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

1, 3, 8, 24, 168, 1848, 35112, 807576, 25034856, 1076498808, 50595443976, 2985131194584, 200003790037128, 14200269092636088, 1121821258318250952, 93111164440414829016, 9590449937362727388648, 1026178143297811830585336, 130324624198822102484337672

Offset: 0

Author

Jianing Song, Oct 18 2021

Keywords

• nonn
• easy

Comments

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.

The number of coprime squares modulo a(n) is given by A046073(a(n)) = A348420(n-2) for n >= 2.

a(n) is the least k such that the Sylow 2-subgroup of (Z/kZ)* is (C_2)^n. - Jianing Song, Aug 13 2023

Examples

			a(2) = 8;
a(3) = 8 * 3 = 24;
a(4) = 8 * 3 * 7 = 168;
a(5) = 8 * 3 * 7 * 11 = 1848;
a(6) = 8 * 3 * 7 * 11 * 19 = 35112.
		

Links

• Jianing Song, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A046072, A046073, A348420, A002145, A102476.

Programs

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

Formula

a(n) = 8 * A078586(n-2) = 8 * (Product_{k=1..n-2} A002145(k)) for n > 2.