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A132213 Number of distinct primes among the squares mod n. 3

%I #21 Mar 02 2018 17:16:50

%S 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,

%T 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,

%U 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

%N Number of distinct primes among the squares mod n.

%C 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?

%C The logarithmic plot of the first 10^6 terms shows seven components.

%C From _Rémy Sigrist_, Nov 28 2017: (Start)

%C Empirically, in the logarithmic plot of the sequence:

%C - the set of indices of the first component (starting from the top), say S_1, is the union of A061345 and of A278568,

%C - 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)),

%C - see logarithmic scatterplot colored according to this scheme in Links section.

%C (End)

%H T. D. Noe, <a href="/A132213/b132213.txt">Table of n, a(n) for n = 1..10000</a>

%H T. D. Noe, <a href="http://www.sspectra.com/math/A132213.gif">Logarithmic plot of 10^6 terms</a>

%H Rémy Sigrist, <a href="/A132213/a132213.png">Colored logarithmic plot of 2*10^6 terms</a>

%e 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.

%t Table[s=Union[Mod[Range[n]^2,n]]; Length[Select[s,PrimeQ]], {n,10000}]

%t Table[Count[Union[PowerMod[Range[n],2,n]],_?PrimeQ],{n,100}] (* _Harvey P. Dale_, Mar 02 2018 *)

%o (Haskell)

%o import Data.List (nub, genericTake)

%o a132213 n = sum $ map a010051' $

%o nub $ genericTake n $ map (`mod` n) $ tail a000290_list

%o -- _Reinhard Zumkeller_, Jun 23 2015, Oct 15 2011

%Y Cf. A000224 (number of squares mod n).

%Y Cf. A000290, A001221, A007814, A010051, A061345, A278568.

%K nice,nonn,look

%O 1,11

%A _T. D. Noe_, Aug 13 2007, Aug 17 2007

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Last modified March 30 01:41 EDT 2024. Contains 371282 sequences. (Running on oeis4.)