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a(n) is the number of Carmichael numbers of the form prime(n)*prime(n')*prime(n") with n < n' < n".
3

%I #14 Sep 25 2019 10:28:59

%S 0,1,3,6,0,5,2,2,1,2,7,5,7,11,3,3,1,10,3,7,4,1,2,5,6,2,5,3,10,5,5,11,

%T 4,6,2,9,11,7,2,3,4,11,6,10,0,7,17,5,4,6,1,5,10,7,5,4,4,14,8,9,2,5,12,

%U 9,16,2,16,15,2,6,5,2,9,8,8,3,1,7,13,7,3,13,5,14,6,8,4,9,6,4,1,1,9,7,3,1

%N a(n) is the number of Carmichael numbers of the form prime(n)*prime(n')*prime(n") with n < n' < n".

%C It is known that there is a finite number of Carmichael numbers with k prime factors if k-2 of the factors are fixed. Here we consider the case k=3 and impose the additional condition that prime(n) be the smallest of the 3 factors.

%C The primes related to the zeros in this sequence are in A051663. - _Jack Brennen_, Jul 01 2008

%H Amiram Eldar, <a href="/A141703/b141703.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..5000 from M. F. Hasler)

%H <a href="/index/Ca#Carmichael">Index entries related to Carmichael numbers</a>.

%F a(n) = # { pqr | p=prime(n) < q=prime(n') < r=prime(n") ; p-1 | pqr-1 ; q-1 | pqr-1 ; r-1 | pqr-1 }

%e a(1)=0 since prime(1)=2 and there is no even Carmichael number.

%e a(2)=1 since prime(2)=3 and 561 is the only Carmichael number of the form 3pq with p,q prime.

%e a(3)=3 since prime(3)=5 and the only Carmichael numbers of the form 5pq are {1105, 2465, 10585}.

%o (PARI) A141703(n,verbose=0) = { /* based on code by J.Brennen (jb AT brennen.net) */ local( V=[], B, p=prime(n), q, r); for( A=1, p-1, B=ceil((p^2+1)/A); while( 1, r=(p*B-p+A*B-B)/(A*B-p*p); q=(A*r-A+1)/p; q<=p && break; denominator(q)==1 && denominator(r)==1 && r>q && isprime(q) && isprime(r) && (p*q*r)%(p-1)==1 && V=concat(V,[p*q*r]); B++ )); verbose && print1(V); #V }

%Y Cf. A002997 and references therein ; A087788 ; A141702 ff.

%K nonn

%O 1,3

%A _M. F. Hasler_, Jul 01 2008