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A346744
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The number of congruencies k^(n-k) + (n-k)^k == 0 (mod n) with 0 < k < n.
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1
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0, 1, 2, 3, 2, 3, 2, 5, 4, 7, 4, 7, 2, 3, 2, 9, 2, 7, 6, 7, 8, 3, 2, 15, 6, 9, 10, 11, 4, 17, 4, 17, 4, 9, 8, 15, 2, 5, 10, 15, 2, 13, 4, 11, 6, 3, 2, 19, 8, 15, 2, 11, 2, 19, 12, 19, 8, 11, 4, 19, 4, 5, 14, 33, 6, 13, 6, 17, 2, 25, 2, 31, 2, 9, 12, 11, 6, 29, 4
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OFFSET
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1,3
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COMMENTS
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Of course for any n, k being equal to either 1 or n-1 would work.
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LINKS
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Seiichi Manyama, Table of n, a(n) for n = 1..10000
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MAPLE
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a:= n-> add(`if`(k&^(n-k)+(n-k)&^k mod n=0, 1, 0), k=1..n-1):
seq(a(n), n=1..100); # Alois P. Heinz, Aug 06 2021
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MATHEMATICA
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f[n_] := Block[{c = 0, k = 1}, While[k < n, If[ Mod[ PowerMod[k, n - k, n] + PowerMod[n - k, k, n], n] == 0, c++]; k++]; c]; Array[f@# &, 100]
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PROG
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(Python)
def a(n): return sum((k**(n-k) + (n-k)**k)%n == 0 for k in range(1, n))
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Jul 31 2021
(PARI) a(n) = sum(k=1, n-1, Mod(k, n)^(n-k) + Mod(n-k, n)^k == 0); \\ Michel Marcus, Aug 06 2021
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CROSSREFS
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Cf. A114977, A114978, A114979, A114980, A114981.
Sequence in context: A077600 A120223 A282691 * A265577 A251103 A065559
Adjacent sequences: A346741 A346742 A346743 * A346745 A346746 A346747
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KEYWORD
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easy,nonn
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AUTHOR
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Robert G. Wilson v, Jul 31 2021
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STATUS
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approved
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