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 18 results. Next

A162903 a(n) = A162531(n)/11.

Original entry on oeis.org

11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 27, 29, 31, 33, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 67, 71, 73, 77, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227
Offset: 1

Views

Author

Omar E. Pol and Zak Seidov, Jul 13 2009

Keywords

Comments

The only composite terms are 12,14,15,16,18,20,21,22,25,27,33,35,49,55,77,121.
After a(42) = 11^2 = 121 all terms are primes: n > 42, a(n) = A000040(n-12).

Crossrefs

Cf. A162531 (Numbers k whose largest divisor <= sqrt(k) equals 11), A000040 (Prime numbers).
Cf. A161344, A162531, A162790, A162792.

A033676 Largest divisor of n <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 4, 1, 3, 1, 4, 3, 2, 1, 4, 5, 2, 3, 4, 1, 5, 1, 4, 3, 2, 5, 6, 1, 2, 3, 5, 1, 6, 1, 4, 5, 2, 1, 6, 7, 5, 3, 4, 1, 6, 5, 7, 3, 2, 1, 6, 1, 2, 7, 8, 5, 6, 1, 4, 3, 7, 1, 8, 1, 2, 5, 4, 7, 6, 1, 8, 9, 2, 1, 7, 5, 2, 3
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) = sqrt(n) is a new record if and only if n is a square. - Zak Seidov, Jul 17 2009
a(n) = A060775(n) unless n is a square, when a(n) = A033677(n) = sqrt(n) is strictly larger than A060775(n). It would be nice to have an efficient algorithm to calculate these terms when n has a large number of divisors, as for example in A060776, A060777 and related problems such as A182987. - M. F. Hasler, Sep 20 2011
a(n) = 1 when n = 1 or n is prime. - Alonso del Arte, Nov 25 2012
a(n) is the smallest central divisor of n. Column 1 of A207375. - Omar E. Pol, Feb 26 2019
a(n^4+n^2+1) = n^2-n+1: suppose that n^2-n+k divides n^4+n^2+1 = (n^2-n+k)*(n^2+n-k+2) - (k-1)*(2*n+1-k) for 2 <= k <= 2*n, then (k-1)*(2*n+1-k) >= n^2-n+k, or n^2 - (2*k-1)*n + (k^2-k+1) = (n-k+1/2)^2 + 3/4 < 0, which is impossible. Hence the next smallest divisor of n^4+n^2+1 than n^2-n+1 is at least n^2-n+(2*n+1) = n^2+n+1 > sqrt(n^4+n^2+1). - Jianing Song, Oct 23 2022

References

  • G. Tenenbaum, pp. 268 ff, in: R. L. Graham et al., eds., Mathematics of Paul Erdős I.

Links

Crossrefs

Cf. A033677 (n/a(n)), A000196 (sqrt), A027750 (list of divisors), A056737 (n/a(n) - a(n)), A219695 (half of this for odd numbers), A207375 (list the central divisor(s)).
The strictly inferior case is A060775. Cf. also A140271.
Indices of given values: A008578 (1 and the prime numbers: a(n) = 1), A161344 (a(n) = 2), A161345 (a(n) = 3), A161424 (4), A161835 (5), A162526 (6), A162527 (7), A162528 (8), A162529 (9), A162530 (10), A162531 (11), A162532 (12), A282668 (indices of primes).

Programs

  • Haskell
    a033676 n = last $ takeWhile (<= a000196 n) $ a027750_row n
-- Reinhard Zumkeller, Jun 04 2012
  • Maple
    A033676 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do if d^2 <= n then a := max(a,d) ; end if; end do: a; end proc: # R. J. Mathar, Aug 09 2009
  • Mathematica
    largestDivisorLEQR[n_Integer] := Module[{dvs = Divisors[n]}, dvs[[Ceiling[Length@dvs/2]]]]; largestDivisorLEQR /@ Range[100] (* Borislav Stanimirov, Mar 28 2010 *)
Table[Last[Select[Divisors[n],#<=Sqrt[n]&]],{n,100}] (* Harvey P. Dale, Mar 17 2017 *)
  • PARI
    A033676(n) = {local(d);if(n<2,1,d=divisors(n);d[(length(d)+1)\2])} \\ Michael B. Porter, Jan 30 2010
  • Python
    from sympy import divisors
def A033676(n):
    d = divisors(n)
    return d[(len(d)-1)//2]  # Chai Wah Wu, Apr 05 2021

Formula

a(n) = n / A033677(n).
a(n) = A161906(n,A038548(n)). - Reinhard Zumkeller, Mar 08 2013
a(n) = A162348(2n-1). - Daniel Forgues, Sep 29 2014

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

A163280 Square array read by antidiagonals where column k lists the numbers j whose largest divisor <= sqrt(j) is k.

