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

0,2

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

Links

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

Formula

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

Example

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.

Prog

(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

Crossrefs

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

Sequence in context: A102919 A363212 A102476 * A302109 A328272 A220486

Adjacent sequences: A348415 A348416 A348417 * A348419 A348420 A348421

Keyword

nonn,easy

Author

Jianing Song, Oct 18 2021

Status

approved