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.

2, 4, 6, 12, 18, 30

Offset

1,1

Comments

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.

Hajdu, Saradha, and Tijdeman have a conditional proof of Pomerance's conjecture, assuming the Riemann Hypothesis.

Shichun Yanga and Alain Togbéb have proved Pomerance's conjecture. - Jonathan Sondow, Jun 14 2014

References

B. M. Recamán, Problem 672, J. Recreational Math. 10 (1978), 283.

Links

Table of n, a(n) for n=1..6.

L. Hajdu, On a conjecture of Pomerance and the Jacobsthal function, 27th Journées Arithmétiques

L. Hajdu and N. Saradha, On a problem of Recaman and its generalization

L. Hajdu, N. Saradha, and R. Tijdeman, On a conjecture of Pomerance, arXiv:1107.5191 [math.NT], 2011.

C. Pomerance, A note on the least prime in an arithmetic progression, J. Number Theory 12 (1980), 218-223.

Shichun Yanga and Alain Togbéb, Proof of the P-integer conjecture of Pomerance, J. Number Theory, 140 (2014), 226-234. DOI: 10.1016/j.jnt.2014.01.014.

Example

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.
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).

Crossrefs

Cf. A000010 (Euler totient function phi), A048669 (Jacobsthal function).

Sequence in context: A332284 A192096 A181740 * A167777 A259941 A007436

Adjacent sequences: A192221 A192222 A192223 * A192225 A192226 A192227

Keyword

nonn,fini,full

Author

Jonathan Sondow, Jun 29 2011

Status

approved