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A348418 a(n) is the smallest k with rank((Z/kZ)*) = n such that there are an odd number of coprime squares modulo k.

%I A348418 #22 Aug 13 2023 08:44:48
%S A348418 1,3,8,24,168,1848,35112,807576,25034856,1076498808,50595443976,
%T A348418 2985131194584,200003790037128,14200269092636088,1121821258318250952,
%U A348418 93111164440414829016,9590449937362727388648,1026178143297811830585336,130324624198822102484337672
%N A348418 a(n) is the smallest k with rank((Z/kZ)*) = n such that there are an odd number of coprime squares modulo k.
%C A348418 The rank of a finitely generated group rank(G) is defined to be the size of the minimal generating sets of G. In particular, rank((Z/kZ)*) = 0 if k <= 2 and A046072(k) otherwise.
%C A348418 The number of coprime squares modulo a(n) is given by A046073(a(n)) = A348420(n-2) for n >= 2.
%C A348418 a(n) is the least k such that the Sylow 2-subgroup of (Z/kZ)* is (C_2)^n. - _Jianing Song_, Aug 13 2023
%H A348418 Jianing Song, <a href="/A348418/b348418.txt">Table of n, a(n) for n = 0..200</a>
%F A348418 a(n) = 8 * A078586(n-2) = 8 * (Product_{k=1..n-2} A002145(k)) for n > 2.
%e A348418 a(2) = 8;
%e A348418 a(3) = 8 * 3 = 24;
%e A348418 a(4) = 8 * 3 * 7 = 168;
%e A348418 a(5) = 8 * 3 * 7 * 11 = 1848;
%e A348418 a(6) = 8 * 3 * 7 * 11 * 19 = 35112.
%o A348418 (PARI) a(n) = if(n<=2, [1, 3, 8][n+1], my(t=8); forprime(p=2, , if(p%4==3, t*=p; if(n--<3, return(t))))) \\ following _Charles R Greathouse IV_'s program for A078586
%Y A348418 Cf. A046072, A046073, A348420, A002145, A102476.
%K A348418 nonn,easy
%O A348418 0,2
%A A348418 _Jianing Song_, Oct 18 2021