A185321 Carmichael numbers congruent to 3 modulo 4.

8911, 1024651, 1152271, 5481451, 10267951, 14913991, 64377991, 67902031, 139952671, 178482151, 368113411, 395044651, 612816751, 652969351, 743404663, 1419339691, 1588247851, 2000436751, 2199931651, 2560600351, 3102234751, 3215031751, 3411338491, 4340265931

Offset: 1

Author

José María Grau Ribas, Jan 27 2012

Keywords

• nonn

Comments

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

Links

• Donovan Johnson and Charles R Greathouse IV, Table of n, a(n) for n = 1..15447 (first 6838 terms from Johnson)
• Charles R Greathouse IV, GP script to compute terms
• Charles R Greathouse IV, Alternate GP script to compute terms
• R. G. E. Pinch, The Carmichael numbers up to 10^15, Mathematics of Computation 61:203 (1993), pp. 381-391.
• Thomas Wright, Infinitely many Carmichael numbers in arithmetic progressions, Bull. London Math. Soc. 45:5 (2013), pp. 943-952.

Crossrefs

Subsequence of A002997, A167181 (and hence A004614), A026424, and A177884.

Programs

Mathematica

Select[4Range[10^4] + 3, (!PrimeQ[#] && IntegerQ[(# - 1)/CarmichaelLambda[#]]) &]

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

Extensions

a(7)-a(40) from Charles R Greathouse IV, Jan 27 2012