A345053 -id:A345053 - cp's OEIS frontend

A345055 Dirichlet inverse of A011772.

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

A345065 Sum of A011772 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 9, 0, 12, 0, 15, 4, 24, 0, -8, 0, 36, 16, 31, 0, 24, 0, -28, 24, 60, 0, 33, 16, 72, 24, -36, 0, -74, 0, 63, 40, 96, 48, 9, 0, 108, 48, 86, 0, -116, 0, -64, 36, 132, 0, 14, 36, 32, 64, -84, 0, 8, 80, 23, 72, 168, 0, 341, 0, 180, 48, 127, 96, -190, 0, -112, 88, -244, 0, -77, 0, 216, 40, -120, 120, -240, 0, -73, 88

Offset: 1

Antti Karttunen, Jun 20 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A011772, A345053, A345055.

up_to = 65537;
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];
A345065(n) = (A011772(n)+A345055(n));

a(n) = A011772(n) + A345055(n).

a(2^i) = 2^(i+1)-1 for i >= 3. See A345053. - Chai Wah Wu, Jul 05 2021

A354877 Positions of zeros in A354875, which is the Dirichlet inverse of A344005.

Original entry on oeis.org

16, 100, 147, 256, 294, 2597, 4096, 5194, 7399, 14798, 17003, 31409, 34006, 36211, 50617, 60221, 62818, 65023, 65536, 72422, 74627, 79429, 101234, 103439, 120442, 122647, 130046, 132251, 137053, 149254, 158858, 175469, 206878, 218687, 223489, 245294, 247499, 264502, 266707, 274106, 276311, 281113, 319529, 324331

Offset: 1

Antti Karttunen, Jun 12 2022

nonn

Cf. A344005, A354875.

Cf. also A345053.

PARI
```
isA354877(n) = (0==A354875(n));
```

cp's OEIS Frontend

A345055 Dirichlet inverse of A011772.

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A345065 Sum of A011772 and its Dirichlet inverse.

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A354877 Positions of zeros in A354875, which is the Dirichlet inverse of A344005.

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