A345065 -id:A345065 - 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

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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

A354876 Sum of A344005 and its Dirichlet inverse, where A344005(n) is the smallest positive m such that n divides the oblong number m*(m+1).

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 5, 4, 8, 0, 10, 0, 12, 16, 15, 0, 12, 0, 20, 24, 20, 0, 12, 16, 24, 24, 30, 0, -6, 0, 35, 40, 32, 48, 20, 0, 36, 48, 20, 0, -12, 0, 50, 36, 44, 0, 9, 36, 32, 64, 60, 0, 28, 80, 32, 72, 56, 0, -63, 0, 60, 48, 71, 96, -18, 0, 80, 88, -20, 0, 32, 0, 72, 40, 90, 120, -24, 0, 10, 88, 80, 0, -98, 128

Offset: 1

Antti Karttunen, Jun 12 2022

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

Cf. A344005, A354875.

Cf. also A345065.

A344005(n) = for(m=1, oo, if((m*(m+1))%n==0, return(m))); \\ From A344005
memoA354875 = Map();
A354875(n) = if(1==n,1,my(v); if(mapisdefined(memoA354875,n,&v), v, v = -sumdiv(n,d,if(dA344005(n/d)*A354875(d),0)); mapput(memoA354875,n,v); (v)));
A354876(n) = (A344005(n)+A354875(n));

a(n) = A344005(n) + A354875(n).

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A344005(d) * A354875(n/d).

cp's OEIS Frontend

A345055 Dirichlet inverse of A011772.

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