A348420 a(n) = Product_{k=1..n} (p_k - 1)/2 where p_1, p_2, ..., p_n are the first n primes congruent to 3 modulo 4.

1, 1, 3, 15, 135, 1485, 22275, 467775, 10758825, 312005925, 10296195525, 360366843375, 14054306891625, 576226582556625, 29387555710387875, 1557540452650557375, 98125048516985114625, 6378128153604032450625, 440090842598678239093125

Offset: 0

Author

Jianing Song, Oct 18 2021

Keywords

• nonn
• easy

Comments

a(n) is the number of coprime squares modulo A348418(n+2), where A348418(n) is the smallest k with rank((Z/kZ)*) = n such that there are an odd number of coprime squares modulo k. (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.)

Examples

			A348418(2) = 8, and the number of coprime squares modulo 8 is a(0) = 1;
A348418(3) = 8 * 3 = 24, and the number of coprime squares modulo 24 is a(1) = (3-1)/2 = 1;
A348418(4) = 8 * 3 * 7 = 168, and the number of coprime squares modulo 168 is a(2) = ((3-1)/2) * ((7-1)/2) = 3;
A348418(5) = 8 * 3 * 7 * 11 = 1848, and the number of coprime squares modulo 1848 is a(3) = ((3-1)/2) * ((7-1)/2) * ((11-1)/2) = 15;
A348418(6) = 8 * 3 * 7 * 11 * 19 = 35112, and the number of coprime squares modulo 35112 is a(4) = ((3-1)/2) * ((7-1)/2) * ((11-1)/2) * ((19-1)/2) = 135.
		

Links

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

Crossrefs

Cf. A002145, A046073, A348418, A046072, A323739.

Programs

PARI

a(n) = my(t=1); forprime(p=2, , if(p%4==3, t*=(p-1)/2; if(n--<1, return(t)))) \\ following Charles R Greathouse IV's program for A078586

Formula

a(n) = Product_{k=1..n} (A002145(k) - 1)/2.

a(n) = A046073(A348418(n+2)).