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A293482
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The number of 5th powers in the multiplicative group modulo n.
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13
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1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 2, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 2, 22, 8, 4, 12, 18, 12, 28, 8, 6, 16, 4, 16, 24, 12, 36, 18, 24, 16, 8, 12, 42, 4, 24, 22, 46, 16, 42, 4, 32, 24, 52, 18, 8, 24, 36, 28, 58, 16, 12, 6, 36, 32, 48, 4, 66, 32, 44, 24, 14, 24, 72, 36, 8, 36, 12, 24, 78, 32, 54, 8, 82, 24
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OFFSET
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1,3
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COMMENTS
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The size of the set of numbers j^5 mod n, gcd(j,n)=1, 1 <= j <= n.
A000010(n) / a(n) is another multiplicative integer sequence.
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LINKS
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FORMULA
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Conjecture: a(2^e) = 1 for e <= 1; a(2^e) = 2^(e-1) for e >= 1; a(5)=4; a(5^e) = 4*5^(e-2) for e > 1; a(p^e) = (p-1)*p^(e-1) for p == {2,3,4} (mod 5); a(p^e) = (p-1)*p^(e-1)/5 for p == 1 (mod 5). - R. J. Mathar, Oct 13 2017
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MAPLE
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local r, j;
r := {} ;
for j from 1 to n do
if igcd(j, n)= 1 then
r := r union { modp(j &^ 5, n) } ;
end if;
end do:
nops(r) ;
end proc:
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MATHEMATICA
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a[n_] := Module[{r, j}, r = {}; For[j = 1, j <= n, j++, If[GCD[j, n] == 1, r = r ~Union~ {PowerMod[j, 5, n]}] ]; Length[r]];
f[p_, e_] := (p - 1)*p^(e - 1)/If[Mod[p, 5] == 1, 5, 1]; f[2, e_] := 2^(e - 1); f[2, 1] = 1; f[5, e_] := 4*5^(e-2); f[5, 1] = 4; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 10 2023 *)
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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