A018253 -id:A018253 - cp's OEIS frontend

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

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

A018261 Divisors of 48.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Offset: 1

N. J. A. Sloane

48 is a highly composite number: A002182(8)=48. - Reinhard Zumkeller, Jun 21 2010

These are the orders, without repetition, of the finite subgroups of GL_3(Z); see Conway and Sloane. - Hal M. Switkay, Nov 06 2023

J. H. Conway and N. J. A. Sloane, Low-dimensional lattices. II. Subgroups of GL(n,Z), Proc. R. Soc. Lond. A 419 (1988), 29-68.
Index entries for sequences related to divisors of numbers

Cf. A018253, A018256, A018266, A018293, A018321, A018350, A018412, A018609, A018676, A178877, A178878, A165412, A178858, A178859, A178860, A178861, A178862, A178863, A178864.

Divisors[48] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)

divisors(48) \\ Charles R Greathouse IV, Jul 10 2016

A018412 Divisors of 360.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360

Offset: 1

N. J. A. Sloane

Comment from J. Lowell: Regular polygons with n sides in which internal angles have integral number of degrees (n >= 3).
360 is a highly composite number: A002182(13) = 360. - Reinhard Zumkeller, Jun 21 2010
There are 22209 ways to represent 360 as a sum of its distinct divisors (A033630). That's more than any smaller number, hence 360 is in A065218. - Alonso del Arte, Oct 09 2017

Index entries for sequences related to divisors of numbers

Cf. A018253, A018256, A018261, A018266, A018293, A018321, A018350, A018609, A018676, A178877, A178878, A165412, A178858, A178859, A178860, A178861, A178862, A178863, A178864.

Divisors[360] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2012 *)

divisors(360) \\ Charles R Greathouse IV, Jun 21 2017

A018256 Divisors of 36.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 12, 18, 36

Offset: 1

N. J. A. Sloane

36 is a highly composite number: A002182(7)=36. - Reinhard Zumkeller, Jun 21 2010

Numbers with all prime indices and exponents <= 2. Reversing inequalities gives A062739, strict A353502. - Gus Wiseman, Jun 28 2022

Index entries for sequences related to divisors of numbers

Cf. A018253, A018261, A018266, A018293, A018321, A018350, A018412, A018609, A018676, A178877, A178878, A165412, A178858, A178859, A178860, A178861, A178862, A178863, A178864.

Cf. A003586, A004709, A018338, A018710, A062739, A353502.

Divisors[36] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)

divisors(36) \\ Charles R Greathouse IV, Jun 21 2017

Intersection of A003586 (3-smooth) and A004709 (cubefree). - Gus Wiseman, Jun 28 2022

A046073 Number of squares in multiplicative group modulo n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 3, 2, 5, 1, 6, 3, 2, 2, 8, 3, 9, 2, 3, 5, 11, 1, 10, 6, 9, 3, 14, 2, 15, 4, 5, 8, 6, 3, 18, 9, 6, 2, 20, 3, 21, 5, 6, 11, 23, 2, 21, 10, 8, 6, 26, 9, 10, 3, 9, 14, 29, 2, 30, 15, 9, 8, 12, 5, 33, 8, 11, 6, 35, 3, 36, 18, 10, 9, 15, 6, 39, 4, 27, 20, 41, 3, 16, 21

Offset: 1

Eric W. Weisstein

a(n) is the number of different diagonal elements in Cayley table for multiplicative group modulo n. But the fact that the same number of different elements are on the diagonal of the Cayley table does not mean in every case that these groups are isomorphic. - Artur Jasinski, Jul 03 2010

The number of quadratic residues modulo n that are coprime to n. These residues are listed in A096103. - Peter Munn, Mar 10 2021

Daniel Shanks, Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 95, 1993.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Steven R. Finch and Pascal Sebah, Square and Cubes Modulo n, arXiv:math/0604465 [math.NT], 2006-2016.
Eric Weisstein's World of Mathematics, Modulo Multiplication Group.
Eric Weisstein's World of Mathematics, Quadratic Residue.

