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).

15, 33, 35, 65, 77, 87, 91, 95, 119, 123, 143, 185, 215, 221, 247, 255, 259, 287, 329, 341, 377, 395, 407, 427, 437, 455, 473, 485, 511, 515, 537, 573, 595, 635, 705, 713, 717, 721, 749, 767, 779, 793, 795, 803, 805, 815, 817, 843, 869, 871, 885, 899, 923, 965, 1001

Offset: 1

Author

Thomas Ordowski, Oct 07 2016

Keywords

• nonn

Comments

Numbers k such that p = A051953(k) < q = A277127(k), and p and q are both primes.

If k is such number, then b^p == b^q (mod k) for every integer b.

Problem: are there infinitely many such numbers?

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

All terms are odd composites. - Robert Israel, Oct 09 2016

It seems that gpf(k) < p = k - phi(k). - Thomas Ordowski, Oct 09 2016

Examples

			For n=15, A051953(15) = 7, A277127(15) = 11, 7 < 11 and both are primes, thus 15 is included in the sequence.
		

Links

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

Crossrefs

Subsequence of A033949 and of A024556.

Cf. A000010, A002322, A050530, A051953, A277127.

Programs

Maple

filter:= proc(n) uses numtheory;
  local p,q;
  p:= n-phi(n);
  q:= n-lambda(n);
  pRobert Israel, Oct 09 2016

Mathematica

Select[Range[10^3], And[#1 < #2, Times @@ Boole@ PrimeQ@ {#1, #2} == 1] & @@ {# - EulerPhi@ #, # - CarmichaelLambda@ #} &] (* Michael De Vlieger, Oct 08 2016 *)

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

Extensions

More terms from Altug Alkan, Oct 07 2016