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 A155918 Number of squared hypotenuses mod n in two dimensions. 5
 1, 2, 3, 3, 5, 6, 7, 5, 7, 10, 11, 9, 13, 14, 15, 9, 17, 14, 19, 15, 21, 22, 23, 15, 25, 26, 21, 21, 29, 30, 31, 17, 33, 34, 35, 21, 37, 38, 39, 25, 41, 42, 43, 33, 35, 46, 47, 27, 43, 50, 51, 39, 53, 42, 55, 35, 57, 58, 59, 45, 61, 62, 49, 33, 65, 66, 67, 51, 69, 70, 71, 35, 73 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Number of images of the map (x,y) -> x^2+y^2 in Z_n. Let n = p^e and k = r*p^b (gcd(r,p) = 1). If p == 1 (mod 4), then x^2 + y^2 == k (mod p) always have solutions; if p == 3 (mod 4), then x^2 + y^2 == k (mod p) is solvable if and only if b is even or b >= e; if p = 2, then x^2 + y^2 == k (mod p) is solvable if and only if r == 1 (mod 4) or b >= e - 1. If 0 <= k < n, then the number of solutions to x^2 + y^2 == k (mod n) is A305191(n,k). - Jianing Song, Apr 20 2019 LINKS Jianing Song, Table of n, a(n) for n = 1..10000 (first 1000 terms from Michel Marcus) FORMULA Multiplicative with a(p^e) = p^e if p == 1 (mod 4); ceiling(p^(e+1)/(p+1)) if p == 3 (mod 4); 2^(e-1) + 1 if p = 2. - Jianing Song, Apr 20 2019 MATHEMATICA (For[v = Table[0, {m, 1, n^2}]; m = 1; i = 0, i < n, i++, For[j = 0, j < n, j++, v[[m]] = Mod[i^2 + j^2, n]; m = m + 1]]; Length[Union[v]]) (* Second program: *) a[n_] := Module[{p, e}, Product[{p, e} = pe; Which[Mod[p, 4] == 1, p^e, Mod[p, 4] == 3, Ceiling[p^(e+1)/(p+1)], p == 2, 2^(e-1) + 1, True, p], {pe, FactorInteger[n]}]]; Array[a, 100] (* Jean-François Alcover, Jul 30 2020 *) PROG (PARI) a(n) = #Set(vector(n^2, i, ((i%n)^2 + (i\n)^2) % n)); \\ Michel Marcus, Jul 08 2017 (PARI) a(n)= {     my(r=1, f=factor(n));     for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);         if(p==2, r*=2^(e-1)+1);         if(p%4==1, r*=p^e);         if(p%4==3, r*=ceil(p^(e+1)/(p+1)));     );     return(r); } \\ Jianing Song, Apr 20 2019 CROSSREFS Cf. A305191. Sequence in context: A085314 A085310 A055653 * A344705 A331170 A325183 Adjacent sequences:  A155915 A155916 A155917 * A155919 A155920 A155921 KEYWORD mult,nonn AUTHOR Steven Finch, Jan 30 2009 STATUS approved

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Last modified June 24 18:46 EDT 2021. Contains 345419 sequences. (Running on oeis4.)