A378218 -id:A378218 - cp's OEIS frontend

A345182 a(1) = 1, a(2) = 0; a(n) = Sum_{d|n, d < n} a(d).

1, 0, 1, 1, 1, 2, 1, 2, 2, 2, 1, 5, 1, 2, 3, 4, 1, 6, 1, 5, 3, 2, 1, 12, 2, 2, 4, 5, 1, 10, 1, 8, 3, 2, 3, 18, 1, 2, 3, 12, 1, 10, 1, 5, 8, 2, 1, 28, 2, 6, 3, 5, 1, 16, 3, 12, 3, 2, 1, 31, 1, 2, 8, 16, 3, 10, 1, 5, 3, 10, 1, 50, 1, 2, 8, 5, 3, 10, 1, 28, 8, 2, 1, 31, 3, 2, 3, 12, 1, 36

Offset: 1

Ilya Gutkovskiy, Jun 10 2021

nonn

From Antti Karttunen, Nov 25 2024: (Start)
a(n) is the number of strict chains of divisors from n to 1 that do not end with 2/1. For example, the a(n) such chains for n = 1, 2, 4, 6, 8, 12, 30 are:
  1 (none) 4/1    6/1     8/1     12/1      30/1
                  6/3/1   8/4/1   12/3/1    30/3/1
                                  12/4/1    30/5/1
                                  12/6/1    30/6/1
                                  12/6/3/1  30/10/1
                                            30/15/1
                                            30/6/3/1
                                            30/10/5/1
                                            30/15/3/1
                                            30/15/5/1
leaving 1, 0, 1, 2, 2, 5, 10 chains out of the 1, 1, 2, 3, 4, 8, 13 chains depicted in the illustration of A074206.
Equally, a(n) is the number of strict chains of divisors from n to 1 where n is not followed by n/2 as the second divisor in the chain, which explains nicely the formula a(n) = A074206(n) - A074206(n/2) when n is even.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A022825 (partial sums), A074206, A167865, A320224, A345138, A345141, A378218 (Dirichlet inverse), A378223 (inverse Möbius transform).

a[1] = 1; a[2] = 0; a[n_] := a[n] = Sum[If[d < n, a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 90}]
nmax = 90; A[] = 0; Do[A[x] = x - x^2 + Sum[A[x^k], {k, 2, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

memoA345182 = Map();
A345182(n) = if(n<=2, n%2, my(v); if(mapisdefined(memoA345182,n,&v), v, v = sumdiv(n,d,if(dA345182(d),0)); mapput(memoA345182,n,v); (v))); \\ Antti Karttunen, Nov 22 2024

up_to = 20000;
A345182list(up_to_n) = { my(v=vector(up_to_n)); v[1] = 1; v[2] = 0; for(n=3,up_to_n,v[n] = sumdiv(n,d,(dA345182list(up_to);
A345182(n) = v345182[n]; \\ Antti Karttunen, Nov 25 2024

G.f. A(x) satisfies: A(x) = x - x^2 + A(x^2) + A(x^3) + A(x^4) + ...

a(n) = A074206(n) if n is odd, otherwise a(n) = A074206(n) - A074206(n/2).

A359508 a(n) = log_2(A359507(n) - 1).

Original entry on oeis.org

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

Offset: 1

Peter Kagey, Jan 03 2023

nonn

Conjecture: A359507(n) is always of the form 2^m + 1.

If log_2(A359507(n) - 1) is not an integer, then define a(n) = -1.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000523, A359506, A359507, A378218.

A359506(n) = if(n==0, return (0), my (x=[n], y); for (m=n+1, oo, if (vecmin(y=[bitxor(v, m) | v<-x])==0, return (m), x=setunion(x, Set(y))))); \\ From A359506.
A359507(n) = (A359506(n)-n);
A359508(n) = (#digits(A359507(n)-1, 2)-1); \\ Antti Karttunen, Nov 22 2024

a(n) = A000523(A359507(n)-1).
Conjecture:
a(4k)   = 1 for k > 0,
a(4k+1) = 2 for k > 0,
a(4k+2) = 1 for k > 0,
a(4k+3) = a(k) + 2 for k > 0.
Apparently, a(n) = abs(A378218(1+n)). [This holds at least up to n=65537] - Antti Karttunen, Nov 22 2024
a(n) = A007814((n - 3*b(n + 1) + 2) mod b(n + 1) + b(n + 2) - 1) + 1, where b(n) = 2^A000523(A002264(n)) for n >= 4. - Alan Michael Gómez Calderón, Feb 25 2025

More terms from Antti Karttunen, Nov 22 2024

A378224 Dirichlet inverse of A378223.

