A290729 -id:A290729 - cp's OEIS frontend

A290730 Analog of A084848, replacing "quadratic residue" (X^2) with "value of X(3X-1)/2". a(n) = A290732(A290729(n)).

1, 3, 4, 6, 7, 9, 10, 12, 11, 12, 18, 21, 24, 28, 36, 40, 42, 44, 66, 77, 72, 84, 108, 120, 126, 162, 168, 216, 240, 252, 280, 264, 308, 396, 440, 462, 594, 504, 648, 720, 756, 840, 1008, 1080, 1134, 1260, 1512, 1512, 1680, 2016

Offset: 1

N. J. A. Sloane, Aug 10 2017

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..182
Andreas Enge, William Hart, Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], 2016-2018. See Table 6.

Cf. A000326, A085635, A084848, A290727, A290728, A290729, A290732.

a290732[n_] := Product[{p, e} = pe; If[p <= 3, p^e, (p^e - p^(e-1))/2 + (p^(e-1) - p^(Mod[e+1, 2]))/(2*(p+1))+1], {pe, FactorInteger[n]}];
r = 2; Reap[For[j = 1, j <= 24001, j = j+1, w = a290732[j]; t = w/j; If[t < r, r = t; Sow[w]]]][[2, 1]] (* Jean-François Alcover, Oct 03 2018, after Hugo Pfoertner *)

a290732(n)={my(f=factor(n));prod(k=1,#f~,my([p,e]=f[k, ]); if(p<=3,p^e,(p^e-p^(e-1))/2+(p^(e-1)-p^((e+1)%2))/(2*(p+1))+1))}
my(r=2);for(j=1,24001,my(w=a290732(j),t=w/j);if(tHugo Pfoertner, Aug 26 2018

More terms from Hugo Pfoertner, Aug 23 2018

a(1), a(19) and a(38) corrected by Hugo Pfoertner, Aug 26 2018

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

Original entry on oeis.org

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

N. J. A. Sloane, Aug 10 2017

			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]
  ...

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.

Cf. A000224 (analog for X^2), A014113, A290729, A290730, A290731, A317623.

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;

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 *)

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

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

a(3^n) = 3^n. - Hugo Pfoertner, Aug 25 2018
a(n) = A317623(n) * A040001(n). - Andrew Howroyd, Oct 27 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

Even terms corrected by Andrew Howroyd, Nov 03 2018

cp's OEIS Frontend

A290730 Analog of A084848, replacing "quadratic residue" (X^2) with "value of X(3X-1)/2". a(n) = A290732(A290729(n)).

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

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cp's OEIS Frontend

A290730 Analog of A084848, replacing "quadratic residue" (X^2) with "value of X(3X-1)/2". a(n) = A290732(A290729(n)).

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Mathematica

PARI

Extensions

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

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Author

Keywords

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Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

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Extensions

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