A345182 -id:A345182 - cp's OEIS frontend

A378218 Dirichlet inverse of A345182.

1, 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

Offset: 1

Antti Karttunen, Nov 22 2024

sign

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

Cf. A345182, A359508.

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)));
memoA378218 = Map();
A378218(n) = if(1==n,1,my(v); if(mapisdefined(memoA378218,n,&v), v, v = -sumdiv(n,d,if(dA345182(n/d)*A378218(d),0)); mapput(memoA378218,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA345182(n/d) * a(d).

Apparently, abs(a(n+1)) = A359508(n).

A378223 Inverse Möbius transform of A345182.

Original entry on oeis.org

1, 1, 2, 2, 2, 4, 2, 4, 4, 4, 2, 10, 2, 4, 6, 8, 2, 12, 2, 10, 6, 4, 2, 24, 4, 4, 8, 10, 2, 20, 2, 16, 6, 4, 6, 36, 2, 4, 6, 24, 2, 20, 2, 10, 16, 4, 2, 56, 4, 12, 6, 10, 2, 32, 6, 24, 6, 4, 2, 62, 2, 4, 16, 32, 6, 20, 2, 10, 6, 20, 2, 100, 2, 4, 16, 10, 6, 20, 2, 56, 16, 4, 2, 62, 6, 4, 6, 24, 2, 72, 6, 10, 6, 4, 6

Offset: 1

Antti Karttunen, Nov 25 2024

nonn

Apparently the Dirichlet convolution of A002131 and A323910. - Antti Karttunen, Nov 30 2024

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

Cf. A002131, A323910, A345182, A378224 (Dirichlet inverse).
Cf. also A067824.
Odd bisection is not equal to A278223.

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));

up_to = 20000;
A378223list(up_to_n) = { my(v=vector(up_to_n)); v[1] = 1; v[2] = 0; for(n=3,up_to_n,v[n] = 1+sumdiv(n,d,(dA378223list(up_to);
A378223(n) = v378223[n];

a(n) = Sum_{d|n} A345182(d).

For n > 2, a(n) = 2*A345182(n).

A323910 Dirichlet inverse of the deficiency of n, A033879.

Original entry on oeis.org

1, -1, -2, 0, -4, 4, -6, 0, -1, 6, -10, 2, -12, 8, 10, 0, -16, 1, -18, 2, 14, 12, -22, 4, -3, 14, -2, 2, -28, -16, -30, 0, 22, 18, 26, 4, -36, 20, 26, 4, -40, -24, -42, 2, 4, 24, -46, 8, -5, -1, 34, 2, -52, 0, 42, 4, 38, 30, -58, 2, -60, 32, 6, 0, 50, -40, -66, 2, 46, -40, -70, 12, -72, 38, 2, 2, 62, -48, -78, 8, -4, 42, -82, -2, 66, 44, 58, 4, -88, 2, 74, 2

Offset: 1

Antti Karttunen, Feb 12 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000
Jon Maiga, Computer-generated formulas for A323910, Sequence Machine.

Cf. A033879, A323911, A323912, A359549 (parity of terms).

Sequences that appear in the convolution formulas: A002033, A008683, A023900, A055615, A046692, A067824, A074206, A174725, A191161, A327960, A328722, A330575, A345182, A349341, A346246, A349387.

b[n_] := 2 n - DivisorSigma[1, n];
a[n_] := a[n] = If[n == 1, 1, -Sum[b[n/d] a[d], {d, Most@ Divisors[n]}]];
Array[a, 100] (* Jean-François Alcover, Feb 17 2020 *)

up_to = 16384;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA033879(n) = (2*n-sigma(n));
v323910 = DirInverse(vector(up_to,n,A033879(n)));
A323910(n) = v323910[n];

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA033879(n/d) * a(d).
From Antti Karttunen, Nov 14 2024: (Start)
Following convolution formulas have been conjectured for this sequence by Sequence Machine, with each one giving the first 10000 terms correctly:
a(n) = Sum_{d|n} A046692(d)*A067824(n/d).
a(n) = Sum_{d|n} A055615(d)*A074206(n/d).
a(n) = Sum_{d|n} A023900(d)*A174725(n/d).
a(n) = Sum_{d|n} A008683(d)*A323912(n/d).
a(n) = Sum_{d|n} A191161(d)*A327960(n/d).
a(n) = Sum_{d|n} A328722(d)*A330575(n/d).
a(n) = Sum_{d|n} A345182(d)*A349341(n/d).
a(n) = Sum_{d|n} A346246(d)*A349387(n/d).
a(n) = Sum_{d|n} A002033(d-1)*A055615(n/d).
(End)

