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A337454
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a(n) is the number of divisors of n such that the ratio (the number of nonnegative m < n such that m^d == m (mod n))/(the number of nonnegative m < n such that -m^d == m (mod n)) is also a divisor of n.
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4
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1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 5, 2, 4, 4, 5, 2, 6, 2, 5, 2, 4, 2, 7, 3, 4, 4, 5, 2, 8, 2, 6, 2, 4, 2, 9, 2, 4, 4, 7, 2, 6, 2, 5, 6, 4, 2, 9, 3, 6, 4, 5, 2, 8, 2, 7, 2, 4, 2, 9, 2, 4, 5, 7, 2, 6, 2, 5, 2, 6, 2, 10, 2, 4, 6, 5, 1, 8, 2, 9, 5, 4, 2, 9, 4, 4, 4, 7, 2, 12, 4, 5, 2, 4, 2, 11
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OFFSET
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1,2
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COMMENTS
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a(n) is the number of ordered pairs (x,y) of divisors of n such that the ratio (the number of nonnegative m < n such that m^x == m (mod n)) / (the number of nonnegative m < n such that -m^x == m (mod n))) is equal to y. These pairs of divisors of each n define the direction of the arcs of some directed graph, the vertices of the number a(n) of which are indicated by the corresponding values of the divisors.
1 <= a(n) <= tau(n) where tau(n) is the number of divisors of n.
The boundary sequences of this relation are A338189 (numbers i such that a(i) = 1) and A338190 (numbers j such that a(j) = tau(j)).
Furthermore, for any nonnegative k, 1 <= the ratio (the number of nonnegative m < n such that m^k == m (mod n)) / (the number of nonnegative m < n such that -m^k == m (mod n)) <= n.
The number of divisors d such that A334006(n,d) is also a divisor of n. - Peter Kagey, Sep 09 2020
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LINKS
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EXAMPLE
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a(1) = 1 solution is pair (x,y) of divisors of n = 1 is (1,1).
a(2) = 2 solutions are pairs (x,y) of divisors of n = 2 are (1,1) and (2,1).
a(3) = 2 solutions are pairs (x,y) of divisors of n = 3 are (1,3) and (3,3).
a(4) = 3 solutions are pairs (x,y) of divisors of n = 4 are (1,2), (2,1) and (4,1).
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PROG
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(Magma) [#[d: d in Divisors(n) | Denominator(n*#[m: m in [0..n-1] | -m^d mod n eq m]/#[m: m in [0..n-1] | m^d mod n eq m]) eq 1]: n in [1..96]];
(PARI) a(n) = sumdiv(n, d, n % (sum(m=0, n-1, Mod(m, n)^d == m)/sum(m=0, n-1, Mod(-m, n)^d == m)) == 0); \\ Michel Marcus, Aug 30 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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