A382120 Numbers k in A024619 such that there exists a prime p | k for which p^(m+1) == r (mod k), where r is also in A024619, and a prime q | k for which q^(m+1) == r (mod k), where r is a prime power.

10, 18, 20, 21, 22, 26, 28, 30, 34, 36, 38, 40, 42, 46, 48, 50, 52, 54, 55, 57, 58, 60, 68, 72, 74, 78, 82, 84, 86, 93, 94, 96, 98, 100, 106, 108, 110, 111, 114, 116, 117, 118, 122, 124, 126, 129, 132, 134, 136, 142, 146, 147, 148, 150, 156, 158, 162, 164, 165

Offset: 1

Author

Michael De Vlieger, Apr 06 2025

Keywords

• nonn

Comments

This sequence intersects neither A381750 nor A381864.

Examples

			Table of a(n) for select n, showing prime decomposition (facs(a(n))), p_x^(m+1) mod n, where x = 1 denotes the smallest prime factor, x = 2, the second smallest prime factor, etc. Brackets appear around residues that are not prime powers.
                          p_x^(m+1) mod n
 n  a(n)  facs(a(n))      p_1   p_2   p_3
-----------------------------------------
 1   10   2 * 5           [6]    5
 2   18   2 * 3^2        [14]    9
 3   20   2^2 * 5        [12]    5
 4   21   3 * 7           [6]    7
 5   22   2 * 11         [10]   11
 6   26   2 * 13          [6]   13
 7   28   2^2 * 7          4   [21]
 8   30   2 * 3 * 5        2   [21]    5
 9   34   2 * 17         [30]   17
10   36   2^2 * 3^2      [28]    9
11   38   2 * 19         [26]   19
22   60   2^2 * 3 * 5      4   [21]    5
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000961, A024619, A381750, A381864.

Programs

Mathematica

nn = 165, Reap[Do[If[! PrimePowerQ[n], If[CountDistinct@ Map[Boole@ PrimePowerQ@ PowerMod[#, 1 + Floor@ Log[#, n], n] &, FactorInteger[n][[All, 1]] ] == 2, Sow[n]]], {n, 2, nn}] ][[-1, 1]]