A160812 -id:A160812 - cp's OEIS frontend

A018253 Divisors of 24.

1, 2, 3, 4, 6, 8, 12, 24

Offset: 1

The divisors of 24 greater than 1 are the only positive integers n with the property m^2 == 1 (mod n) for all integer m coprime to n. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 10 2001
Numbers n for which all Dirichlet characters are real. - Benoit Cloitre, Apr 21 2002
These are the numbers n that are divisible by all numbers less than or equal to the square root of n. - Tanya Khovanova, Dec 10 2006 [For a proof, see the Tauvel paper in references. - Bernard Schott, Dec 20 2012]
Also, numbers n such that A160812(n) = 0. - Omar E. Pol, Jun 19 2009
It appears that these are the only positive integers n such that A160812(n) = 0. - Omar E. Pol, Nov 17 2009
24 is a highly composite number: A002182(6)=24. - Reinhard Zumkeller, Jun 21 2010
Chebolu points out that these are exactly the numbers for which the multiplication table of the integers mod n have 1s only on their diagonal, i.e., ab == 1 (mod n) implies a = b (mod n). - Charles R Greathouse IV, Jul 06 2011
It appears that 3, 4, 6, 8, 12, 24 (the divisors >= 3 of 24) are also the only numbers n whose proper non-divisors k are prime numbers if k = d-1 and d divides n. - Omar E. Pol, Sep 23 2011
About the last Pol's comment: I have searched to 10^7 and have found no other terms. - Robert G. Wilson v, Sep 23 2011
Sum_{i=1..8} A000005(a(i))^3 = (Sum_{i=1..8} A000005(a(i)))^2, see Kordemsky in References and Barbeau et al. in Links section. - Bruno Berselli, Dec 29 2014

			Square root of 12 = 3.46... and 1, 2 and 3 divide 12.
From the tenth comment: 1^3 + 2^3 + 2^3 + 3^3 + 4^3 + 4^3 + 6^3 + 8^3 = (1+2+2+3+4+4+6+8)^2 = 900. - _Bruno Berselli_, Dec 28 2014

Harvey Cohn, "Advanced Number Theory", Dover, chap.II, p. 38
Boris A. Kordemsky, The Moscow Puzzles: 359 Mathematical Recreations, C. Scribner's Sons (1972), Chapter XIII, Paragraph 349.
Patrick Tauvel, "Exercices d'algèbre générale et d'arithmétique", Dunod, 2004, exercice 70 page 368.

John Baez, My Favorite Number: 24, The Rankin Lectures (September 19, 2008).
Edward Barbeau and Samer Seraj, Sum of Cubes is Square of Sum, arXiv:1306.5257 [math.NT], 2013.
Paul T. Bateman and Marc E. Low, Prime numbers in arithmetic progressions with difference 24, The American Mathematical Monthly 72:2 (Feb., 1965), pp. 139-143.
Sunil K. Chebolu, What is special about the divisors of 24?, Math. Mag., 85 (2012), 366-372.
M. H. Eggar, A curious property of the integer 24, Math. Gazette 84 (2000), pp. 96-97.
J. C. Lagarias (proposer), Problem 11747, Amer. Math. Monthly, 121 (2014), 83.
Eric Weisstein's World of Mathematics, Modulo Multiplication Group
Index entries for sequences related to divisors of numbers

Cf. A174228, A018256, A018261, A018266, A018293, A018321, A018350, A018412, A018609, A018676, A178877, A178878, A165412, A178858-A178864.
Cf. A000005, A158649. - Bruno Berselli, Dec 29 2014
Cf. A303704 (with respect to Astudillo's 2001 comment above).

DivisorsInt(24); # Bruno Berselli, Feb 13 2018

Divisors[24] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)

divisors(24) \\ Charles R Greathouse IV, Apr 28 2011

divisors(24); # Zerinvary Lajos, Jun 13 2009

a(n) = A161710(n-1). - Reinhard Zumkeller, Jun 21 2009

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

Omar E. Pol, Jun 20 2009

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

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

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.

