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Numbers n such that x - lambda(x) = n has no solution, where lambda(x) = A002322(x).
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%I #51 Apr 27 2017 23:18:00

%S 21,28,45,46,51,64,65,77,82,85,91,106,111,126,129,133,136,148,155,166,

%T 172,175,185,189,205,208,209,217,221,225,231,232,235,237,244,247,267,

%U 273,274,276,286,291,298,305,316,319,326,333,339,341,358,362,364,365,371

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

%C Problem: are there infinitely many such numbers?

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

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

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

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

%o (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

%o (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

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

%K nonn

%O 1,1

%A _Thomas Ordowski_, Oct 03 2016

%E More terms from _Michel Marcus_, Oct 03 2016