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A294169
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Semiprimes k = pq such that p^k == p (mod k) and q^k == q (mod k).
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2
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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).
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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