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

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

1,1

Comments

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

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

Problem: are there infinitely many such numbers?

Suppose p^2 divides n. Then p divides n - phi(n), and so the only way n - phi(n) can be prime is if n = p^2. But then n - phi(n) = n - A002322(n). 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(n) < p = n - phi(n). - Thomas Ordowski, Oct 09 2016

Links

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

Example

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

Maple

filter:= proc(n) uses numtheory;
  local p, q;
  p:= n-phi(n);
  q:= n-lambda(n);
  p<q and isprime(p) and isprime(q);
end proc:
select(filter, [seq(i, i=3..10000, 2)]); # Robert 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 *)

Prog

(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

Crossrefs

Subsequence of A033949 and of A024556.

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

Sequence in context: A068081 A089967 A064900 * A157137 A141230 A140608

Adjacent sequences: A277251 A277252 A277253 * A277255 A277256 A277257

Keyword

nonn

Author

Thomas Ordowski, Oct 07 2016

Extensions

More terms from Altug Alkan, Oct 07 2016

Status

approved