cp's OEIS Frontend

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.

%I A381525 #33 Mar 29 2025 18:17:28
%S A381525 6,10,15,22,26,35,38,51,58,74,85,87,106,115,118,119,122,134,143,159,
%T A381525 166,185,187,202,214,215,235,247,262,265,267,278,298,303,319,326,339,
%U A381525 346,358,362,365,391,394,411,415,422,427,447,451,454,481,485,515,519,527
%N 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.
%C A381525 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}.
%C A381525 This sequence contains squarefree semiprimes p*q such that card(S(p*q),p) = q and card(S(p*q),q) = p.
%H A381525 Michael De Vlieger, <a href="/A381525/b381525.txt">Table of n, a(n) for n = 1..10000</a>
%H A381525 Michael De Vlieger, <a href="/A381525/a381525.png">Plot p*q at (x,y) = (pi(p),pi(q))</a> 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.
%H A381525 Michael De Vlieger, <a href="/A381525/a381525_1.png">Plot p*q at (x,y) = (pi(p),pi(q))</a> for terms whose prime factors do not exceed prime(256) = 1619, 4X exaggeration.
%H A381525 Michael De Vlieger, <a href="/A381525/a381525_2.png">Plot p*q at (x,y) = (pi(p),pi(q))</a> for terms whose prime factors do not exceed prime(1024) = 8161. Demonstrates density of terms in this sequence among squarefree semiprimes.
%F A381525 A381804(a(n)) = 0.
%e A381525 a(1) = 6 since S(6,2) = {1,2,4} and S(6,3) = {1,3}.
%e A381525 a(2) = 10 is a term since S(10,2) = {1,2,4,8,6} and S(10,5) = {1,5}.
%e A381525 14 = 2*7 is not a term since S(14,2) = {1,2,4,8}, with only 4 terms.
%e A381525 a(3) = 15 since S(15,3) = {1,3,9,12,6} and S(15,5) = {1,5,10}.
%e A381525 21 = 3*7 is not a term since S(21,7) = {1,7}, with only 2 terms.
%e A381525 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}.
%e A381525 34 = 2*17 is not a term since S(34,2) = {1,2,4,8,16,32,30,26,18}, with only 9 terms.
%e A381525 a(16) = 119 = 7*17 since card(S(119,7)) = 17 and card(S(119,17)) = 7, etc.
%t A381525 nn = 530;
%t A381525 s = Join @@ Table[{p, q},
%t A381525   {p, Prime@ Range[PrimePi@ Sqrt[nn]]},
%t A381525   {q, Prime@ Range[PrimePi[p] + 1, PrimePi[nn/p]]}];
%t A381525 Union@ Reap[Do[
%t A381525   n = Times @@ s[[i]];
%t A381525   Set[{p, q}, s[[i]]];
%t A381525     If[DuplicateFreeQ@ Map[PowerMod[p, #, n] &, Divisors[q - 1]],
%t A381525       If[DuplicateFreeQ@ Map[PowerMod[q, #, n] &, Divisors[p - 1]],
%t A381525       Sow[n] ] ], {i, Length[s]}] ][[-1, 1]]
%Y A381525 Cf. A007947, A006881, A381804.
%K A381525 nonn,easy
%O A381525 1,1
%A A381525 _Michael De Vlieger_, Mar 24 2025