A185321 - OEIS

%I #49 Aug 06 2024 10:29:02

%S 8911,1024651,1152271,5481451,10267951,14913991,64377991,67902031,

%T 139952671,178482151,368113411,395044651,612816751,652969351,

%U 743404663,1419339691,1588247851,2000436751,2199931651,2560600351,3102234751,3215031751,3411338491,4340265931

%N Carmichael numbers congruent to 3 modulo 4.

%C Most Carmichael numbers are congruent to 1 modulo 4.

%C 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

%C 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

%C 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

%H Donovan Johnson and Charles R Greathouse IV, <a href="/A185321/b185321.txt">Table of n, a(n) for n = 1..15447</a> (first 6838 terms from Johnson)

%H Charles R Greathouse IV, <a href="/A185321/a185321.gp.txt">GP script to compute terms</a>

%H Charles R Greathouse IV, <a href="/A185321/a185321_1.gp.txt">Alternate GP script to compute terms</a>

%H R. G. E. Pinch, <a href="https://citeseerx.ist.psu.edu/pdf/276fe2afb2d34bbc05a740eb1641b76f16cad625">The Carmichael numbers up to 10^15</a>, Mathematics of Computation 61:203 (1993), pp. 381-391.

%H Thomas Wright, <a href="http://arxiv.org/abs/1212.5850">Infinitely many Carmichael numbers in arithmetic progressions</a>, Bull. London Math. Soc. 45:5 (2013), pp. 943-952.

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

%o (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

%o 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

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

%K nonn

%O 1,1

%A _José María Grau Ribas_, Jan 27 2012

%E a(7)-a(40) from _Charles R Greathouse IV_, Jan 27 2012