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 5, 8, 12, 16, 7, 10, 15, 20, 25, 11, 14, 18, 24, 30, 36, 13, 22, 21, 28, 35, 42, 49, 17, 26, 27, 32, 40, 48, 56, 64, 19, 34, 33, 44, 45, 54, 63, 72, 81, 23, 38, 39, 52, 50, 60, 70, 80, 90, 100, 29, 46, 51, 68, 55, 66, 77, 88, 99, 110, 121, 31, 58, 57, 76, 65, 78, 84, 96, 108, 120, 132, 144
Offset: 1

Views

Author

Omar E. Pol, Aug 07 2009

Keywords

Comments

This sequence is a permutation of the natural numbers A000027. Note that the first column is formed by 1 together with the prime numbers.
Column k contains exactly those numbers j=k*m where m is either a prime >= j or one of the numbers in row k of A163925. - Franklin T. Adams-Watters, Aug 12 2009

Examples

			Array begins:
   1,  4,  9,  16,  25,  36,  49,  64,  81, 100, 121, 144, ...
   2,  6, 12,  20,  30,  42,  56,  72,  90, 110, 132, 156, ...
   3,  8, 15,  24,  35,  48,  63,  80,  99, 120, 143, 168, ...
   5, 10, 18,  28,  40,  54,  70,  88, 108, 130, 154, 180, ...
   7, 14, 21,  32,  45,  60,  77,  96, 117, 140, 165, 192, ...
  11, 22, 27,  44,  50,  66,  84, 104, 126, 150, 176, 204, ...
  13, 26, 33,  52,  55,  78,  91, 112, 135, 160, 187, 216, ...
  17, 34, 39,  68,  65, 102,  98, 128, 153, 170, 198, 228, ...
  19, 38, 51,  76,  75, 114, 105, 136, 162, 190, 209, 264, ...
  23, 46, 57,  92,  85, 138, 119, 152, 171, 200, 220, 276, ...
  29, 58, 69, 116,  95, 174, 133, 184, 189, 230, 231, 348, ...
  31, 62, 87, 124, 115, 186, 147, 232, 207, 250, 242, 372, ...
  ...

Links

Crossrefs

Rows 1-12: A000290, A002378, A005563, A164004, A100451, A164006, A164007, A164008, A164009, A164010, A164011, A164012.
Columns 1-12: A008578, A161344, A161345, A161424, A161835, A162526, A162527, A162528, A162529, A162530, A162531, A162532.
Another version: A163990.
Cf. A000027, A000040, A033676, A147861, A163100, A164000.

Programs

  • Maple
    A163280 := proc(n,k) local r,T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then RETURN(T) ; fi; fi; od: end: # R. J. Mathar, Aug 09 2009
  • Mathematica
    nmax = 12;
pm = Prime[nmax];
sDiv[n_] := Select[Divisors[n], #^2 <= n&][[-1]];
Clear[col]; col[k_] := col[k] = Select[Range[k pm], sDiv[#] == k&];
T[n_, k_ /; 1 <= k <= Length[col[k]]] := col[k][[n]];
Table[T[n-k+1, k], {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 15 2019 *)

Formula

Column k lists the numbers j such that A033676(j)=k.

Extensions

Edited by R. J. Mathar, Aug 01 2010
Example edited by Jean-François Alcover, Dec 15 2019

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

A161835 Numbers k whose largest divisor <= sqrt(k) is 5.

Original entry on oeis.org

25, 30, 35, 40, 45, 50, 55, 65, 75, 85, 95, 115, 125, 145, 155, 185, 205, 215, 235, 265, 295, 305, 335, 355, 365, 395, 415, 445, 485, 505, 515, 535, 545, 565, 635, 655, 685, 695, 745, 755, 785, 815, 835, 865, 895, 905, 955, 965, 985, 995, 1055, 1115, 1135, 1145, 1165, 1195
Offset: 1

Views

Author

Omar E. Pol, Jun 20 2009

Keywords

Comments

See A161344 for more information. - Omar E. Pol, Jul 05 2009

Crossrefs

Cf. A033676, A008578, A161344, A161345, A161424, A161835, A162526, A162527, A162528, A162529, A162530, A162531, A162532.

Programs

  • Mathematica
    Select[Range[1, 1000], Function[m, Max[Select[Divisors[m], # <= Sqrt[m] &]] == 4]] (* Ashton Baker, Nov 03 2013 *)
  • PARI
    is(n)=divisors(n)[(numdiv(n)+1)\2]==5 \\ - M. F. Hasler, Nov 03 2013

Formula

Numbers k such that A033676(k)=5. - Omar E. Pol, Jul 05 2009

Extensions

Definition and more terms added by R. J. Mathar, Jun 28 2009

A162527 Numbers k whose largest divisor <= sqrt(k) equals 7.