Cf. A046072, A007735, A060594, A000010, A087692, A000224.
Row lengths of A096103.
Positions of ones: A018253.

F:= n -> nops({seq}(`if`(igcd(t,n)=1,t^2 mod n,NULL), t=1..floor(n/2))):
1, seq(F(n), n=2..100); # Robert Israel, Jan 04 2015
# 2nd program
A046073 := proc(n)
    local a,p,e,pf;
    a := 1;
    for pf in ifactors(n)[2] do
        p := op(1,pf) ;
        e := op(2,pf) ;
        if p = 2 then
            a := a*p^max(e-3,0) ;
        else
            a := a*(p-1)/2*p^(e-1) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Oct 03 2016

Table[EulerPhi[n]/Sum[Boole[Mod[k^2, n] == 1] + Boole[n == 1], {k, n}], {n, 86}] (* or *)
Table[Apply[Times, FactorInteger[n] /. {p_, e_} /; p > 0 :> Which[p == 1, 1, p == 2, 2^Max[e - 3, 0], True, (p - 1) p^(e - 1)/2]], {n, 86}] (* Michael De Vlieger, Jul 18 2017 *)

A060594(n) = if(n<=2, 1, 2^#znstar(n)[3]); \\ This function from Joerg Arndt, Jan 06 2015
A046073(n) = eulerphi(n)/A060594(n); \\ Antti Karttunen, Jul 17 2017, after Sharon Sela's Mar 09 2002 formula.

A046073(n)=if(n>4,(n=znstar(n))[1]>>#n[3],1) \\ Avoids duplicate computation of phi(n). - M. F. Hasler, Nov 27 2017, typo fixed Mar 11 2021

from sympy import factorint, prod
def a(n): return 1 if n==1 else prod([2**max(e - 3, 0) if p==2 else (p - 1)*p**(e - 1)//2 for p, e in factorint(n).items()])
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 17 2017

(define (A046073 n) (cond ((= 1 n) n) ((even? n) (* (A000079 (max (- (A007814 n) 3) 0)) (A046073 (A028234 n)))) (else (* (/ 1 2) (- (A020639 n) 1) (/ (A028233 n) (A020639 n)) (A046073 (A028234 n)))))) ;; Antti Karttunen, Jul 17 2017, after the given multiplicative formula.

a(n) * A060594(n) = A000010(n) = phi(n) (This gives a formula for a(n) using the one in A060594(n) ). - Sharon Sela (sharonsela(AT)hotmail.com), Mar 09 2002
Multiplicative with a(2^e) = 2^max(e-3,0), a(p^e) = (p-1)*p^(e-1)/2 for p an odd prime.
Sum_{k=1..n} a(k) ~ c * n^2/sqrt(log(n)), where c = (43/(80*sqrt(Pi))) * Product_{p prime} (1+1/(2*p))*sqrt(1-1/p) = 0.24627260085060864229... (Finch and Sebah, 2006). - Amiram Eldar, Oct 18 2022

Edited and verified by Franklin T. Adams-Watters, Nov 07 2006

A018266 Divisors of 60.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Offset: 1

N. J. A. Sloane

Sequence is finite with last term a(12) = 60; A000005(60) = 12. - Reinhard Zumkeller, Dec 08 2009.
60 is a highly composite number: A002182(9) = 60. - Reinhard Zumkeller, Jun 21 2010
There are 35 ways to partition 60 as a sum of its distinct divisors (see A033630). This is more than any smaller number (hence 60 is listed in A065218). - Alonso del Arte, Oct 12 2017

Index entries for sequences related to divisors of numbers

Cf. A035303, A171425. - Reinhard Zumkeller, Dec 08 2009

Cf. A018253, A018256, A018261, A018293, A018321, A018350, A018412, A018609, A018676, A178877, A178878, A165412, A178858, A178859, A178860, A178861, A178862, A178863, A178864. - Reinhard Zumkeller, Jun 21 2010

Divisors[60] (* Vladimir Joseph Stephan Orlovsky, Dec 21 2008 *)

