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A215202 Irregular triangle in which n-th row gives m in 1, ..., n-1 such that m^2 == m (mod n).

%I A215202 #32 Sep 08 2022 08:46:03
%S A215202 1,1,1,1,1,3,4,1,1,1,1,5,6,1,1,4,9,1,1,7,8,1,6,10,1,1,1,9,10,1,1,5,16,
%T A215202 1,7,15,1,11,12,1,1,9,16,1,1,13,14,1,1,8,21,1,1,6,10,15,16,21,25,1,1,
%U A215202 1,12,22,1,17,18,1,15,21,1,9,28,1,1,19,20,1,13
%N A215202 Irregular triangle in which n-th row gives m in 1, ..., n-1 such that m^2 == m (mod n).
%C A215202 The n-th row has length A034444(n) - 1.
%C A215202 If m appears in row n, then gcd(n,m) appears in the n-th row of A077610. Moreover, if m', distinct from m, also appears in row n, then gcd(n, m) does not equal gcd(n, m').
%C A215202 For odd n and any integer m, m^2 == m (mod n) iff m^2 == m (mod 2n).
%C A215202 Let P(1)={1} and for integers x > 1, let P(x) be the set of distinct prime divisors of x. We can define an equivalence relation ~ on the set of elements in the ring (Z_n, +mod n,*mod n): for all a,b in Z_n (where a,b are the least nonnegative residues modulo n) a ~ b iff P(gcd(a,n)) intersect P(n) is equal to P(gcd(b,n)) intersect P(n). If we include 0 in each row then these elements can represent the equivalence classes. They form a commutative monoid. - _Geoffrey Critzer_, Feb 13 2016
%H A215202 Eric M. Schmidt, <a href="/A215202/b215202.txt">Rows 2..2000, flattened</a>
%e A215202 Triangle begins:
%e A215202 1;
%e A215202 1;
%e A215202 1;
%e A215202 1;
%e A215202 1, 3, 4;
%e A215202 1;
%e A215202 1;
%e A215202 1;
%e A215202 1, 5, 6;
%e A215202 1;
%e A215202 1, 4, 9;
%e A215202 1;
%e A215202 1, 7, 8;
%e A215202 1, 6, 10;
%e A215202 1;
%e A215202 1;
%e A215202 1, 9, 10; etc.  - _Bruno Berselli_, Aug 06 2012
%t A215202 Table[Select[Range[n], Mod[#^2, n] == # &], {n, 2, 30}] // Grid (* _Geoffrey Critzer_, May 26 2015 *)
%o A215202 (Sage) def A215202(n) : return [m for m in range(1, n) if m^2 % n == m];
%o A215202 (Magma) [m: m in [1..n-1], n in [2..40] | m^2 mod n eq m]; // _Bruno Berselli_, Aug 06 2012
%Y A215202 For m^2 == m (mod n), see: n=2: A001477; n=3: A032766; n=4: A042948; n=5: A008851; n=6: A032766; n=7: A047274; n=8: A047393; n=9: A090570; n=10: A008851; n=11: A112651; n=12: A112652; n=13: A112653; n=14: A047274; n=15: A151972; n=16: A151977; n=17: A151978; n=18: A090570; n=19: A151979; n=20: A151980; n=21: A151971; n=22: A112651; n=24: A151973; n=26: A112653; n=30: A151972; n=32: A151983; n=34: A151978; n=38: A151979; n=42: A151971; n=48: A151981; n=64: A151984; n=100: A008852; n=1000: A008853.
%K A215202 nonn,tabf
%O A215202 2,6
%A A215202 _Eric M. Schmidt_, Aug 05 2012