

A123017


Numbers n such that n and n+3 are both semiprimes.


3



6, 22, 35, 46, 55, 62, 74, 82, 91, 115, 118, 119, 142, 143, 155, 158, 166, 202, 203, 206, 214, 215, 218, 259, 262, 295, 298, 299, 302, 323, 326, 355, 358, 362, 391, 395, 451, 466, 478, 482, 502, 511, 514, 526, 535, 542, 551, 559, 562, 583, 586, 611, 623, 626
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OFFSET

1,1


COMMENTS

When a(n+1) = a(n) + 3 we have that a(n) is a semiprime such that a(n) and a(n)+3 and a(n) + 3 + 3 are all semiprimes, hence at least 3 semiprimes in arithmetic progression with common difference 3. This subsequence begins 115, 155. There cannot be 4 semiprimes in arithmetic progression with common difference 3, starting with k, because modulo 4 we have {k, k+3, k+6, k+9} == {k+0, k+3, k+2, k+1} and one of these must be divisible by 4, hence a nonsemiprime (eliminating k = 4 by inspection).


LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000


FORMULA

{a(n)} = {k such that k is in A001358 and k+3 is in A001358}.


EXAMPLE

a(1) = 6 because 6 = 2 * 3 is semiprime and 6 + 3 = 9 = 3^2 is semiprime.
a(2) = 22 because 22 = 2 * 11 ans 22 + 3 = 25 = 5^2.
a(3) = 35 because 35 = 5 * 7 and 35 + 3 = 38 = 2 * 19.
a(4) = 46 because 46 = 2 * 23 and 46 + 3 = 49 = 7^2.
a(5) = 55 because 55 = 5 * 11 and 55 + 3 = 58 = 2 * 29.


MATHEMATICA

semiprimeQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; Select[ Range@ 670, semiprimeQ[ # ] && semiprimeQ[ # + 3] &]  Robert G. Wilson v, Aug 31 2007
SequencePosition[Table[If[PrimeOmega[n]==2, 1, 0], {n, 700}], {1, _, _, 1}][[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 03 2017 *)


CROSSREFS

Cf. A000040, A001358, A056809, A070552, A092125, A092126, A092127, A092128, A092129.
Sequence in context: A075799 A046392 A046408 * A264043 A056821 A031083
Adjacent sequences: A123014 A123015 A123016 * A123018 A123019 A123020


KEYWORD

easy,nonn


AUTHOR

Jonathan Vos Post, Nov 04 2006


EXTENSIONS

More terms from Robert G. Wilson v, Aug 31 2007


STATUS

approved