PARI
```
divisors(60)
```

a(n) = n + floor((n-1)/6)*(60/(13-n)-n). - Aaron J Grech, Aug 11 2024

A018609 Divisors of 720.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 360, 720

Offset: 1

N. J. A. Sloane

720 is a highly composite number: A002182(14)=720. - Reinhard Zumkeller, Jun 21 2010

Index entries for sequences related to divisors of numbers

Cf. A018253, A018256, A018261, A018266, A018293, A018321, A018350, A018412, A018676, A178877, A178878, A165412, A178858, A178859, A178860, A178861, A178862, A178863, A178864. - Reinhard Zumkeller, Jun 21 2010

Divisors[720] (* Vladimir Joseph Stephan Orlovsky, Nov 19 2010 *)

PARI
```
divisors(720)
```

A161710 a(n) = (-6n^7 + 154n^6 - 1533n^5 + 7525n^4 - 18879n^3 + 22561n^2 - 7302*n + 2520)/2520.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 24, 39, -2, -295, -1308, -3980, -9996, -22150, -44808, -84483, -150534, -256001, -418588, -661806, -1016288, -1521288, -2226376, -3193341, -4498314, -6234123, -8512892, -11468896, -15261684, -20079482, -26142888

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 8} = divisors of 24:

a(n) = A027750(A006218(23) + k + 1), 0 <= k < A000005(24).

			Differences of divisors of 24 to compute the coefficients of their interpolating polynomial, see formula:
1 ... 2 ... 3 ... 4 ... 6 ... 8 .. 12 .. 24
.. 1 ... 1 ... 1 ... 2 ... 2 ... 4 .. 12
..... 0 ... 0 ... 1 ... 0 ... 2 ... 8
........ 0 ... 1 .. -1 ... 2 ... 6
........... 1 .. -2 ... 3 ... 4
............. -3 ... 5 ... 1
................. 8 .. -4
.................. -12.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Reinhard Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A005408, A000124, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A080856, A161711, A161712, A161713, A161715, A006261, A018253, A161700, A161856.

[(-6*n^7 + 154*n^6 - 1533*n^5 + 7525*n^4 - 18879*n^ 3 + 22561*n^2 - 7302*n + 2520)/2520: n in [0..40]]; // Vincenzo Librandi, Jul 17 2011

Table[(-6n^7+154n^6-1533n^5+7525n^4-18879n^3+22561n^2-7302n+2520)/2520,{n,0,40}] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,2,3,4,6,8,12,24},40] (* Harvey P. Dale, Jul 15 2012 *)

a(n)=(-6*n^7+154*n^6-1533*n^5+7525*n^4-18879*n^3+22561*n^2-7302*n+2520)/2520 \\ Charles R Greathouse IV, Sep 24 2015

A161710_list, m = [1], [-12, 80, -223, 333, -281, 127, -23, 1]
for _ in range(1,10**2):
    for i in range(7):
        m[i+1]+= m[i]
    A161710_list.append(m[-1]) # Chai Wah Wu, Nov 09 2014

a(n) = C(n,0) + C(n,1) + C(n,4) - 3*C(n,5) + 8*C(n,6) - 12*C(n,7).
G.f.: (1-6*x+15*x^2-20*x^3+16*x^4-12*x^5+18*x^6-24*x^7)/(1-x)^8. - Bruno Berselli, Jul 17 2011
a(0)=1, a(1)=2, a(2)=3, a(3)=4, a(4)=6, a(5)=8, a(6)=12, a(7)=24, a(n)=8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*a(n-6)+ 8*a(n-7)- a(n-8). - Harvey P. Dale, Jul 15 2012

cp's OEIS Frontend

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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A018261 Divisors of 48.

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A018412 Divisors of 360.

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A018256 Divisors of 36.

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A046073 Number of squares in multiplicative group modulo n.

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A161710 a(n) = (-6n^7 + 154n^6 - 1533n^5 + 7525n^4 - 18879n^3 + 22561n^2 - 7302*n + 2520)/2520.