 11, 9, 13, 15, 11, 10, 13, 14, 11, 21, 13, 5, 17, 9, 11, 11, 9, 19, 12, 5, 18, 19, 11, 3, 13, 7, 18, 15, 4, 7, 1, 7, 13, 11, 16, 19, 14, 13, 17, 19, 15, 29, 18, 7, 24, 13, 16, 17, 14, 30, 19, 8, 29, 31, 18, 5, 22, 12, 31, 26, 7

Offset: 0

Antti Karttunen, Feb 20 2019

nonn

The motivation for this kind of sequence was a question: what kind of simply defined non-injective functions f exist such that this sequence can be defined as their function, e.g., as a(n) = g(f(n)), where g is a nontrivial integer-valued function? The same question can also be asked about A324288, A324337 and A324338. Note that A005187, A283477 and A006068 used in their definitions are all injections. Of course, A324377(n) = A000265(A005187(n)) fills the bill as A002487(n) = A002487(A000265(n)), but are there any less obvious solutions? - Antti Karttunen, Feb 28 2019

Antti Karttunen, Table of n, a(n) for n = 0..16385
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to Stern's sequences

Cf. A000265, A002487, A005187, A006068, A283477, A324286, A324288, A324293, A324337, A324338, A324377.

A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); }; \\ Modified from the one given in A002487, sign not actually needed here.
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A324287(n) = A002487(A005187(n));

from functools import reduce
def A324287(n): return sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin((n<<1)-n.bit_count())[-1:2:-1],(1,0))) if n else 0 # Chai Wah Wu, May 05 2023

a(n) = A002487(A005187(n)).
a(n) = A324286(A283477(n)).
a(n) = A002487(A324377(n)).

A324378 a(n) = A000265(1+A005187(n)).

Original entry on oeis.org

1, 1, 1, 5, 1, 9, 11, 3, 1, 17, 19, 5, 23, 3, 13, 27, 1, 33, 35, 9, 39, 5, 21, 43, 47, 3, 25, 51, 27, 55, 57, 29, 1, 65, 67, 17, 71, 9, 37, 75, 79, 5, 41, 83, 43, 87, 89, 45, 95, 3, 49, 99, 51, 103, 105, 53, 55, 111, 113, 57, 117, 59, 15, 121, 1, 129, 131, 33, 135, 17, 69, 139, 143, 9, 73, 147, 75, 151, 153, 77, 159, 5, 81

Offset: 0

Antti Karttunen, Feb 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A000265, A002487, A005187, A324288, A324377.

A000265(n) = (n/2^valuation(n, 2));
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A324378(n) = A000265(1+A005187(n));

a(n) = A000265(1+A005187(n)).
A002487(a(n)) = A324288(n).
a(2^n) = 1 for all n >= 0.

A324379 a(n) = A007814(A005187(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A005187, A007814, A277808, A324377.

A324379(n) = { my(s=n); while(n>>=1, s+=n); valuation(s,2); };

a(n) = A007814(A005187(n)).

cp's OEIS Frontend

A324287 a(n) = A002487(A005187(n)).

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A324378 a(n) = A000265(1+A005187(n)).

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A324379 a(n) = A007814(A005187(n)).

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