A344767 -id:A344767 - cp's OEIS frontend

A344768 a(n) = n - A344767(n); Möbius transform of A344765.

0, 0, 2, 0, 2, 7, 2, 0, 3, 12, 2, 11, 2, 15, 15, 0, 2, 19, 2, 13, 22, 23, 2, 25, 5, 28, 9, 32, 2, 19, 2, 0, 33, 36, 30, 41, 2, 39, 40, 48, 2, 28, 2, 27, 47, 47, 2, 47, 7, 50, 51, 29, 2, 53, 58, 23, 58, 60, 2, 72, 2, 63, 48, 0, 55, 66, 2, 72, 69, 63, 2, 24, 2, 76, 76, 43, 71, 77, 2, 47, 27, 84, 2, 57, 70, 87, 87, 96, 2, 73

Offset: 1

Antti Karttunen, May 30 2021

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A008683, A011772, A344765, A344767.

A344767(n) = sumdiv(n,d,moebius(n/d)*A011772(d));
A344768(n) = (n-A344767(n));

a(n) = n - A344767(n) = n - Sum_{d|n} A008683(n/d) * A011772(d).

a(n) = Sum_{d|n} A008683(n/d) * A344765(d).

A345055 Dirichlet inverse of A011772.

Original entry on oeis.org

1, -3, -2, 2, -4, 9, -6, 0, -4, 20, -10, -16, -12, 29, 11, 0, -16, 16, -18, -43, 18, 49, -22, 18, -8, 60, -2, -43, -28, -89, -30, 0, 29, 80, 34, 1, -36, 89, 36, 71, -40, -136, -42, -96, 27, 109, -46, -18, -12, 8, 47, -123, -52, -19, 70, -25, 54, 140, -58, 326, -60, 149, 21, 0, 71, -201, -66, -128, 65, -264, -70, -140, -72, 180, 16

Offset: 1

Antti Karttunen, Jun 20 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A011772, A345053 (positions of zeros), A345065.

Cf. also A344767.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1])*sumdiv(n, d, if(dA011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
v345055 = DirInverseCorrect(vector(up_to,n,A011772(n)));
A345055(n) = v345055[n];
(Python 3.8+)
from itertools import combinations
from math import prod
from sympy import factorint, divisors
from sympy.ntheory.modular import crt
def A011772(n):
    plist = [p**q for p, q in factorint(2*n).items()]
    return 2*n-1 if len(plist) == 1 else min(min(crt([m,2*n//m],[0,-1])[0],crt([2*n//m,m],[0,-1])[0]) for m in (prod(d) for l in range(1,len(plist)//2+1) for d in combinations(plist,l)))
def A345055(n): return 1 if n == 1 else -sum(A011772(n//d)*A345055(d) for d in divisors(n, generator=True) if d < n) # Chai Wah Wu, Jun 20 2021

a(2^i) = 0 for i >= 3. See A345053. - Chai Wah Wu, Jul 05 2021

A346614 Inverse Moebius transform of A011772.

Original entry on oeis.org

1, 4, 3, 11, 5, 9, 7, 26, 11, 12, 11, 24, 13, 17, 12, 57, 17, 25, 19, 34, 15, 25, 23, 54, 29, 28, 37, 31, 29, 37, 31, 120, 24, 36, 25, 48, 37, 41, 27, 64, 41, 48, 43, 64, 29, 49, 47, 117, 55, 60, 36, 74, 53, 78, 25, 94, 39, 60, 59, 82, 61, 65, 50, 247, 42, 52, 67, 59, 48

Offset: 1

N. J. A. Sloane, Aug 18 2021

nonn

N. J. A. Sloane, Transforms.

Cf. A011772, A344767, A344005, A346615, A346616.

A346615 Moebius transform of A344005.

Original entry on oeis.org

1, 0, 1, 2, 3, 0, 5, 4, 6, 0, 9, -1, 11, 0, 0, 8, 15, 0, 17, -2, -1, 0, 21, 1, 20, 0, 18, -1, 27, 0, 29, 16, 0, 0, 5, -1, 35, 0, -1, 7, 39, 0, 41, -1, -2, 0, 45, -1, 42, 0, 0, -2, 51, 0, -3, -4, -1, 0, 57, 11, 59, 0, 15, 32, 10, 0, 65, -2, 0, 0, 69, -5, 71, 0, -1, -1, 6, 0, 77, -8

Offset: 0

N. J. A. Sloane, Aug 18 2021

sign

N. J. A. Sloane, Transforms.

Cf. A011772, A344767, A346614, A344005, A346616.

f(n) = my(m=1); while ((m*(m+1))%n, m++); m; \\ A344005
a(n) = sumdiv(n, d, moebius(n/d)*f(d)); \\ Michel Marcus, Aug 19 2021

A346616 Inverse Moebius transform of A344005.

Original entry on oeis.org

1, 2, 3, 5, 5, 6, 7, 12, 11, 10, 11, 12, 13, 14, 12, 27, 17, 22, 19, 17, 15, 22, 23, 27, 29, 26, 37, 24, 29, 24, 31, 58, 24, 34, 25, 36, 37, 38, 27, 39, 41, 30, 43, 36, 29, 46, 47, 57, 55, 58, 36, 41, 53, 74, 25, 38, 39, 58, 59, 49, 61, 62, 50, 121, 42, 48, 67, 53, 48, 50, 71, 59

Offset: 1

N. J. A. Sloane, Aug 18 2021

nonn

N. J. A. Sloane, Transforms.

Cf. A011772, A344767, A346614, A344005, A346615.

cp's OEIS Frontend

A344768 a(n) = n - A344767(n); Möbius transform of A344765.

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A345055 Dirichlet inverse of A011772.

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A346614 Inverse Moebius transform of A011772.

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A346615 Moebius transform of A344005.

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PARI

A346616 Inverse Moebius transform of A344005.

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Author

Keywords

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