A381525 Squarefree semiprimes k = p*q, primes p < q, with q distinct power residues r (mod k) congruent to p^m, m >= 0, and p distinct power residues r (mod k) congruent to q^m, m >= 0.

6, 10, 15, 22, 26, 35, 38, 51, 58, 74, 85, 87, 106, 115, 118, 119, 122, 134, 143, 159, 166, 185, 187, 202, 214, 215, 235, 247, 262, 265, 267, 278, 298, 303, 319, 326, 339, 346, 358, 362, 365, 391, 394, 411, 415, 422, 427, 447, 451, 454, 481, 485, 515, 519, 527

Offset: 1

Author

Michael De Vlieger, Mar 24 2025

Keywords

• nonn
• easy

Comments

Define S(n,p) to be the set of distinct power residues r (mod n) beginning with 1 and thereafter multiplying by prime divisor p | n. For example, S(10,2) = {1, 2, 4, 8, 6}, S(10,5) = {1, 5}.

This sequence contains squarefree semiprimes p*q such that card(S(p*q),p) = q and card(S(p*q),q) = p.

Examples

			a(1) = 6 since S(6,2) = {1,2,4} and S(6,3) = {1,3}.
a(2) = 10 is a term since S(10,2) = {1,2,4,8,6} and S(10,5) = {1,5}.
14 = 2*7 is not a term since S(14,2) = {1,2,4,8}, with only 4 terms.
a(3) = 15 since S(15,3) = {1,3,9,12,6} and S(15,5) = {1,5,10}.
21 = 3*7 is not a term since S(21,7) = {1,7}, with only 2 terms.
a(4) = 22 = 2*11 since S(22,2) = {1,2,4,8,16,10,20,18,14,6,12} and S(22,11) = {1,11}.
34 = 2*17 is not a term since S(34,2) = {1,2,4,8,16,32,30,26,18}, with only 9 terms.
a(16) = 119 = 7*17 since card(S(119,7)) = 17 and card(S(119,17)) = 7, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Plot p*q at (x,y) = (pi(p),pi(q)) for terms whose prime factors do not exceed prime(48) = 223, labeling terms with p*q < 200, or pi(p) over pi(q) in italics otherwise. Light gray dots indicate squarefree semiprimes that are not in the sequence, principally for clarity.
• Michael De Vlieger, Plot p*q at (x,y) = (pi(p),pi(q)) for terms whose prime factors do not exceed prime(256) = 1619, 4X exaggeration.
• Michael De Vlieger, Plot p*q at (x,y) = (pi(p),pi(q)) for terms whose prime factors do not exceed prime(1024) = 8161. Demonstrates density of terms in this sequence among squarefree semiprimes.

Crossrefs

Cf. A007947, A006881, A381804.

Programs

Mathematica

nn = 530;
s = Join @@ Table[{p, q},
  {p, Prime@ Range[PrimePi@ Sqrt[nn]]},
  {q, Prime@ Range[PrimePi[p] + 1, PrimePi[nn/p]]}];
Union@ Reap[Do[
  n = Times @@ s[[i]];
  Set[{p, q}, s[[i]]];
    If[DuplicateFreeQ@ Map[PowerMod[p, #, n] &, Divisors[q - 1]],
      If[DuplicateFreeQ@ Map[PowerMod[q, #, n] &, Divisors[p - 1]],
      Sow[n] ] ], {i, Length[s]}] ][[-1, 1]]

Formula

A381804(a(n)) = 0.