A294169 - OEIS

%I #38 Apr 10 2019 10:21:25

%S 65,133,301,793,2041,2413,2501,2701,3781,4699,5617,5963,7081,7991,

%T 9073,9881,9937,10261,10349,12209,13213,13333,14111,14981,18721,20737,

%U 24727,27133,31201,31621,35431,40321,47197,49141,49591,49601,54913,60701,64079,65869,67721,70801

%N Semiprimes k = pq such that p^k == p (mod k) and q^k == q (mod k).

%C The number k = pq is a weak pseudoprime to prime bases p and q.

%C Problem: are there infinitely many such numbers?

%C All the terms are odd squarefree semiprimes.

%C Semiprimes pq such that p^(p-1) == 1 (mod q) and q^(q-1) == 1 (mod p).

%C Odd semiprimes pq such that (q-p)^(q-p) == 1 (mod pq).

%C Semiprimes pq > 6 such that (q-p)^(q-p) == 1 (mod pq).

%C Odd semiprimes pq pseudoprime to base q-p.

%H Robert G. Wilson v, <a href="/A294169/b294169.txt">Table of n, a(n) for n = 1..2812</a>

%e 65 = 5*13 is a term since 5^65 == 5 (mod 65) and 13^65 == 13 (mod 65).

%e Equivalently: 5^(5-1) == 1 (mod 13) and 13^(13-1) == 1 (mod 5).

%e Also (13-5)^(5*13-1) == 1 (mod 5*13) or (13-5)^(13-5) == 1 (mod 5*13).

%t k = 4; lst = {}; NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k], sp}, sp = n + sgn; While[c < Abs[k], While[ PrimeOmega@ sp != 2, If[ sgn < 0, sp--, sp++]]; If[sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]]; fQ[k_] := Block[{fi = First@# & /@ FactorInteger@ k}, PowerMod[#, k, k] & /@ fi == fi]; While[k < 100000, If[ fQ@ k, AppendTo[lst, k]]; k = NextSemiPrime@ k] (* _Robert G. Wilson v_, Feb 10 2018 *)

%o (PARI) lista(nn) = {for (n=1, nn, if (bigomega(n) == 2, if (omega(n) == 2, p = factor(n)[1,1]; q = factor(n)[2,1];, p = factor(n)[1,1]; q = factor(n)[1,1];); mp = Mod(p, n); mq = Mod(q, n); if ((mp^n == mp) && (mq^n == mq), print1(n, ", "));););} \\ _Michel Marcus_, Feb 10 2018

%Y Cf. A001358.

%Y Subsequence of A046388.

%K nonn

%O 1,1

%A _Thomas Ordowski_, Feb 10 2018

%E More terms from _Michel Marcus_, Feb 10 2018

%E Edited by _Thomas Ordowski_, Mar 12 2019