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A213381
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a(n) = n^n mod (n+2).
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6
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1, 1, 0, 2, 4, 3, 0, 7, 6, 5, 4, 6, 8, 13, 0, 8, 16, 9, 4, 19, 12, 11, 16, 17, 14, 7, 4, 14, 16, 15, 0, 31, 18, 13, 16, 18, 20, 37, 24, 20, 16, 21, 4, 7, 24, 23, 16, 17, 6, 49, 4, 26, 34, 3, 8, 55, 30, 29, 4, 30, 32, 61, 0, 57, 16, 33, 4, 67, 46, 35, 16, 36, 38
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OFFSET
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0,4
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COMMENTS
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Conjectures:
1. Indices of zeros: 2^(x+2)-2, x >= 0.
3. Every integer k >= 0 appears in a(n) at least once.
4. Every k >= 0 appears in a(n) infinitely many times.
Conjecture 1) is true: with m = n+2, a(n) = (-2)^(m-2) mod m = 0 iff m divides 2^(m-2), i.e., m = 2^k for some k with k <= m-2 (which is true for k >= 2).
Conjecture 2) is true: if n = 3*p-2 where p is prime, then n == 1 (mod 3) so n^n == n (mod 3), and n^(p-1) == 1 (mod p) so n^n == n (mod p), and therefore (if p <> 3) n^n == n (mod 3*p). A separate computation verifies the case p=3.
If p is an odd prime, then a(p+2) = (p-1)/2. (End)
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LINKS
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FORMULA
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a(n) = (n^n) mod (n+2).
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EXAMPLE
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a(5) = 5^5 mod 7 = 3125 mod 7 = 3.
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MATHEMATICA
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a[n_] := PowerMod[-2, n, n+2];
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PROG
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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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