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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. 1
2, 4, 6, 12, 18, 30 (list; graph; refs; listen; history; text; internal format)



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


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


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

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


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).


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

Sequence in context: A215821 A192096 A181740 * A167777 A259941 A007436

Adjacent sequences:  A192221 A192222 A192223 * A192225 A192226 A192227




Jonathan Sondow, Jun 29 2011



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