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%I #47 Mar 30 2017 08:21:52
%S 21,33,57,65,69,77,91,93,105,129,133,141,145,161,177,185,189,201,209,
%T 213,217,225,237,249,253,265,273,297,301,305,309,321,329,341,345,369,
%U 377,381,385,393,413,417,437,441,451,453,465,469,473,481,489,497,501,505,513,517,537,545,553,559,573
%N Numbers n such that A187730(n) < A049559(n).
%C Terms are not of the form p^k, where p is a prime.
%C There are no terms of the form 2p+1, where p is a prime.
%C The sequence contains all Carmichael numbers except A264012.
%C If n is in the sequence, then n-1 is not squarefree. - _Thomas Ordowski_, Jan 02 2017
%C Theorem: the set of such numbers has natural density 0. Proof: Let y = y(n) = loglog n /logloglog n. Using part 1 of Lemma 2.1 in paper 199 on my home page (joint with Luca), applied to the residue class 1: But for a set of n of density 0, for each integer d < y, there is a prime p|n with p == 1 (mod d). In particular, lambda(n) is divisible by every integer d up to y. Suppose now that gcd(lambda(n),n-1) < gcd(phi(n),n-1). Then there is a prime power q^a such that q^a | phi(n), q^a | n-1, and q^a does not divide lambda(n). Then, but for a set of n of density 0, we would have q^a > y. Since q | lambda(n), we have a at least 2. So, n-1 is divisible by some q^a > y with a >= 2. The set of such n has density 0. QED. - Carl Pomerance, Jan 02 2017
%C Number of terms < 10^k: 0, 8, 112, 1258, 13069, 132262, 1324263, 13229372, 132009236, ..., . _Robert G. Wilson v_, Jan 04 2017
%C If p and q are distinct primes == 3 (mod 4), then p*q is in the sequence. - _Thomas Ordowski_, Mar 30 2017
%H Robert Israel, <a href="/A280262/b280262.txt">Table of n, a(n) for n = 1..10000</a>
%p select(t -> igcd(numtheory:-lambda(t),t-1) < igcd(numtheory:-phi(t),t-1), [$1..1000]);
%t Select[Range@ 600, GCD[CarmichaelLambda@ #, # - 1] < GCD[# - 1, EulerPhi@ #] &] (* _Michael De Vlieger_, Dec 31 2016 *)
%Y Subsequence of A033949.
%Y Cf. A000010, A002322, A049559, A187730.
%K nonn
%O 1,1
%A _Thomas Ordowski_ and _Robert Israel_, Dec 30 2016
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