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A289559 Number of modulo n residues among sums of two fourth powers. 2
1, 2, 3, 3, 3, 6, 7, 3, 7, 6, 11, 9, 10, 14, 9, 3, 13, 14, 19, 9, 21, 22, 23, 9, 11, 20, 19, 21, 22, 18, 31, 6, 33, 26, 21, 21, 37, 38, 30, 9, 41, 42, 43, 33, 21, 46, 47, 9, 43, 22, 39, 30, 53, 38, 33, 21, 57, 44, 59, 27, 61, 62, 49, 11, 30, 66, 67 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Conjecture: the only primes p for which a(p) < p are 5, 13, 17, 29. - Robert Israel, Jul 09 2017

Conjecture is true: see Math Overflow link. - Robert Israel, Apr 01 2020

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

Mathematics Overflow, Does the expression x^4+y^4 take on all values in Z/pZ (see answer by J. Silverman)

EXAMPLE

a(7) = 7 because (j^4 + k^4) mod 7, where j and k are integers, can take on all 7 values 0..6; e.g.:

   (0^4 + 0^4) mod 7 = ( 0 +  0) mod 7 =  0 mod 7 = 0;

   (0^4 + 1^4) mod 7 = ( 0 +  1) mod 7 =  1 mod 7 = 1;

   (1^4 + 1^4) mod 7 = ( 1 +  1) mod 7 =  2 mod 7 = 2;

   (1^4 + 2^4) mod 7 = ( 1 + 16) mod 7 = 17 mod 7 = 3;

   (2^4 + 2^4) mod 7 = (16 + 16) mod 7 = 32 mod 7 = 4;

   (1^4 + 3^4) mod 7 = ( 1 + 81) mod 7 = 82 mod 7 = 5;

   (2^4 + 3^4) mod 7 = (16 + 81) mod 7 = 97 mod 7 = 6.

a(16) = 3 because (j^4 + k^4) mod 16 can take on only the three values 0, 1, and 2. (This is because j^4 mod 16 = 0 for all even j and 1 for all odd j.)

MAPLE

f1:= proc(n) option remember; local S;

    S:= {seq(x^4 mod n, x=0..n-1)};

  nops({seq(seq(S[i]+S[j] mod n, i=1..j), j=1..nops(S))});

end proc:

f:= proc(n) local t;

mul(f1(t[1]^t[2]), t = ifactors(n)[2])

end proc:

map(f, [$1..100]); # Robert Israel, Jul 09 2017

MATHEMATICA

f1[n_] := f1[n] = Module[{S = Table[Mod[x^4, n], {x, 0, n-1}] // Union}, Table[Mod[S[[i]] + S[[j]], n], {j, 1, Length[S]}, {i, 1, j}] // Flatten // Union // Length];

f[n_] := Module[{p, e}, Product[{p, e} = pe; f1[p^e], {pe, FactorInteger[n]}]];

Array[f, 100] (* Jean-Fran├žois Alcover, Jul 30 2020, after Maple *)

PROG

(PARI) a(n) = #Set(vector(n^2, i, ((i%n)^4 + (i\n)^4) % n)); \\ Michel Marcus, Jul 08 2017

CROSSREFS

Cf. A155918 (gives number of modulo n residues among sums of two squares).

Sequence in context: A205442 A049982 A245642 * A070167 A168113 A170895

Adjacent sequences:  A289556 A289557 A289558 * A289560 A289561 A289562

KEYWORD

nonn,mult

AUTHOR

Jon E. Schoenfield, Jul 08 2017

STATUS

approved

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Last modified May 18 23:41 EDT 2021. Contains 344009 sequences. (Running on oeis4.)