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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). 3
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, 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, 1, 12, 22, 1, 17, 18, 1, 15, 21, 1, 9, 28, 1, 1, 19, 20, 1, 13 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,6

COMMENTS

The n-th row has length A034444(n) - 1.

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').

For odd n and any integer m, m^2 == m (mod n) iff m^2 == m (mod 2n).

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

LINKS

Eric M. Schmidt, Rows 2..2000, flattened

EXAMPLE

Triangle begins:

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; etc.  - Bruno Berselli, Aug 06 2012

MATHEMATICA

Table[Select[Range[n], Mod[#^2, n] == # &], {n, 2, 30}] // Grid (* Geoffrey Critzer, May 26 2015 *)

PROG

(Sage) def A215202(n) : return [m for m in range(1, n) if m^2 % n == m];

(MAGMA) [m: m in [1..n-1], n in [2..40] | m^2 mod n eq m]; // Bruno Berselli, Aug 06 2012

CROSSREFS

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.

Sequence in context: A176979 A299989 A058022 * A139344 A137925 A171528

Adjacent sequences:  A215199 A215200 A215201 * A215203 A215204 A215205

KEYWORD

nonn,tabf

AUTHOR

Eric M. Schmidt, Aug 05 2012

STATUS

approved

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Last modified July 12 16:03 EDT 2020. Contains 335665 sequences. (Running on oeis4.)