cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 14 results. Next

A160812 a(n) = A161205(n)-A000005(n).

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 2, 0, 2, 2, 4, 0, 4, 2, 2, 2, 6, 2, 6, 2, 4, 4, 6, 0, 6, 6, 6, 4, 8, 2, 8, 4, 6, 6, 6, 2, 10, 8, 8, 4, 10, 4, 10, 6, 6, 8, 10, 2, 10, 8, 10, 8, 12, 6, 10, 6, 10, 10, 12, 2, 12, 10, 8, 8, 12, 8, 14, 10, 12, 8, 14, 4, 14, 12, 10, 10, 12, 8, 14, 6, 12, 14, 16, 6, 14, 14, 14, 10, 16, 6
Offset: 1

Views

Author

Omar E. Pol, Jun 19 2009

Keywords

Comments

It appears that a(n)= 0 if and only if n divides 24 (See also the comments in A018253).

Links

Crossrefs

Cf. A000005, A000720, A018253, A160811, A161205.

Formula

a(n) = 2*(A000196(n) - A038548(n)) = 2*A236627(n). - Omar E. Pol, Feb 05 2014

Extensions

Edited by Omar E. Pol, Aug 02 2009

A161339 Partial sums of A161205.

Original entry on oeis.org

1, 3, 5, 8, 12, 16, 20, 24, 29, 35, 41, 47, 53, 59, 65, 72, 80, 88, 96, 104, 112, 120, 128, 136, 145, 155, 165, 175, 185, 195, 205, 215, 225, 235, 245, 256, 268, 280, 292, 304, 316, 328, 340, 352, 364, 376, 388, 400, 413, 427, 441, 455, 469, 483, 497, 511
Offset: 1

Views

Author

Omar E. Pol, Jun 19 2009

Keywords

Links

Crossrefs

Cf. A000005, A000720, A018253, A160811, A160812, A161205.

Programs

  • Maple
    A161339 := proc(n) option remember: local s: if(n=1)then return 1: fi: s:=sqrt(n): if(frac(s)=0)then return procname(n-1)+2*s-1: else return procname(n-1)+2*floor(s): fi: end: seq(A161339(n), n=1..60); # Nathaniel Johnston, May 06 2011

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

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Jun 20 2009

Keywords

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

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

A161345 Numbers k whose largest divisor <= sqrt(k) is 3.

Original entry on oeis.org

9, 12, 15, 18, 21, 27, 33, 39, 51, 57, 69, 87, 93, 111, 123, 129, 141, 159, 177, 183, 201, 213, 219, 237, 249, 267, 291, 303, 309, 321, 327, 339, 381, 393, 411, 417, 447, 453, 471, 489, 501, 519, 537, 543, 573, 579, 591, 597, 633, 669, 681, 687, 699, 717, 723
Offset: 1

Views

Author

Omar E. Pol, Jun 20 2009

Keywords

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=3 and cannot be eliminated by any sieve s >= 4. - R. J. Mathar, Jun 24 2009
See A161344 for more information. - Omar E. Pol, Jul 05 2009
See also the array in A163280, the main entry for this sequence. - Omar E. Pol, Oct 24 2009
Union of {12, 18, 27} and all the numbers of the form 3*p, where p is an odd prime. - Amiram Eldar, Apr 17 2024

Links

Crossrefs

Cf. A000005, A018253, A160811, A160812, A161205, A161344, A161346, A033676, A008578, A161424, A161835, A162526, A162527, A162528, A162529, A162530, A162531, A162532.
Third column of the array in A163280. Also, third row of array in A163990. - Omar E. Pol, Oct 24 2009

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: isA161345 := proc(n) for s from 4 to n do if isA(n,s) then RETURN(false); fi; od: isA(n,3) ; end: for n from 1 to 3000 do if isA161345(n) then printf("%d,",n) ; fi; od; # R. J. Mathar, Jun 24 2009
  • Mathematica
    md3Q[n_]:=Max[Select[Divisors[n],#<=Sqrt[n]&]]==3; Select[Range[800],md3Q] (* Harvey P. Dale, Aug 12 2013 *)

Formula

Numbers k such that A033676(k)=3. - Omar E. Pol, Jul 05 2009

Extensions

Terms beyond a(10) from R. J. Mathar, Jun 24 2009
Definition added by R. J. Mathar, Jun 28 2009

A161424 Numbers k whose largest divisor <= sqrt(k) equals 4.