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

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 *)

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

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

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

Omar E. Pol, Jun 20 2009

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

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Omar E. Pol, Illustration: Divisors and pi(x).
Omar E. Pol, Illustration for A008578, A161344, A161345 and A161424. [From _Omar E. Pol_, Oct 24 2009]

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

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

md3Q[n_]:=Max[Select[Divisors[n],#<=Sqrt[n]&]]==3; Select[Range[800],md3Q] (* Harvey P. Dale, Aug 12 2013 *)

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

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

Omar E. Pol, Jun 20 2009

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

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos
Omar E. Pol, Illustration: Divisors and pi(x)
Omar E. Pol, Illustration for A008578, A161344, A161345 and A161424 [From _Omar E. Pol_, Oct 24 2009]

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

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

Select[Range[1, 1000], Function[m, Max[Select[Divisors[m], # <= Sqrt[m] &]] == 4]] (* Ashton Baker, Nov 03 2013 *)

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

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

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

A161205 Triangle read by rows in which row n lists 2n-1 followed by 2n numbers 2n.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 4, 5, 6, 6, 6, 6, 6, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 16, 16, 16, 16, 16, 16, 16, 16

Offset: 1

Omar E. Pol, Jun 19 2009

Row sums: A125202(n+1). - R. J. Mathar, Feb 16 2010

			Triangle begins:
  1,  2,  2;
  3,  4,  4,  4,  4;
  5,  6,  6,  6,  6,  6,  6;
  7,  8,  8,  8,  8,  8,  8,  8,  8;
  9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10;

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos
Omar E. Pol, Illustration: Divisors and pi(x)

Cf. A000005, A000720, A018253, A125202, A160811, A160812, A161339.

A161205 := proc(n,k) if k=1 then 2*n-1; else 2*n; end if; end proc: seq(seq(A161205(n,k),k=1..2*n+1),n=1..12) ; # R. J. Mathar, Feb 16 2010

If n is a perfect square, then a(n) = 2*sqrt(n)-1; otherwise a(n) = 2*floor(sqrt(n)). - Nathaniel Johnston, May 06 2011

a(n) = A000196(n-1) + A000196(n) = floor(sqrt(n-1)) + floor(sqrt(n)). - Ridouane Oudra, Jun 07 2019

More terms from R. J. Mathar, Feb 16 2010

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

Omar E. Pol, Jun 20 2009

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

Michael De Vlieger, Table of n, a(n) for n = 1..10000

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

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

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

A162348 List of pairs (i,j) of central factors of n, such that i*j = n, where i is the largest divisor of n <= sqrt(n) and j is the smallest divisor of n >= sqrt(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 2, 1, 5, 2, 3, 1, 7, 2, 4, 3, 3, 2, 5, 1, 11, 3, 4, 1, 13, 2, 7, 3, 5, 4, 4, 1, 17, 3, 6, 1, 19, 4, 5, 3, 7, 2, 11, 1, 23, 4, 6, 5, 5, 2, 13, 3, 9, 4, 7, 1, 29, 5, 6, 1, 31, 4, 8, 3, 11, 2, 17, 5, 7, 6, 6, 1, 37, 2, 19, 3, 13, 5, 8, 1, 41, 6, 7, 1, 43, 4, 11, 5, 9, 2, 23, 1, 47, 6, 8, 7

Offset: 1

Omar E. Pol, Jul 04 2009

Note that if n is a square then the square root of n appears repeated: i = j = sqrt(n).

Squarest (least oblong) integral rectangle with area n. This has minimal semiperimeter (A063655), since s = i + j = i + n/i is minimal when ds/di = 1 - n/i^2 = 0, i.e., n = i^2. - Daniel Forgues, Sep 29 2014

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A033676, A033677, A018253, A160812, A161344, A006446, A162190.

f[n_] := Block[{d = Divisors@n}, len = Length[d]/2; {d[[Ceiling@len]], d[[Floor[len + 1]] ]}]; f[1] = {1, 1}; Array[f, 49] // Flatten (* Robert G. Wilson v, Aug 17 2009 *)

a(35) and further terms from Robert G. Wilson v, Aug 17 2009; corrected Aug 18 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

Omar E. Pol, Jun 20 2009

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

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

Omar E. Pol, Jun 19 2009

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

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

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

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

Omar E. Pol, Jun 20 2009

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

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

cp's OEIS Frontend

A018253 Divisors of 24.

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

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

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

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A161205 Triangle read by rows in which row n lists 2n-1 followed by 2n numbers 2n.

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

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A162348 List of pairs (i,j) of central factors of n, such that i*j = n, where i is the largest divisor of n <= sqrt(n) and j is the smallest divisor of n >= sqrt(n).

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