login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A102574 a(n) is the sum of the distinct norms of the divisors of n over the Gaussian integers. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Wikipedia, Gaussian integer

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

EXAMPLE

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.

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-Fran├žois Alcover, Sep 20 2019, from PARI *)

PROG

(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

CROSSREFS

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

Sequence in context: A280966 A174466 A070422 * A317797 A284418 A244165

Adjacent sequences:  A102571 A102572 A102573 * A102575 A102576 A102577

KEYWORD

nonn,mult

AUTHOR

Yasutoshi Kohmoto, Feb 25 2005

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 19 22:55 EDT 2019. Contains 328244 sequences. (Running on oeis4.)