Original entry on oeis.org

16, 20, 24, 28, 32, 44, 52, 68, 76, 92, 116, 124, 148, 164, 172, 188, 212, 236, 244, 268, 284, 292, 316, 332, 356, 388, 404, 412, 428, 436, 452, 508, 524, 548, 556, 596, 604, 628, 652, 668, 692, 716, 724, 764, 772, 788, 796, 844, 892, 908, 916, 932, 956, 964
Offset: 1

Views

Author

Omar E. Pol, Jun 20 2009

Keywords

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=4 and cannot be eliminated by any sieve s >= 5. - R. J. Mathar, Jun 24 2009
See A161344 for more information. - Omar E. Pol, Jul 05 2009
See also the array in A163280, the main entry for this sequence. - Omar E. Pol, Oct 24 2009

Links

Crossrefs

Cf. A000005, A018253, A160811, A160812, A161205, A161344, A161345, A161346, A161425, A161428, A033676, A008578, A161835, A162526, A162527, A162528, A162529, A162530, A162531, A162532.
Cf. Fourth column of array in A163280. Also, fourth row of array in A163990. - Omar E. Pol, Oct 24 2009

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: isA161424 := proc(n) for s from 5 to n do if isA(n,s) then RETURN(false); fi; od: isA(n,4) ; end: for n from 1 to 3000 do if isA161424(n) then printf("%d,",n) ; fi; od; # R. J. Mathar, Jun 24 2009
  • Mathematica
    Select[Range[1, 1000], Function[m, Max[Select[Divisors[m], # <= Sqrt[m] &]] == 4]] (* Ashton Baker, Nov 03 2013 *)

Formula

Numbers n such that A033676(n)=4. - Omar E. Pol, Jul 05 2009

Extensions

Terms beyond a(8) from R. J. Mathar, Jun 24 2009
Definition added by R. J. Mathar, Jun 28 2009

A160811 Numbers not dividing 24.

Original entry on oeis.org

5, 7, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77
Offset: 1

Views

Author

Omar E. Pol, Jun 19 2009, Jun 28 2009

Keywords

Comments

These terms m > 5 can be called "triphile" or "3-phile" numbers, because there are 3 positive integers b_1 < b_2 < b_3 such that b_1 divides b_2, b_2 divides b_3 and m = b_1 + b_2 + b_3. A number that is not "triphile" is called "triphobe" or "3-phobe" (A019532). The smallest triphile number is 7 = 1 + 2 + 4 and the largest triphobe is 24. See A348517 for more explanations and link. - Bernard Schott, Oct 21 2021

Links

Crossrefs

Complement of A018253.
Cf. A000005, A000720, A161205.
Cf. A019532, A348517.

Programs

Formula

a(n) = n + 8 for n > 16. [Charles R Greathouse IV, Oct 26 2011]

Extensions

Definition corrected by Omar E. Pol, Nov 17 2009

A161346 a(n) = A161345(n)/3.

Original entry on oeis.org

3, 4, 5, 6, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271
Offset: 1

Views

Author

Omar E. Pol, Jun 20 2009

Keywords

Comments

Union of {4, 6, 9} and all the odd primes. - Amiram Eldar, Apr 17 2024

Links

Crossrefs

Cf. A018253, A160811, A160812, A161205, A161344, A161345.

Programs

  • Mathematica
    Select[Range[271], Function[{n, s}, Max[TakeWhile[Divisors[n], # <= s &]] == 3] @@ {#, Sqrt@ #} &[3 #] &] (* Michael De Vlieger, Feb 14 2020 *)

Extensions

Terms beyond a(10) from R. J. Mathar, Jun 24 2009

A161425 a(n) = A161424(n)/2.

Original entry on oeis.org

8, 10, 12, 14, 16, 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

Views

Author

Omar E. Pol, Jun 20 2009

Keywords

Crossrefs

Cf. A018253, A160811, A160812, A161205, A161344, A161345, A161346, A161424, A161428.

Extensions

Terms beyond a(8) from R. J. Mathar, Jun 24 2009

A161428 a(n) = A161424(n)/4.

Original entry on oeis.org

4, 5, 6, 7, 8, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277
Offset: 1

Views

Author

Omar E. Pol, Jun 20 2009

Keywords

Crossrefs

Cf. A018253, A160811, A160812, A161205, A161344, A161345, A161346, A161424, A161425.

Extensions

Terms beyond a(8) from R. J. Mathar, Jun 24 2009

A161827 Complement of A006446.

Original entry on oeis.org

5, 7, 10, 11, 13, 14, 17, 18, 19, 21, 22, 23, 26, 27, 28, 29, 31, 32, 33, 34, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 98
Offset: 1

Views

Author

Omar E. Pol, Jun 21 2009, Jun 28 2009, Feb 08 2010

Keywords

Comments

The asymptotic density of this sequence is 1 (Cooper and Kennedy, 1989). - Amiram Eldar, Jul 10 2020

Links

Crossrefs

Cf. A000005, A018253, A161205, A161344, A161345, A161424, A006446, A161828, A161835.

Extensions

More terms from N. J. A. Sloane, Feb 08 2010
Showing 1-10 of 14 results. Next

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