

A111173


Sophie Germain triprimes: k and 2k + 1 are both the product of 3 primes, not necessarily distinct.


9



52, 76, 130, 171, 172, 212, 238, 318, 322, 325, 332, 357, 370, 387, 388, 402, 423, 430, 436, 442, 465, 507, 508, 556, 604, 610, 654, 665, 670, 710, 722, 747, 759, 762, 772, 775, 786, 790, 805, 814, 822, 826, 847, 874, 885, 902, 906, 916, 927, 942, 987, 1004
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OFFSET

1,1


COMMENTS

There should also be triprime chains of length j analogous to Cunningham chains of the first kind and Tomaszewski chains of the first kind. A triprime chain of length j is a sequence of triprimes a(1) < a(2) < ... < a(j) such that a(i+1) = 2*a(i) + 1 for i = 1, ..., j1. The first of these are: Length 3: 332, 665, 1331 = 11^3; 387, 775, 1551 = 3 * 11 * 47.


LINKS



FORMULA

{a(n)} = a(n) is an element of A014612 and 2*a(n)+1 is an element of A014612.


EXAMPLE

n k = a(n) 2k + 1
= ================ ================
1 52 = 2^2 * 13 105 = 3 * 5 * 7
2 76 = 2^2 * 19 153 = 3^2 * 17
3 130 = 2 * 5 * 13 261 = 3^2 * 29
4 171 = 3^2 * 19 343 = 7^3
5 172 = 2^2 * 43 345 = 3 * 5 * 23
6 212 = 2^2 * 53 425 = 5^2 * 17


MATHEMATICA

fQ[n_]:=PrimeOmega[n] == 3 == PrimeOmega[2 n + 1]; Select[Range@1100, fQ] (* Vincenzo Librandi, Aug 19 2018 *)


PROG

(Magma) Is3primes:=func<i&+[d[2]: d in Factorization(i)] eq 3>; [n: n in [2..1200]  Is3primes(n) and Is3primes(2*n+1)]; // Vincenzo Librandi, Aug 19 2018


CROSSREFS



KEYWORD

easy,nonn


AUTHOR



EXTENSIONS



STATUS

approved



