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A290732
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Number of distinct values of X*(3*X-1)/2 mod n.
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6
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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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
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EXAMPLE
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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]
...
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MAPLE
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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;
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MATHEMATICA
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a[n_] := Table[PolynomialMod[X(3X-1)/2, n], {X, 0, 2*n-1}]// Union // Length;
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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