

A293620


Numbers n such that f(n), f(n+1) and f(n+2) are all primes, where f(k) = (2k+1)^2  2 (A073577).


0



1, 2, 16, 58, 149, 177, 534, 681, 954, 1045, 1052, 1255, 1367, 1563, 2046, 2074, 2515, 2557, 2564, 2788, 3586, 3593, 3908, 4062, 4552, 5252, 5371, 5385, 6400, 6729, 7443, 7478, 9305, 9375, 9942, 10355, 10411, 10726, 10740, 11286, 11545, 11559, 11832, 11965
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OFFSET

1,2


COMMENTS

Sierpiński proved that under Schinzel's hypothesis H this sequence is infinite.
Sierpiński showed that the only quadruple of consecutive primes of the form (2n+1)^2  2 are for n = 1 (i.e. 1 and 2 are the only consecutive terms in this sequence).
Numbers n such that the 3 consecutive integers n, n+1 and n+2 belong to A088572.  Michel Marcus, Oct 13 2017


LINKS

Table of n, a(n) for n=1..44.
Wacław Sierpiński, Remarque sur la distribution de nombres premiers, Matematički Vesnik, Vol. 2(17), Issue 31 (1965), pp. 7778.
Eric W. Weisstein, NearSquare Prime.
Wikipedia, Schinzel's hypothesis H.


EXAMPLE

The first triples are: n = 1: (7, 23, 47), n = 2: (23, 47, 79), n = 16: (1087, 1223, 1367).


MATHEMATICA

Select[Range[10^4], AllTrue[{(2#+1)^22, (2#+3)^22, (2#+5)^22}, PrimeQ] &]


PROG

(PARI) f(n) = 4*n^2 + 4*n  1;
isok(n) = isprime(f(n)) && isprime(f(n+1)) && isprime(f(n+2)); \\ Michel Marcus, Oct 13 2017


CROSSREFS

Cf. A088572, A008865, A028870, A028871, A073577, A088572.
Sequence in context: A107610 A091914 A123791 * A206980 A170991 A209219
Adjacent sequences: A293617 A293618 A293619 * A293621 A293622 A293623


KEYWORD

nonn


AUTHOR

Amiram Eldar, Oct 13 2017


STATUS

approved



