A102574 a(n) is the sum of the distinct norms of the divisors of n over the Gaussian integers.

1, 7, 10, 31, 31, 70, 50, 127, 91, 217, 122, 310, 183, 350, 310, 511, 307, 637, 362, 961, 500, 854, 530, 1270, 781, 1281, 820, 1550, 871, 2170, 962, 2047, 1220, 2149, 1550, 2821, 1407, 2534, 1830, 3937, 1723, 3500, 1850, 3782, 2821, 3710, 2210, 5110, 2451

Offset: 1

Author

Yasutoshi Kohmoto, Feb 25 2005

Keywords

• nonn
• mult
• easy

Comments

Also sum of divisors of n^2 which are the sum of two squares (A001481). For example the divisors of 3^2 are 1, 3, 9 of which only 1 and 9 are in A001481 and a(3) = 1 + 9 = 10. - Jianing Song, Aug 03 2018

Examples

			Let ||i|| denote the norm of i.
a(2) = 1 + ||1+i|| + 2^2 = 1 + 2 + 4 = 7.
a(5) = 1 + ||1+2i|| + 5^2 = 1 + 5 + 25 = 31. Note that ||1+2i|| = ||2+i|| so their norm (5) is only counted once.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Andrew Howroyd)
• Wikipedia, Gaussian integer.

Crossrefs

Cf. A000203 (sigma), A001157 (sigma_2), A001481, A097706, A103230, A243380.

Programs

Mathematica

b[n_] := Product[{p, e} = pe; If[Mod[p, 4] == 3, p^e, 1], {pe, FactorInteger[n]}];
a[n_] := With[{r = b[n]}, DivisorSigma[2, r] DivisorSigma[1, (n/r)^2]];
a /@ Range[50] (* Jean-François Alcover, Sep 20 2019, from PARI *)

PARI

\\ here b(n) is A097706.
b(n)={my(f=factor(n)); my(r=prod(i=1, #f~, my([p,e]=f[i,]); if(p%4==3, p^e, 1))); r}
a(n)={my(r=b(n)); sigma(r,2)*sigma((n/r)^2)} \\ Andrew Howroyd, Aug 03 2018

Python

from math import prod
from sympy import factorint
def A102574(n): return prod((q := int(p & 3 == 3))*(p**(2*(e+1))-1)//(p**2-1) + (1-q)*(p**(2*e+1)-1)//(p-1) for p, e in factorint(n).items()) # Chai Wah Wu, Jun 28 2022

Formula

a(n) = sigma_2(A097706(n)) * sigma((n/A097706(n))^2). - Andrew Howroyd, Aug 03 2018

Multiplicative with a(p^e) = sigma(p^(2e)) = (p^(2e+1) - 1)/(p - 1) if p = 2 or p == 1 (mod 4); sigma_2(p^e) = (p^(2e+2) - 1)/(p^2 - 1) if p == 3 (mod 4). - Jianing Song, Aug 03 2018

Sum_{k=1..n} a(k) ~ c * n^3, where c = (5/12) * zeta(3) * A243380 = 0.52812367275583317729... . - Amiram Eldar, Feb 13 2024

Extensions

Corrected and extended by David Wasserman, Apr 08 2008

Keyword:mult added by Andrew Howroyd, Aug 03 2018

Name clarified by Jianing Song, Aug 03 2018