A280262 Numbers n such that A187730(n) < A049559(n).

21, 33, 57, 65, 69, 77, 91, 93, 105, 129, 133, 141, 145, 161, 177, 185, 189, 201, 209, 213, 217, 225, 237, 249, 253, 265, 273, 297, 301, 305, 309, 321, 329, 341, 345, 369, 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

Offset: 1

Author

Thomas Ordowski and Robert Israel, Dec 30 2016

Keywords

• nonn

Comments

Terms are not of the form p^k, where p is a prime.

There are no terms of the form 2p+1, where p is a prime.

The sequence contains all Carmichael numbers except A264012.

If n is in the sequence, then n-1 is not squarefree. - Thomas Ordowski, Jan 02 2017

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

Number of terms < 10^k: 0, 8, 112, 1258, 13069, 132262, 1324263, 13229372, 132009236, ..., . Robert G. Wilson v, Jan 04 2017

If p and q are distinct primes == 3 (mod 4), then p*q is in the sequence. - Thomas Ordowski, Mar 30 2017

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A033949.

Cf. A000010, A002322, A049559, A187730.

Programs

Maple

select(t -> igcd(numtheory:-lambda(t),t-1) < igcd(numtheory:-phi(t),t-1), [$1..1000]);

Mathematica

Select[Range@ 600, GCD[CarmichaelLambda@ #, # - 1] < GCD[# - 1, EulerPhi@ #] &] (* Michael De Vlieger, Dec 31 2016 *)