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A185321
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Carmichael numbers congruent to 3 modulo 4.
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5
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8911, 1024651, 1152271, 5481451, 10267951, 14913991, 64377991, 67902031, 139952671, 178482151, 368113411, 395044651, 612816751, 652969351, 743404663, 1419339691, 1588247851, 2000436751, 2199931651, 2560600351, 3102234751, 3215031751, 3411338491, 4340265931
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OFFSET
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1,1
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COMMENTS
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Most Carmichael numbers are congruent to 1 modulo 4.
This is a subsequence of A167181: if a prime p | a(n), (p-1) | (a(n)-1) by Korselt's criterion. But a(n)-1 is 2 mod 4, so p-1 cannot be 0 mod 4. Hence all primes dividing a(n) are 3 mod 4. - Charles R Greathouse IV, Jan 27 2012
Pinch call the intersection of A007304 with this sequence C3, which are precisely those numbers which pass a Rabin-Miller test to a random base with probability 1/4. The first member of this sequence not in C3 is a(16) = 7 * 11 * 19 * 103 * 9419. - Charles R Greathouse IV, Jan 27 2012
Wright proves that this sequence is infinite, and in particular there are more than x^(k/(log log log x)^2) terms up to x for some k and large enough x. - Charles R Greathouse IV, Nov 09 2015
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LINKS
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MATHEMATICA
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Select[4Range[10^4] + 3, (!PrimeQ[#] && IntegerQ[(# - 1)/CarmichaelLambda[#]]) &]
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PROG
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(PARI) Korselt(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]>1||(n-1)%(f[i, 1]-1), return(0))); 1
p=5; forprime(q=7, 1e7, forstep(n=if(p%4==3, p+4, p+2), q-2, 4, if(Korselt(n), print1(n", "))); p=q) \\ Charles R Greathouse IV, Jan 27 2012
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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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