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 A290731 Number of distinct values of X*(X+1) mod n. 7
 1, 1, 2, 2, 3, 2, 4, 4, 4, 3, 6, 4, 7, 4, 6, 8, 9, 4, 10, 6, 8, 6, 12, 8, 11, 7, 11, 8, 15, 6, 16, 16, 12, 9, 12, 8, 19, 10, 14, 12, 21, 8, 22, 12, 12, 12, 24, 16, 22, 11, 18, 14, 27, 11, 18, 16, 20, 15, 30, 12, 31, 16, 16, 32, 21, 12, 34, 18, 24, 12, 36, 16, 37, 19, 22, 20, 24, 14, 40, 24 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 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 5. FORMULA Multiplicative with a(2^e) = 2^(e-1), a(p^2) = 1 + floor(p^(e+1)/(2*p+2)) for odd prime p. - Andrew Howroyd, Aug 01 2018 EXAMPLE The values taken by X^2+X mod n for small n are: 1, [0] 2, [0] 3, [0, 2] 4, [0, 2] 5, [0, 1, 2] 6, [0, 2] 7, [0, 2, 5, 6] 8, [0, 2, 4, 6] 9, [0, 2, 3, 6] 10, [0, 2, 6] 11, [0, 1, 2, 6, 8, 9] 12, [0, 2, 6, 8] ... MAPLE a:=[]; M:=80; for n from 1 to M do q1:={}; for i from 0 to n-1 do q1:={op(q1), (i^2+i) mod n}; od; s1:=sort(convert(q1, list)); a:=[op(a), nops(s1)]; od: a; MATHEMATICA a[n_] := Product[{p, e} = pe; If[p==2, 2^(e-1), 1+Quotient[p^(e+1), (2p+2)]], {pe, FactorInteger[n]}]; Array[a, 100] (* Jean-François Alcover, Aug 05 2018, after Andrew Howroyd *) PROG (PARI) a(n)={my(v=vector(n)); for(i=0, n-1, v[i*(i+1)%n + 1]=1); vecsum(v)} \\ Andrew Howroyd, Aug 01 2018 (PARI) a(n)={my(f=factor(n)); prod(i=1, #f~, my([p, e]=f[i, ]); if(p==2, 2^(e-1), 1 + p^(e+1)\(2*p+2)))} \\ Andrew Howroyd, Aug 01 2018 CROSSREFS Cf. A000224 (analog for X^2), A290732. Sequence in context: A066589 A007897 A180783 * A106289 A332436 A165418 Adjacent sequences:  A290728 A290729 A290730 * A290732 A290733 A290734 KEYWORD nonn,mult AUTHOR N. J. A. Sloane, Aug 10 2017 STATUS approved

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Last modified April 21 11:10 EDT 2021. Contains 343150 sequences. (Running on oeis4.)