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

1,1

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.

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A206235 A329799 A329468 * A267462 A256237 A065235

Adjacent sequences: A185318 A185319 A185320 * A185322 A185323 A185324

Keyword

nonn

Author

José María Grau Ribas, Jan 27 2012

Extensions

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

Status

approved