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A074379
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Carmichael numbers with exactly 4 prime factors.
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10
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41041, 62745, 63973, 75361, 101101, 126217, 172081, 188461, 278545, 340561, 449065, 552721, 656601, 658801, 670033, 748657, 838201, 852841, 997633, 1033669, 1082809, 1569457, 1773289, 2100901, 2113921, 2433601, 2455921
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OFFSET
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1,1
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COMMENTS
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Original name was: "Super-Carmichael numbers with exactly 4 factors", and a comment explained that the prefix "super" means that the Moebius function (A008683) equals mu(N) = +1 for these. But for squarefree numbers such as Carmichael numbers (A002997), this just means that they have an even number of prime factors, which is trivial if that number is 4.
In the literature there are other definitions of "super-Carmichael numbers", see the McIntosh and Meštrović references, so we prefer not to use this terminology at all.
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LINKS
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FORMULA
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EXAMPLE
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41041 = 7 * 11 * 13 * 41.
62745 = 3 * 5 * 47 * 89.
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MATHEMATICA
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p = Table[ Prime[i], {i, 1, 10}]; f[n_] := Union[ PowerMod[ Select[p, GCD[ #, n] == 1 & ], n - 1, n]]; Select[ Range[2500000], !PrimeQ[ # ] && OddQ[ # ] && Length[ FactorInteger[ # ]] == 4 && MoebiusMu[ # ] == 1 && f[ # ] == {1} & ]
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PROG
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(PARI) list(lim)=my(v=List()); forprime(p=3, sqrtnint(lim\=1, 4), forprime(q=p+2, sqrtnint(lim\p, 3), if(q%p==1, next); forprime(r=q+2, sqrtint(lim\p\q), if(r%p==1 || r%q==1, next); my(m=lcm([p-1, q-1, r-1]), pqr=p*q*r, t=Mod(1, m)/pqr, L=lim\pqr); fordiv(pqr-1, d, my(s=d+1); if(s>L, break); if(s==t && s>r && isprime(s), listput(v, pqr*s)))))); Set(v) \\ Charles R Greathouse IV, Apr 23 2022
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CROSSREFS
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Cf. A002997 (Carmichael numbers), A006931 (least Carmichael with n prime factors), A046386 (products of four distinct primes).
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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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