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

0,2

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)

Links

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

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A002104 A102919 A363212 * A348418 A302109 A328272

Adjacent sequences: A102473 A102474 A102475 * A102477 A102478 A102479

Keyword

easy,nonn

Author

David W. Wilson, Jan 10 2005

Status

approved