A364567 -id:A364567 - cp's OEIS frontend

A364568 a(n) = A290077(n) - A364567(n).

1, 0, 0, 0, 0, 0, 2, 0, -2, 0, 4, 0, 12, 2, 10, 0, -6, -2, 4, 0, 16, 4, 16, 0, 26, 12, 32, 4, 84, 10, 38, 0, -20, -6, 4, -4, 24, 4, 20, 0, 44, 16, 40, 8, 104, 16, 56, 0, 78, 26, 68, 24, 152, 32, 104, 8, 262, 84, 184, 20, 468, 38, 130, 0, -48, -20, -8, -12, 16, 4, 28, -8, 40, 24, 64, 8, 168, 20, 76, 0, 88, 44, 104, 32

Offset: 0

Antti Karttunen, Aug 05 2023

sign

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A000010, A005940, A033265, A290077, A297112, A364558, A364567.

Cf. also A364499.

A290077(n) = { my(p=2,z=1); while(n, if(!(n%2), p=nextprime(1+p), z *= (p-(1==(n%4)))); n>>=1); (z); };
A364567(n) = if(!n,n, my(i=1); while(n>1, if((n%4)!=1, i<<=1); n >>= 1); (i));
A364568(n) = (A290077(n) - A364567(n));

For n > 0, a(n) = -A364558(A005940(1+n)) = A000010(A005940(1+n)) - 2^A033265(n).

A033265 Number of i such that d(i) >= d(i-1), where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

			The base-2 representation of n=4 is 100 with d(0)=0, d(1)=0, d(2)=1. There are two rise-or-equal, one from d(0) to d(1) and one from d(1) to d(2), so a(4)=2. - _R. J. Mathar_, Oct 16 2015

Antti Karttunen, Table of n, a(n) for n = 1..65537
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Index entries for sequences related to binary expansion of n

Cf. A014081, A023416, A037800, A037809, A005940, A156552, A297113, A364567 [= 2^a(n)].

A033265 := proc(n)
    a := 0 ;
    dgs := convert(n,base,2);
    for i from 2 to nops(dgs) do
        if op(i,dgs)>=op(i-1,dgs) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Oct 16 2015

A033265(n) = { my(i=0); while(n>1, if((n%4)!=1, i++); n >>= 1); (i); }; \\ Antti Karttunen, Aug 06 2023

From Ralf Stephan, Oct 05 2003: (Start)
a(0) = 0, a(2n) = a(n) + 1, a(2n+1) = a(n) + [n odd].
a(n) = A014081(n) + A023416(n).
G.f.: 1/(1-x) * Sum_{k>=0} (t^2 + t^3 + t^4)/((1+t)*(1+t^2)), t=x^2^k. (End)
a(n) = -1 + A297113(A005940(1+n)). - Antti Karttunen, Dec 30 2017

Sign in Name corrected by R. J. Mathar, Oct 16 2015

A364571 a(n) = A297171(A163511(n)), where A297171 is the Möbius transform of the inverse permutation of A163511.

Original entry on oeis.org

0, 1, 1, 3, 2, 2, 2, 7, 4, 4, 2, 4, 5, 3, 6, 15, 8, 8, 4, 8, 4, 4, 4, 8, 10, 10, 4, 5, 13, 11, 14, 31, 16, 16, 8, 16, 8, 8, 8, 16, 8, 8, 4, 8, 8, 8, 8, 16, 20, 20, 10, 20, 7, 9, 6, 9, 26, 26, 12, 21, 29, 27, 30, 63, 32, 32, 16, 32, 16, 16, 16, 32, 16, 16, 8, 16, 16, 16, 16, 32, 16, 16, 8, 16, 8, 8, 8, 16, 16, 16

Offset: 0

Antti Karttunen, Aug 05 2023

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A163511, A243071, A297171.

Cf. also A364567.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A054429(n) = ((3<<#binary(n\2))-n-1); \\ From A054429
A163511(n) = if(!n,1,A005940(1+A054429(n)));
A243071(n) = if(n<=2, n-1, my(f=factor(n), p, 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]); ((3<<#binary(res\2))-res-1)); \\ (Combining programs given in A156552 and A054429) - Antti Karttunen, Aug 05 2023
A297171(n) = sumdiv(n,d,moebius(n/d)*A243071(d));
A364571(n) = A297171(A163511(n));

a(n) = A297171(A163511(n)).

A364953 a(n) = A364952(A005940(1+n)), where A364952 is Dirichlet inverse of A364557, which is Möbius transform of A005941 [the inverse permutation of A005940].

Original entry on oeis.org

1, -1, -2, -1, -4, 2, 0, -1, -8, 4, 12, 2, 8, 0, 0, -1, -16, 8, 24, 4, 56, -12, -8, 2, 48, -8, -40, 0, -16, 0, 0, -1, -32, 16, 48, 8, 112, -24, -16, 4, 240, -56, -232, -12, -208, 8, 0, 2, 224, -48, -208, -8, -528, 40, 64, 0, -288, 16, 112, 0, 32, 0, 0, -1, -64, 32, 96, 16, 224, -48, -32, 8, 480, -112, -464, -24

Offset: 0

Antti Karttunen, Aug 29 2023

sign

Cf. A005940, A005941, A364557, A364952.

Cf. also A364567, A364575.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A364557(n) = if(1==n, 1, 2^(primepi(vecmax(factor(n)[, 1]))+(bigomega(n)-omega(n))-1));
memoA364952 = Map();
A364952(n) = if(1==n,1,my(v); if(mapisdefined(memoA364952,n,&v), v, v = -sumdiv(n,d,if(dA364557(n/d)*A364952(d),0)); mapput(memoA364952,n,v); (v)));
A364953(n) = A364952(A005940(1+n));

cp's OEIS Frontend

A364568 a(n) = A290077(n) - A364567(n).

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A033265 Number of i such that d(i) >= d(i-1), where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.

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A364571 a(n) = A297171(A163511(n)), where A297171 is the Möbius transform of the inverse permutation of A163511.

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PARI

Formula

A364953 a(n) = A364952(A005940(1+n)), where A364952 is Dirichlet inverse of A364557, which is Möbius transform of A005941 [the inverse permutation of A005940].

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PARI