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 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 (* Jean-François 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

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