A290732 Number of distinct values of X*(3*X-1)/2 mod n.
1, 2, 3, 4, 3, 6, 4, 8, 9, 6, 6, 12, 7, 8, 9, 16, 9, 18, 10, 12, 12, 12, 12, 24, 11, 14, 27, 16, 15, 18, 16, 32, 18, 18, 12, 36, 19, 20, 21, 24, 21, 24, 22, 24, 27, 24, 24, 48, 22, 22, 27, 28, 27, 54, 18, 32, 30, 30, 30, 36
Offset: 1
Examples
The values taken by (3*X^2-X)/2 mod n for small n are: 1, [0] 2, [0, 1] 3, [0, 1, 2] 4, [0, 1, 2, 3] 5, [0, 1, 2] 6, [0, 1, 2, 3, 4, 5] 7, [0, 1, 2, 5] 8, [0, 1, 2, 3, 4, 5, 6, 7] 9, [0, 1, 2, 3, 4, 5, 6, 7, 8] 10, [0, 1, 2, 5, 6, 7] 11, [0, 1, 2, 4, 5, 7] 12, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..10000
- Andreas Enge, William Hart, Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], (24-August-2016). See Table 6.
Programs
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Maple
a:=[]; M:=80; for n from 1 to M do q1:={}; for i from 0 to 2*n-1 do q1:={op(q1), i*(3*i-1)/2 mod n}; od; s1:=sort(convert(q1,list)); a:=[op(a),nops(s1)]; od: a; -
Mathematica
a[n_] := Table[PolynomialMod[X(3X-1)/2, n], {X, 0, 2*n-1}]// Union // Length; Array[a, 60] (* Jean-François Alcover, Sep 01 2018 *) -
PARI
a(n)={my(v=vector(n)); for(i=0, 2*n-1, v[i*(3*i-1)/2%n + 1]=1); vecsum(v)} \\ Andrew Howroyd, Oct 27 2018 -
PARI
a(n)={my(f=factor(n)); prod(i=1, #f~, my([p,e]=f[i,]); if(p<=3, p^e, 1 + p^(e+1)\(2*p+2)))} \\ Andrew Howroyd, Nov 03 2018
Formula
a(3^n) = 3^n. - Hugo Pfoertner, Aug 25 2018
Multiplicative with a(2^e) = 2^e, a(3^e) = 3^e, a(p^e) = 1 + floor( p^(e+1)/(2*p+2) ) for prime p >= 5. - Andrew Howroyd, Nov 03 2018
Extensions
Even terms corrected by Andrew Howroyd, Nov 03 2018