Original entry on oeis.org

49, 56, 63, 70, 77, 84, 91, 98, 105, 119, 133, 147, 161, 175, 203, 217, 245, 259, 287, 301, 329, 343, 371, 413, 427, 469, 497, 511, 553, 581, 623, 679, 707, 721, 749, 763, 791, 889, 917, 959, 973, 1043, 1057, 1099, 1141, 1169, 1211, 1253, 1267, 1337, 1351
Offset: 1

Views

Author

Omar E. Pol, Jul 05 2009

Keywords

Comments

See A161344 for more information.

Links

Crossrefs

Cf. A033676, A008578, A161344, A161345, A161424, A161835, A162526, A162528, A162529, A162530, A162531, A162532.

Programs

  • Maple
    A033676 := proc(n) local dvs; dvs := sort(convert(numtheory[divisors](n),list)) ; op(floor((nops(dvs)+1)/2) ,dvs) ; end: for n from 1 to 2000 do if A033676(n) = 7 then printf("%d,",n) ; fi; od: # R. J. Mathar, Jul 13 2009
  • Mathematica
    ld = 7;
selQ[n_] := AllTrue[Divisors[n], # <= ld || #^2 > n&];
Select[Range[ld, 200] ld, selQ] (* Jean-François Alcover, Apr 14 2020 *)
ld7Q[n_]:=Select[Divisors[n],#<=Sqrt[n]&][[-1]]==7; Select[Range[1400],ld7Q] (* Harvey P. Dale, Jan 13 2023 *)

Formula

Numbers k such that A033676(k)=7.

Extensions

More terms from R. J. Mathar, Jul 13 2009

A162528 Numbers k whose largest divisor <= sqrt(k) equals 8.

Original entry on oeis.org

64, 72, 80, 88, 96, 104, 112, 128, 136, 152, 184, 232, 248, 296, 328, 344, 376, 424, 472, 488, 536, 568, 584, 632, 664, 712, 776, 808, 824, 856, 872, 904, 1016, 1048, 1096, 1112, 1192, 1208, 1256, 1304, 1336, 1384, 1432, 1448, 1528, 1544, 1576, 1592, 1688
Offset: 1

Views

Author

Omar E. Pol, Jul 05 2009

Keywords

Comments

See A161344 for more information.

Crossrefs

Cf. A033676, A008578, A161344, A161345, A161424, A161835, A162526, A162527, A162529, A162530, A162531, A162532.

Programs

  • Maple
    A033676 := proc(n) local dvs; dvs := sort(convert(numtheory[divisors](n),list)) ; op(floor((nops(dvs)+1)/2) ,dvs) ; end: for n from 1 to 2000 do if A033676(n) = 8 then printf("%d,",n) ; fi; od: # R. J. Mathar, Jul 13 2009
  • Mathematica
    ld8Q[n_]:=Last[Select[Divisors[n],#<=Sqrt[n]&]]==8; Select[Range[ 2000], ld8Q] (* Harvey P. Dale, Apr 08 2017 *)

Formula

Numbers k such that A033676(k)=8.

Extensions

More terms from R. J. Mathar, Jul 13 2009

A162530 Numbers k whose largest divisor <= sqrt(k) equals 10.

Original entry on oeis.org

100, 110, 120, 130, 140, 150, 160, 170, 190, 200, 230, 250, 290, 310, 370, 410, 430, 470, 530, 590, 610, 670, 710, 730, 790, 830, 890, 970, 1010, 1030, 1070, 1090, 1130, 1270, 1310, 1370, 1390, 1490, 1510, 1570, 1630, 1670, 1730, 1790, 1810, 1910, 1930
Offset: 1

Views

Author

Omar E. Pol, Jul 05 2009

Keywords

Comments

See A161344 for more information.

Links

Crossrefs

Cf. A033676, A008578, A161344, A161345, A161424, A161835, A162526, A162527, A162528, A162529, A162531, A162532.

Programs

  • Maple
    filter:= n -> andmap(t -> t<=10 or t^2 > n, numtheory:-divisors(n)):
select(filter, [seq(n,n=100..10000,10)]); # Robert Israel, Aug 16 2018
  • Mathematica
    ld10Q[n_]:=Last[Select[Divisors[n],#<=Sqrt[n]&]]==10; Select[Range[2000],ld10Q]  (* Harvey P. Dale, Jan 30 2011 *)

Formula

Numbers k such that A033676(k) = 10.

Extensions

More terms from Gerard P. Michon, Jul 12 2009
Showing 1-10 of 18 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.