A132213 Number of distinct primes among the squares mod n.

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

Offset: 1

Author

T. D. Noe, Aug 13 2007, Aug 17 2007

Keywords

• nice
• nonn
• look

Comments

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.

From Rémy Sigrist, Nov 28 2017: (Start)

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)

Examples

			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.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• T. D. Noe, Logarithmic plot of 10^6 terms
• Rémy Sigrist, Colored logarithmic plot of 2*10^6 terms

Crossrefs

Cf. A000224 (number of squares mod n).

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

Programs

Haskell

import Data.List (nub, genericTake)
a132213 n = sum $ map a010051' $
            nub $ genericTake n $ map (`mod` n) $ tail a000290_list
-- Reinhard Zumkeller, Jun 23 2015, Oct 15 2011

Mathematica

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 *)