Original entry on oeis.org

1, -1, -2, -1, -2, 0, -2, -1, 0, 0, -2, 0, -2, 0, 2, -1, -2, 0, -2, 0, 2, 0, -2, 0, 0, 0, 0, 0, -2, 0, -2, -1, 2, 0, 2, 0, -2, 0, 2, 0, -2, 0, -2, 0, 0, 0, -2, 0, 0, 0, 2, 0, -2, 0, 2, 0, 2, 0, -2, 0, -2, 0, 0, -1, 2, 0, -2, 0, 2, 0, -2, 0, -2, 0, 0, 0, 2, 0, -2, 0, 0, 0, -2, 0, 2, 0, 2, 0, -2, 0, 2, 0, 2, 0, 2, 0, -2

Offset: 1

Antti Karttunen, Nov 25 2024

sign

Möbius transform of A378218.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A008683, A345182, A378218, A378223.

Cf. also A378225.

memoA345182 = Map();
A345182(n) = if(n<=2, n%2, my(v); if(mapisdefined(memoA345182,n,&v), v, v = sumdiv(n,d,if(dA345182(d),0)); mapput(memoA345182,n,v); (v)));
A378223(n) = sumdiv(n,d,A345182(d));
memoA378224 = Map();
A378224(n) = if(1==n,1,my(v); if(mapisdefined(memoA378224,n,&v), v, v = -sumdiv(n,d,if(dA378223(n/d)*A378224(d),0)); mapput(memoA378224,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA378223(n/d) * a(d).

a(n) = Sum_{d|n} A008683(n/d)*A378218(d).

A378532 Dirichlet convolution of A296075 and A378525.

Original entry on oeis.org

1, 0, 0, -2, 0, -3, 0, -2, -2, -3, 0, -4, 0, -3, -3, -2, 0, -4, 0, -4, -3, -3, 0, -2, -2, -3, -2, -4, 0, -6, 0, -2, -3, -3, -3, -1, 0, -3, -3, -2, 0, -6, 0, -4, -4, -3, 0, -2, -2, -4, -3, -4, 0, -2, -3, -2, -3, -3, 0, 0, 0, -3, -4, -2, -3, -6, 0, -4, -3, -6, 0, 0, 0, -3, -4, -4, -3, -6, 0, -2, -2, -3, 0, 0, -3, -3, -3, -2

Offset: 1

Antti Karttunen, Dec 01 2024

sign

Inverse Möbius transform of A378534.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A033879, A296075, A378531 (Dirichlet inverse), A378534 (Möbius transform), A378525, A378542.

Cf. also A378218.

A033879(n) = ((2*n)-sigma(n));
A296075(n) = sumdiv(n,d,A033879(d));
A378542(n) = sumdiv(n,d,d*!(bigomega(n/d)%2));
memoA378525 = Map();
A378525(n) = if(1==n,1,my(v); if(mapisdefined(memoA378525,n,&v), v, v = -sumdiv(n,d,if(dA378542(n/d)*A378525(d),0)); mapput(memoA378525,n,v); (v)));
A378532(n) = sumdiv(n,d,A296075(d)*A378525(n/d));

a(n) = Sum_{d|n} A296075(d)*A378525(n/d).

a(n) = Sum_{d|n} A378534(d).

cp's OEIS Frontend

A345182 a(1) = 1, a(2) = 0; a(n) = Sum_{d|n, d < n} a(d).

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A359508 a(n) = log_2(A359507(n) - 1).

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A378224 Dirichlet inverse of A378223.

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A378532 Dirichlet convolution of A296075 and A378525.

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