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A132213
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Number of distinct primes among the squares mod n.
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3
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0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 2, 0, 1, 3, 0, 0, 2, 2, 4, 1, 1, 3, 3, 0, 2, 4, 3, 0, 4, 1, 4, 1, 2, 4, 2, 1, 3, 6, 2, 0, 5, 2, 6, 2, 2, 7, 5, 0, 6, 5, 3, 3, 8, 6, 3, 0, 3, 6, 8, 0, 6, 8, 3, 2, 2, 3, 7, 3, 3, 2, 7, 0, 9, 10, 3, 4, 6, 4, 9, 1, 10, 10, 11, 1, 2, 13, 3, 0, 10, 4, 5, 4, 4, 13, 4, 1, 11, 10, 4, 4
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listen;
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OFFSET
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1,11
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COMMENTS
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It appears that a(n)=0 for only the 30 numbers in A065428, which appears to be related to idoneal numbers, A000926. The graph shows a(n) can be quite small even for large n. For example, a(9240)=7. Observe that the graph up to n=10000 appears to have 5 components. Why?
The logarithmic plot of the first 10^6 terms shows seven components.
Empirically, in the logarithmic plot of the sequence:
- the set of indices of the first component (starting from the top), say S_1, is the union of A061345 and of A278568,
- the set of indices of the n-th component (for n > 1), say S_n, contains the numbers k not in a previous component and such that (omega(k) = n-1) or (omega(k) = n and val(k) = 0 or 2) or (omega(k) = n+1 and val(k) = 1) (where omega(k) = A001221(k) and val(k) = A007814(k)),
- see logarithmic scatterplot colored according to this scheme in Links section.
(End)
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LINKS
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EXAMPLE
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For n=14, the squares (mod n) repeat 0,1,4,9,2,11,8,7,8,11,2,9,4,1,0,..., a sequence containing three distinct primes: 2, 7 and 11. Hence a(14)=3.
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MATHEMATICA
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Table[s=Union[Mod[Range[n]^2, n]]; Length[Select[s, PrimeQ]], {n, 10000}]
Table[Count[Union[PowerMod[Range[n], 2, n]], _?PrimeQ], {n, 100}] (* Harvey P. Dale, Mar 02 2018 *)
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PROG
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(Haskell)
import Data.List (nub, genericTake)
a132213 n = sum $ map a010051' $
nub $ genericTake n $ map (`mod` n) $ tail a000290_list
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CROSSREFS
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Cf. A000224 (number of squares mod n).
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KEYWORD
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AUTHOR
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STATUS
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approved
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