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Trace of addition table of group of divisors of n.
2

%I #52 Dec 22 2023 15:57:22

%S 1,2,2,7,2,4,2,10,13,4,2,14,2,4,4,31,2,26,2,14,4,4,2,20,31,4,20,14,2,

%T 8,2,42,4,4,4,91,2,4,4,20,2,8,2,14,26,4,2,62,57,62,4,14,2,40,4,20,4,4,

%U 2,28,2,4,26,127,4,8,2,14,4,8,2,130,2,4,62,14,4,8,2,62,121,4,2,28

%N Trace of addition table of group of divisors of n.

%C Let v_p(n) denote the valuation of n at the prime p. The set D_n of divisors of n can be made into an abelian (D_n, +) group by setting: d + e := Product_{p|n} p^((v_p(d)+v_p(e)) mod (v_p(n)+1)), since for each divisor d of n we have the inequalities 0 <= v_p(d) <= v_p(n), so v_p(d) can be seen as a number mod (v_p(n)+1). This group is isomorphic to the abelian group (Z/(a_1+1))x...x(Z/(a_r+1)) where n := p_1^a_1 * ... * p_r^a_r is the prime factorization of n. The number a(n) is the trace of the addition table of this group for n, where the divisors are ordered by their absolute value.

%C If n = q^e * m^2 (by Euler) is an odd perfect number, with q prime and q == e == 1 (mod 4), then n = ((q+1)/2) * (a(n)/2) where the factors in the right-hand side are natural numbers.

%H MathOverflow, <a href="https://mathoverflow.net/questions/369941/">Sum of divisors and unitary divisors as the eigenvalue and the spectral norm of some addition matrix?</a>

%F a(n) is multiplicative with a(p^v) = sigma(p^v)*(p^v+p^(2*floor(v/2)+1))/(p^v*(p+1)), where sigma(n) = A000203(n) is the sum of divisors of n.

%F If we denote eta(n,d):=Product_{p|n, 2*v_p(d)>=v_p(n)+1} p^(v_p(n)+1) for a divisor d|n, then we have a(n) = Sum_{d|n} d^2/eta(n,d).

%F A048250(n)*n*a(n) = A000203(n) * A007947(n) * A367203(n).

%F Multiplicative with a(p^v) = (p^(v+1)-1)/(p-1) if v is even and a(p^v) = 2*(p^(v+1)-1)/(p^2-1) otherwise. - _Chai Wah Wu_, Nov 19 2023

%F a(n) = A000203(n) if and only if n is a square. - _Chai Wah Wu_, Nov 20 2023

%e For n = 3 the divisors are {1,3} and the addition table is given by [[1,3],[3,1]] with trace equal to the sum of the elements on the diagonal which are 1+1 = 2.

%t f[p_, e_] := If[EvenQ[e], (p^(e + 1) - 1)/(p - 1), 2*(p^(e + 1) - 1)/(p^2 - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 22 2023 *)

%o (SageMath)

%o def a(n):

%o return sigma(n)*prod((p**valuation(n,p)+p**(2*floor(valuation(n,p)/2)+1))/(p**valuation(n,p)*(p+1)) for p in prime_divisors(n))

%o (PARI) b(n,d) = my(f=factor(n)); prod(k=1,#f~, if(2*valuation(d,f[k,1])>=valuation(n,f[k,1])+1,{f[k,1]^(valuation(n,f[k,1])+1)},{1}));

%o a(n) = my(d=divisors(n)); sumdiv(n, d, d^2/b(n,d));

%o (PARI) a(n)=my(f=factor(n)); sigma(n)*prod(k=1,#f~, (f[k,1]^valuation(n,f[k,1])+f[k,1]^(2*floor(valuation(n,f[k,1])/2)+1))/(f[k,1]^valuation(n,f[k,1])*(f[k,1]+1) ));

%o (PARI) a(n) = my(d=divisors(n), f=factor(n), m=matrix(#d, #d, i, j, prod(k=1, #f~, f[k,1]^((valuation(d[i],f[k,1])+valuation(d[j],f[k,1])) % (1+f[k,2]))))); trace(m); \\ _Michel Marcus_, Nov 10 2023

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def A367197(n): return prod((p**(e+1)-1<<1)//(p**2-1) if e&1 else (p**(e+1)-1)//(p-1) for p,e in factorint(n).items()) # _Chai Wah Wu_, Nov 19 2023

%Y Cf. A000203, A048250, A367203, A007947.

%K nonn,mult

%O 1,2

%A _Orges Leka_, Nov 10 2023