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A293484 The number of 7th powers in the multiplicative group modulo n.

%I A293484 #34 Aug 10 2023 02:21:05
%S A293484 1,1,2,2,4,2,6,4,6,4,10,4,12,6,8,8,16,6,18,8,12,10,22,8,20,12,18,12,4,
%T A293484 8,30,16,20,16,24,12,36,18,24,16,40,12,6,20,24,22,46,16,6,20,32,24,52,
%U A293484 18,40,24,36,4,58,16,60,30,36,32,48,20,66,32,44,24,10,24,72,36,40,36
%N A293484 The number of 7th powers in the multiplicative group modulo n.
%C A293484 The size of the set of numbers j^7 mod n, gcd(j,n)=1, 1 <= j <= n.
%C A293484 A000010(n) / a(n) is another multiplicative integer sequence (size of the kernel of the isomorphism of the multiplicative group modulo n to the multiplicative group of 7th powers modulo n).
%H A293484 R. J. Mathar, <a href="/A293484/b293484.txt">Table of n, a(n) for n = 1..10116</a>
%H A293484 Richard J. Mathar, <a href="/A293482/a293482.pdf">Size of the Set of Residues of Integer Powers of Fixed Exponent</a>, 2017.
%F A293484 Conjecture: a(2^e) = 1 for e <= 1; a(2^e) = 2^(e-1) for e >= 1; a(7^e) = 6 for e=1; a(7^e) = 6*7^(e-2) for e >= 2; a(p^e) = (p-1)*p^(e-1) for p == {2,3,4,5,6} (mod 7); a(p^e) = (p-1)*p^(e-1)/7 for p == 1 (mod 7). - _R. J. Mathar_, Oct 13 2017
%F A293484 a(n) = A000010(n)/A319101(n). This implies that the conjecture above is true. - _Jianing Song_, Nov 10 2019
%p A293484 A293484 := proc(n)
%p A293484     local r,j;
%p A293484     r := {} ;
%p A293484     for j from 1 to n do
%p A293484         if igcd(j,n)= 1 then
%p A293484             r := r union { modp(j &^ 7,n) } ;
%p A293484         end if;
%p A293484     end do:
%p A293484     nops(r) ;
%p A293484 end proc:
%p A293484 seq(A293484(n),n=1..120) ;
%t A293484 a[n_] := EulerPhi[n]/Count[Range[0, n - 1]^7 - 1, k_ /; Divisible[k, n]];
%t A293484 Array[a, 100] (* _Jean-François Alcover_, May 24 2023 *)
%t A293484 f[p_, e_] := (p-1)*p^(e-1)/If[Mod[p, 7] == 1, 7, 1]; f[2, e_] := 2^(e-1); f[7, 1] = 6; f[7, e_] := 6*7^(e-2); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Aug 10 2023 *)
%Y A293484 The number of k-th powers in the multiplicative group modulo n: A046073 (k=2), A087692 (k=3), A250207 (k=4), A293482 (k=5), A293483 (k=6), this sequence (k=7), A293485 (k=8).
%Y A293484 Cf. A085310, A319101, A000010.
%K A293484 nonn,mult
%O A293484 1,3
%A A293484 _R. J. Mathar_, Oct 10 2017