A161344 Numbers k with A033676(k)=2, where A033676 is the largest divisor <= sqrt(k).

4, 6, 8, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514

Offset: 1

Author

Omar E. Pol, Jun 20 2009

Keywords

• easy
• nonn

Comments

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

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

From Omar E. Pol, Jul 18 2009: (Start)

A classification of the natural numbers A000027.

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Numbers k whose largest divisor <= sqrt(k) equals j

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j Sequence Comment

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1 ..... A008578 1 together with the prime numbers

2 ..... A161344 This sequence

3 ..... A161345

4 ..... A161424

5 ..... A161835

6 ..... A162526

7 ..... A162527

8 ..... A162528

9 ..... A162529

10 .... A162530

11 .... A162531

12 .... A162532

... And so on. (End)

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

See also A163280, the main entry for this sequence. - Omar E. Pol, Oct 24 2009

Also A100484 UNION 8. - Omar E. Pol, Nov 29 2012 (after Seidov and Hasler)

Is this the union of {4} and A073582? - R. J. Mathar, May 30 2025

Links

• Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos
• Omar E. Pol, Illustration: Divisors and pi(x)
• Omar E. Pol, Illustration of initial terms
• Omar E. Pol. Illustration of initial terms of A008578, A161344, A161345, A161424

Crossrefs

Cf. A000005, A018253, A160811, A160812, A161205, A161346, A033676, A008578, A161345, A161424, A161835, A162526, A162527, A162528, A162529, A162530, A162531, A162532, A163925.

Second column of array in A163280. Also, second row of array in A163990.

Programs

Maple

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

Mathematica

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 *)

PARI

a(n)=if(n>3,prime(n-1),n+1)*2 \\ M. F. Hasler, Nov 27 2012

Formula

Equals 2*A000040 union {8}. - M. F. Hasler, Nov 27 2012

a(n) = 2*A046022(n+1) = 2*A175787(n). - Omar E. Pol, Nov 27 2012

Extensions

More terms from R. J. Mathar, Jun 24 2009

Definition added by R. J. Mathar, Jun 28 2009