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A290732 Number of distinct values of X*(3*X-1)/2 mod n.

%I A290732 #33 Nov 03 2018 19:45:34
%S A290732 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,
%T A290732 15,18,16,32,18,18,12,36,19,20,21,24,21,24,22,24,27,24,24,48,22,22,27,
%U A290732 28,27,54,18,32,30,30,30,36
%N A290732 Number of distinct values of X*(3*X-1)/2 mod n.
%H A290732 Andrew Howroyd, <a href="/A290732/b290732.txt">Table of n, a(n) for n = 1..10000</a>
%H A290732 Andreas Enge, William Hart, Fredrik Johansson, <a href="http://arxiv.org/abs/1608.06810">Short addition sequences for theta functions</a>, arXiv:1608.06810 [math.NT], (24-August-2016). See Table 6.
%F A290732 a(3^n) = 3^n. - _Hugo Pfoertner_, Aug 25 2018
%F A290732 a(n) = A317623(n) * A040001(n). - _Andrew Howroyd_, Oct 27 2018
%F A290732 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
%e A290732 The values taken by (3*X^2-X)/2 mod n for small n are:
%e A290732    1, [0]
%e A290732    2, [0, 1]
%e A290732    3, [0, 1, 2]
%e A290732    4, [0, 1, 2, 3]
%e A290732    5, [0, 1, 2]
%e A290732    6, [0, 1, 2, 3, 4, 5]
%e A290732    7, [0, 1, 2, 5]
%e A290732    8, [0, 1, 2, 3, 4, 5, 6, 7]
%e A290732    9, [0, 1, 2, 3, 4, 5, 6, 7, 8]
%e A290732   10, [0, 1, 2, 5, 6, 7]
%e A290732   11, [0, 1, 2, 4, 5, 7]
%e A290732   12, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]
%e A290732   ...
%p A290732 a:=[]; M:=80;
%p A290732 for n from 1 to M do
%p A290732 q1:={};
%p A290732 for i from 0 to 2*n-1 do q1:={op(q1), i*(3*i-1)/2 mod n}; od;
%p A290732 s1:=sort(convert(q1,list));
%p A290732 a:=[op(a),nops(s1)];
%p A290732 od:
%p A290732 a;
%t A290732 a[n_] := Table[PolynomialMod[X(3X-1)/2, n], {X, 0, 2*n-1}]// Union // Length;
%t A290732 Array[a, 60] (* _Jean-François Alcover_, Sep 01 2018 *)
%o A290732 (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
%o A290732 (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
%Y A290732 Cf. A000224 (analog for X^2), A014113, A290729, A290730, A290731, A317623.
%K A290732 nonn,mult
%O A290732 1,2
%A A290732 _N. J. A. Sloane_, Aug 10 2017
%E A290732 Even terms corrected by _Andrew Howroyd_, Nov 03 2018