A022825 a(n) = a([ n/2 ]) + a([ n/3 ]) + . . . + a([ n/n ]) for n > 1, a(1) = 1.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 9, 11, 13, 14, 19, 20, 22, 25, 29, 30, 36, 37, 42, 45, 47, 48, 60, 62, 64, 68, 73, 74, 84, 85, 93, 96, 98, 101, 119, 120, 122, 125, 137, 138, 148, 149, 154, 162, 164, 165, 193, 195, 201, 204, 209, 210, 226, 229, 241, 244, 246, 247, 278, 279

Offset: 1

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A025523, A078346, A345182.

a:= proc(n) option remember; `if`(n<2, 1,
      add(a(iquo(n,j)), j=2..n))
    end:
seq(a(n), n=1..63);  # Alois P. Heinz, Mar 31 2021

Fold[Append[#1, Total[#1[[Quotient[#2, Range[2, #2]]]]]] &, {1}, Range[2, 60]] (* Ivan Neretin, Aug 24 2016 *)

from functools import lru_cache
@lru_cache(maxsize=None)
def A022825(n):
    if n <= 1:
        return n
    c, j = 0, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A022825(k1)
        j, k1 = j2, n//j2
    return c+n+1-j # Chai Wah Wu, Mar 31 2021

G.f. A(x) satisfies: A(x) = x + (1/(1 - x)) * Sum_{k>=2} (1 - x^k) * A(x^k). - Ilya Gutkovskiy, Feb 21 2022

Offset corrected by Alois P. Heinz, Mar 31 2021

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).

A378531 Dirichlet convolution of A378432 and A378542.

Original entry on oeis.org

1, 0, 0, 2, 0, 3, 0, 2, 2, 3, 0, 4, 0, 3, 3, 6, 0, 4, 0, 4, 3, 3, 0, 14, 2, 3, 2, 4, 0, 6, 0, 10, 3, 3, 3, 18, 0, 3, 3, 14, 0, 6, 0, 4, 4, 3, 0, 30, 2, 4, 3, 4, 0, 14, 3, 14, 3, 3, 0, 30, 0, 3, 4, 22, 3, 6, 0, 4, 3, 6, 0, 48, 0, 3, 4, 4, 3, 6, 0, 30, 6, 3, 0, 30, 3, 3, 3, 14, 0, 30, 3, 4, 3, 3, 3, 74, 0, 4, 4, 18

Offset: 1

Antti Karttunen, Dec 01 2024

nonn

Möbius transform of A378533.

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

Cf. A008683, A378532 (Dirichlet inverse), A378432, A378533 (inverse Möbius transform), A378542.

Cf. also A345182.

A033879(n) = ((2*n)-sigma(n));
A296075(n) = sumdiv(n,d,A033879(d));
memoA378432 = Map();
A378432(n) = if(1==n,1,my(v); if(mapisdefined(memoA378432,n,&v), v, v = -sumdiv(n,d,if(dA296075(n/d)*A378432(d),0)); mapput(memoA378432,n,v); (v)));
A378542(n) = sumdiv(n,d,d*!(bigomega(n/d)%2));
A378531(n) = sumdiv(n,d,A378432(d)*A378542(n/d));

a(n) = Sum_{d|n} A378432(d)*A378542(n/d).

a(n) = Sum_{d|n} A008683(d)*A378533(n/d).

cp's OEIS Frontend

A378218 Dirichlet inverse of A345182.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A378223 Inverse Möbius transform of A345182.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

A323910 Dirichlet inverse of the deficiency of n, A033879.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A022825 a(n) = a([ n/2 ]) + a([ n/3 ]) + . . . + a([ n/n ]) for n > 1, a(1) = 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

Extensions

A378224 Dirichlet inverse of A378223.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A378531 Dirichlet convolution of A378432 and A378542.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula