A276674 Numbers n such that x - lambda(x) = n has no solution, where lambda(x) = A002322(x).

21, 28, 45, 46, 51, 64, 65, 77, 82, 85, 91, 106, 111, 126, 129, 133, 136, 148, 155, 166, 172, 175, 185, 189, 205, 208, 209, 217, 221, 225, 231, 232, 235, 237, 244, 247, 267, 273, 274, 276, 286, 291, 298, 305, 316, 319, 326, 333, 339, 341, 358, 362, 364, 365, 371

Offset: 1

Author

Thomas Ordowski, Oct 03 2016

Keywords

• nonn

Comments

Problem: are there infinitely many such numbers?

Note that all these numbers are composite, because p - lambda(p) = 1 and p^2 - lambda(p^2) = p prime.

If x - lambda(x) = n > 1, then x <= n^2.

Conjecture: if x - lambda(x) = 2*m > 0, then x <= 4*m.

Noncototients among these numbers are 172, 232, 244, 274, 298, 326, 362, ...

Crossrefs

Cf. A002322, A005278 (see links). Complement of A277127.

Programs

PARI

lista(nn) = {v = vecsort(vector(nn^2, n, n - lcm(znstar(n)[2])), ,8); for (n=1, nn, if (! vecsearch(v, n), print1(n, ", ")););} \\ Michel Marcus, Oct 03 2016

Perl

use ntheory ":all"; sub A { my $l=shift; my %C; undef $C{$-carmichael_lambda($)} for 1..$l*$l; my @R = grep { !exists $C{$} } 1..$l; @R; } say for A(500); # _Dana Jacobsen, Apr 27 2017

Extensions

More terms from Michel Marcus, Oct 03 2016