A106628 Anomalous prime numbers

[Davide Rotondo]

A106628

Anomalous prime numbers.

1

199, 211, 283, 317, 337, 389, 491, 509, 547, 577, 619, 683, 701, 773, 787, 797, 863, 887, 1069, 1109, 1129, 1153, 1163, 1373, 1381, 1409, 1459, 1523, 1531, 1571, 1627, 1637, 1669, 1709, 1723, 1733, 1759, 1831, 1889, 1913, 1933, 1951, 1979, 2003, 2017

Question: are these primes infinite?

See you soon 

Davide

[Geoffrey Caveney]

Based on their frequency in the list of the first 10,000 of them, I would make the even stronger conjecture that the sum of their reciprocals diverges. (For reference, the sum of the reciprocals of the prime numbers diverges; the sum of the reciprocals of the twin primes does not.) See Greathouse's comment conjecturing that the anomalous primes' asymptotic density is the same as that of all primes (as stated in the prime number theorem). One may even rather ask whether the non-anomalous primes are infinite and whether the sum of their reciprocals diverges. Perhaps the behavior of the primes < 199 gives a misleading sense of the actual rarity of the anomalous primes, which are not so rare among larger primes. It makes sense that as the average prime gaps increase among larger numbers, it will become more and more common for neither q-1 nor q+1 to be divisible by the prime gap q-p. When this prime gap is <= 6, the anomalous prime property is impossible, but as the prime gaps grow, the property is possible and increasingly likely.

[Tomasz Ordowski]

Hi Davide! 

Yes, there are probably infinitely many such primes. Without proof. See: 

Conjecture: a(n) ~ n log n. - Charles R Greathouse IV, Aug 20 2017

If so, then they may be dense in the set of all primes. 

Best, 

Tom Ordo 

_____________

Subsequence of A083371.

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