A294169 Semiprimes k = pq such that p^k == p (mod k) and q^k == q (mod k).

65, 133, 301, 793, 2041, 2413, 2501, 2701, 3781, 4699, 5617, 5963, 7081, 7991, 9073, 9881, 9937, 10261, 10349, 12209, 13213, 13333, 14111, 14981, 18721, 20737, 24727, 27133, 31201, 31621, 35431, 40321, 47197, 49141, 49591, 49601, 54913, 60701, 64079, 65869, 67721, 70801

Offset: 1

Author

Thomas Ordowski, Feb 10 2018

Keywords

• nonn

Comments

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

Problem: are there infinitely many such numbers?

All the terms are odd squarefree semiprimes.

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

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

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

Odd semiprimes pq pseudoprime to base q-p.

Examples

			65 = 5*13 is a term since 5^65 == 5 (mod 65) and 13^65 == 13 (mod 65).
Equivalently: 5^(5-1) == 1 (mod 13) and 13^(13-1) == 1 (mod 5).
Also (13-5)^(5*13-1) == 1 (mod 5*13) or (13-5)^(13-5) == 1 (mod 5*13).
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..2812

Crossrefs

Cf. A001358.

Subsequence of A046388.

Programs

Mathematica

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 *)

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

Extensions

More terms from Michel Marcus, Feb 10 2018

Edited by Thomas Ordowski, Mar 12 2019