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%I #51 Jan 05 2019 05:15:58
%S 4,6,8,10,14,22,26,34,38,46,58,62,74,82,86,94,106,118,122,134,142,146,
%T 158,166,178,194,202,206,214,218,226,254,262,274,278,298,302,314,326,
%U 334,346,358,362,382,386,394,398,422,446,454,458,466,478,482,502,514
%N Numbers k with A033676(k)=2, where A033676 is the largest divisor <= sqrt(k).
%C Define a sieve operation with parameter s that eliminates integers of the form s^2 + s*i (i >= 0) from the set A000027 of natural numbers. The sequence lists those natural numbers that are eliminated by the sieve s=2 and cannot be eliminated by any sieve s >= 3. - _R. J. Mathar_, Jun 24 2009
%C After a(3)=8 all terms are 2*prime; for n > 3, a(n) = 2*prime(n-1) = 2*A000040(n-1). - _Zak Seidov_, Jul 18 2009
%C From _Omar E. Pol_, Jul 18 2009: (Start)
%C A classification of the natural numbers A000027.
%C =============================================================
%C Numbers k whose largest divisor <= sqrt(k) equals j
%C =============================================================
%C j Sequence Comment
%C =============================================================
%C 1 ..... A008578 1 together with the prime numbers
%C 2 ..... A161344 This sequence
%C 3 ..... A161345
%C 4 ..... A161424
%C 5 ..... A161835
%C 6 ..... A162526
%C 7 ..... A162527
%C 8 ..... A162528
%C 9 ..... A162529
%C 10 .... A162530
%C 11 .... A162531
%C 12 .... A162532
%C ... And so on. (End)
%C The numbers k whose largest divisor <= sqrt(k) is j are exactly those numbers j*m where m is either a prime >= k or one of the numbers in row j of A163925. - _Franklin T. Adams-Watters_, Aug 06 2009
%C See also A163280, the main entry for this sequence. - _Omar E. Pol_, Oct 24 2009
%C Also A100484 UNION 8. - _Omar E. Pol_, Nov 29 2012 (after Seidov and Hasler)
%H Omar E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polprdipi.jpg">Illustration: Divisors and pi(x)</a>
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv08.jpg">Illustration of initial terms</a>
%H Omar E. Pol. <a href="http://www.polprimos.com/imagenespub/poldiv06.jpg">Illustration of initial terms of A008578, A161344, A161345, A161424</a>
%F Equals 2*A000040 union {8}. - _M. F. Hasler_, Nov 27 2012
%F a(n) = 2*A046022(n+1) = 2*A175787(n). - _Omar E. Pol_, Nov 27 2012
%p isA := proc(n,s) if n mod s <> 0 then RETURN(false); fi; if n/s-s >= 0 then RETURN(true); else RETURN(false); fi; end: isA161344 := proc(n) for s from 3 to n do if isA(n,s) then RETURN(false); fi; od: isA(n,2) ; end: for n from 1 to 3000 do if isA161344(n) then printf("%d,",n) ; fi; od; # _R. J. Mathar_, Jun 24 2009
%t a[n_] := If[n <= 3, 2n+2, 2*Prime[n-1]]; Table[a[n], {n, 1, 56}] (* _Jean-François Alcover_, Nov 26 2012, after _Zak Seidov_ *)
%o (PARI) a(n)=if(n>3,prime(n-1),n+1)*2 \\ _M. F. Hasler_, Nov 27 2012
%Y Cf. A000005, A018253, A160811, A160812, A161205, A161346, A033676, A008578, A161345, A161424, A161835, A162526, A162527, A162528, A162529, A162530, A162531, A162532, A163925.
%Y Second column of array in A163280. Also, second row of array in A163990.
%K easy,nonn
%O 1,1
%A _Omar E. Pol_, Jun 20 2009
%E More terms from _R. J. Mathar_, Jun 24 2009
%E Definition added by _R. J. Mathar_, Jun 28 2009
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