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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 (list; graph; refs; listen; history; text; internal format)
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. 77-78.

Eric W. Weisstein, Near-Square 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)^2-2, (2#+3)^2-2, (2#+5)^2-2}, 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

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Last modified November 25 10:26 EST 2020. Contains 338623 sequences. (Running on oeis4.)