A333848 a(n) gives the sum of the odd numbers of the smallest nonnegative reduced residue system modulo 2*n + 1, for n >= 0.

0, 1, 4, 9, 13, 25, 36, 32, 64, 81, 66, 121, 124, 121, 196, 225, 170, 216, 324, 240, 400, 441, 272, 529, 513, 416, 676, 560, 522, 841, 900, 570, 792, 1089, 770, 1225, 1296, 752, 1170, 1521, 1093, 1681, 1376, 1232, 1936, 1656, 1410, 1728, 2304, 1490, 2500

Offset: 0

Author

Wolfdieter Lang, May 01 2020

Keywords

• nonn
• easy

Comments

The smallest nonnegative reduced residue system modulo N is the ordered set RRS(N) (written as a list) with integers k from {0, 1, ..., N-1} satisfying gcd(k, N) = 1, for N >= 1. See A038566 (with A038566(1) = 0).

If only odd members of RRS(N) are considered, name this list RRSodd(N), e.g., RRSodd(1) = [], the empty list, RRSodd(2) = [1], etc. See A216319 (but there A216319(1) = 1). The number of elements of RRSodd(N) is delta(N) = A055034(N), for N >= 2, and 0 for N = 1.

Here only numbers N = 2*n + 1 >= 1 are considered, and for the empty list RRSodd(1) a(0) is set to 0.

a(n) gives for n >= 1 also the sum of the numbers of the primitive period of the unsigned Schick sequences SBB(2*n+1, q0 = 1) (BB for Brändli and Beyne), for which 2*n + 1 satisfies A135303(n) = 1 (in Schick's notation B(2*n+1) = 1, implying initial value q0 = 1). The numbers n satisfying A135303(n) = 1 are given in A333854.

The sequence with members gcd(a(n), 2*(2*n+1)) = A333849(n) is important for a length formula for the Euler tours ET(2*n+1, q0 = 1) given in A332441(n), for n >= 1 (but A333849(n) is used only for 2*n+1 values from A333854).

Examples

			n = 4: RRSodd(9) = {1, 5, 7} with sum a(4) = 13. Schick's unsigned cycle is SBB(9, 1) = (1, 7, 5). Because A135303(4) = B(9) = 1 there is only this cycle for n = 9.
		

References

• Carl Schick, Trigonometrie und unterhaltsame Zahlentheorie, Bokos Druck, Zürich, 2003 (ISBN 3-9522917-0-6). Tables 3.1 to 3.10, for odd p = 3..113 (with gaps), pp. 158-166.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• Gerold Brändli and Tim Beyne, Modified Congruence Modulo n with Half the Amount of Residues, arXiv:1504.02757 [math.NT], 2016.
• Wolfdieter Lang, On the Equivalence of Three Complete Cyclic Systems of Integers, arXiv:2008.04300 [math.NT], 2020.

Crossrefs

Cf. A038566, A055034, A135303, A216319, A333849, A333854.

Programs

Mathematica

{0}~Join~Table[Total@ Select[Range[1, m, 2], GCD[#, m] == 1 &], {m, Array[2 # + 1 &, 50]}] (* Michael De Vlieger, Oct 15 2020 *)

PARI

a(n) = if (n==0, 0, my(m=2*n+1); vecsum(select(x->((gcd(m, x)==1) && (x%2)), [1..m]))); \\ Michel Marcus, May 05 2020

PARI

apply( {A333848(n)=vecsum([2*m-1|m<-[1..n],gcd(m*2-1,n*2+1)==1])}, [0..50]) \\ M. F. Hasler, Jun 04 2020

Formula

a(n) = Sum_{j=1..delta(2*n+1)} RRSodd(2*n+1)_j, for n >= 1, with delta(k) = A055034(k). a(0) = 0 (undefined case).