

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).


1



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
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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:= nphi(n);
q:= nlambda(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=neulerphi(f), q=nlcm(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



