cp's OEIS Frontend

A277254 Numbers k such that p = k - phi(k) < q = k - lambda(k), and p and q are both primes, where phi(k) = A000010(k) and lambda(k) = A002322(k).

%I A277254 #40 Jul 22 2025 21:13:40
%S A277254 15,33,35,65,77,87,91,95,119,123,143,185,215,221,247,255,259,287,329,
%T A277254 341,377,395,407,427,437,455,473,485,511,515,537,573,595,635,705,713,
%U A277254 717,721,749,767,779,793,795,803,805,815,817,843,869,871,885,899,923,965,1001
%N A277254 Numbers k such that p = k - phi(k) < q = k - lambda(k), and p and q are both primes, where phi(k) = A000010(k) and lambda(k) = A002322(k).
%C A277254 Numbers k such that p = A051953(k) < q = A277127(k), and p and q are both primes.
%C A277254 If k is such number, then b^p == b^q (mod k) for every integer b.
%C A277254 Problem: are there infinitely many such numbers?
%C A277254 Suppose p^2 divides k. Then p divides k - phi(k), and so the only way k - phi(k) can be prime is if k = p^2. But then k - phi(k) = k - A002322(k). Hence all terms in this sequence are squarefree. - _Charles R Greathouse IV_, Oct 08 2016
%C A277254 All terms are odd composites. - _Robert Israel_, Oct 09 2016
%C A277254 It seems that gpf(k) < p = k - phi(k). - _Thomas Ordowski_, Oct 09 2016
%H A277254 Robert Israel, <a href="/A277254/b277254.txt">Table of n, a(n) for n = 1..10000</a>
%e A277254 For n=15, A051953(15) = 7, A277127(15) = 11, 7 < 11 and both are primes, thus 15 is included in the sequence.
%p A277254 filter:= proc(n) uses numtheory;
%p A277254   local p,q;
%p A277254   p:= n-phi(n);
%p A277254   q:= n-lambda(n);
%p A277254   p<q and isprime(p) and isprime(q);
%p A277254 end proc:
%p A277254 select(filter, [seq(i,i=3..10000,2)]); # _Robert Israel_, Oct 09 2016
%t A277254 Select[Range[10^3], And[#1 < #2, Times @@ Boole@ PrimeQ@ {#1, #2} == 1] & @@ {# - EulerPhi@ #, # - CarmichaelLambda@ #} &] (* _Michael De Vlieger_, Oct 08 2016 *)
%o A277254 (PARI) is(n)=my(f=factor(n),p=n-eulerphi(f),q=n-lcm(znstar(f)[2])); p < q && isprime(p) && isprime(q) \\ _Charles R Greathouse IV_, Oct 08 2016
%Y A277254 Subsequence of A033949 and of A024556.
%Y A277254 Cf. A000010, A002322, A050530, A051953, A277127.
%K A277254 nonn
%O A277254 1,1
%A A277254 _Thomas Ordowski_, Oct 07 2016
%E A277254 More terms from _Altug Alkan_, Oct 07 2016