A102476 Least modulus with 2^n square roots of 1.

1, 3, 8, 24, 120, 840, 9240, 120120, 2042040, 38798760, 892371480, 25878772920, 802241960520, 29682952539240, 1217001054108840, 52331045326680120, 2459559130353965640, 130356633908760178920, 7691041400616850556280

Offset: 0

Author

David W. Wilson, Jan 10 2005

Keywords

• easy
• nonn

Comments

The number of square roots of 1 in any modulus is a power of 2.

Another way of expressing the same: These are also the record setting values of m for the number of solutions to "m*k+1 is a square", for some k, 0<=k<=m. There is 1 solution for a(0)=m=1, and for m = a(n), n>0, there is the first occurrence of 2^n solutions. Compare with A006278. - Richard R. Forberg, Mar 18 2016

Also a(n) is the least k such that the proportion of squares in a reduced residue system modulo n is 1/2^n, i.e. A046073(k)/A000010(k) = 1/2^n. - Jianing Song, Nov 12 2019

From Jianing Song, Oct 18 2021: (Start)

a(n) is the smallest k such that rank((Z/kZ)*) = n. 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)) = A323739(n-1) for n >= 2. (End)

Examples

			a(3) = 24 because 24 is the least modulus with 2^3 square roots of 1, namely 1,5,7,11,13,17,19,23.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..351

Crossrefs

Cf. A060594, A002110, A006278, A046072, A046073, A272590, A323739.

Programs

Mathematica

{1, 3}~Join~Table[4 Product[Prime[k], {k, n}], {n, 17}] (* Michael De Vlieger, Mar 27 2016 *)
nxt[{a_, p_}] := {a*NextPrime[p], NextPrime[p]}; Join[{1,3},NestList[nxt,{8,2},20][[All,1]]] (* or *) Join[{1,3},4*FoldList[ Times, Prime[ Range[ 21]]]](* Harvey P. Dale, Dec 18 2016 *)

PARI

a(n) = if(n<=1, [1,3][n+1], 4*factorback(primes(n-1))) \\ Jianing Song, Oct 19 2021, following David A. Corneth's program for A002110

Formula

a(n) = 4(prime(n-1))# = 4*A002110(n-1) for n >= 2. Least k with A060594(k) = 2^n.