cp's OEIS Frontend

A192224 P-integers: n such that the first phi(n) primes coprime to n form a reduced residue system modulo n, where phi is Euler's totient function A000010.

%I A192224 #31 Jan 22 2020 20:09:56
%S A192224 2,4,6,12,18,30
%N A192224 P-integers: n such that the first phi(n) primes coprime to n form a reduced residue system modulo n, where phi is Euler's totient function A000010.
%C A192224 Pomerance proved that the sequence is finite and conjectured that 30 is the largest element. Hajdu and Saradha proved Recamán's conjecture that 2 is the only prime P-integer. Both proofs use Jacobsthal's function A048669.
%C A192224 Hajdu, Saradha, and Tijdeman have a conditional proof of Pomerance's conjecture, assuming the Riemann Hypothesis.
%C A192224 Shichun Yanga and Alain Togbéb have proved Pomerance's conjecture. - _Jonathan Sondow_, Jun 14 2014
%D A192224 B. M. Recamán, Problem 672, J. Recreational Math. 10 (1978), 283.
%H A192224 L. Hajdu, <a href="http://atlas-conferences.com/cgi-bin/abstract/cbbv-77">On a conjecture of Pomerance and the Jacobsthal function</a>, 27th Journées Arithmétiques
%H A192224 L. Hajdu and N. Saradha, <a href="http://www.math.klte.hu/~hajdul/recamanjntrev2.pdf">On a problem of Recaman and its generalization</a>
%H A192224 L. Hajdu, N. Saradha, and R. Tijdeman, <a href="http://arxiv.org/abs/1107.5191">On a conjecture of Pomerance</a>, arXiv:1107.5191 [math.NT], 2011.
%H A192224 C. Pomerance, <a href="http://dx.doi.org/10.1016/0022-314X(80)90056-6">A note on the least prime in an arithmetic progression</a>, J. Number Theory 12 (1980), 218-223.
%H A192224 Shichun Yanga and Alain Togbéb, <a href="https://doi.org/10.1016/j.jnt.2014.01.014">Proof of the P-integer conjecture of Pomerance</a>, J. Number Theory, 140 (2014), 226-234. DOI: 10.1016/j.jnt.2014.01.014.
%e A192224 12 is a P-integer because phi(12) = 4 and the first four primes coprime to 12 are 5, 7, 11, 13, which are pairwise incongruent modulo 12.
%e A192224 8 is not a P-integer because phi(8) = 4 and the first four primes coprime to 8 are 3, 5, 7, 11, but 3 == 11 (mod 8).
%Y A192224 Cf. A000010 (Euler totient function phi), A048669 (Jacobsthal function).
%K A192224 nonn,fini,full
%O A192224 1,1
%A A192224 _Jonathan Sondow_, Jun 29 2011