A116882 -id:A116882 - cp's OEIS frontend

A363121 Primitive terms of A116882: terms k of A116882 such that k/2 is not a term of A116882.

1, 12, 40, 56, 144, 176, 208, 240, 544, 608, 672, 736, 800, 864, 928, 992, 2112, 2240, 2368, 2496, 2624, 2752, 2880, 3008, 3136, 3264, 3392, 3520, 3648, 3776, 3904, 4032, 8320, 8576, 8832, 9088, 9344, 9600, 9856, 10112, 10368, 10624, 10880, 11136, 11392, 11648, 11904

Offset: 1

Amiram Eldar, May 16 2023

If k is a term of this sequence then k*2^m is a term of A116882 for any m >= 0.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A070941, A116882.

q[n_] := 2^(2*IntegerExponent[n, 2]) >= n; Join[{1}, Select[Range[2, 12000, 2], q[#] && !q[#/2] &]]
(* or *)
a[1] = 1; a[n_] := (2*n - 1)*2^IntegerLength[2*n - 1, 2]; Array[a, 100]

a(n) = if(n == 1, 1, (2*n - 1)*2^length(binary(2*n - 1)));

def A363121(n): return (m:=2*n-1)<1 else 1 # Chai Wah Wu, May 17 2023

a(n) = (2*n-1)*2^A070941(n-1), for n > 1.

A033677 Smallest divisor of n >= sqrt(n).

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 4, 3, 5, 11, 4, 13, 7, 5, 4, 17, 6, 19, 5, 7, 11, 23, 6, 5, 13, 9, 7, 29, 6, 31, 8, 11, 17, 7, 6, 37, 19, 13, 8, 41, 7, 43, 11, 9, 23, 47, 8, 7, 10, 17, 13, 53, 9, 11, 8, 19, 29, 59, 10, 61, 31, 9, 8, 13, 11, 67, 17, 23, 10, 71, 9, 73, 37, 15, 19, 11, 13, 79, 10

Offset: 1

N. J. A. Sloane

a(n) is the smallest k such that n appears in the k X k multiplication table and A027424(k) is the number of n with a(n) <= k.
a(n) is the largest central divisor of n. Right border of A207375. - Omar E. Pol, Feb 26 2019
If we define a divisor d|n to be superior if d >= n/d, then superior divisors are counted by A038548 and listed by A161908. This sequence selects the smallest superior divisor of n. - Gus Wiseman, Feb 19 2021
a(p) = p for p a prime or 1, these are also the record high points in this sequence. - Charles Kusniec, Aug 26 2022
a(n^4+n^2+1) = n^2+n+1 (see A033676). - Jianing Song, Oct 23 2022

			From _Gus Wiseman_, Feb 19 2021: (Start)
The divisors of 36 are {1,2,3,4,6,9,12,18,36}. Of these {1,2,3,4,6} are inferior and {6,9,12,18,36} are superior, so a(36) = 6.
The divisors of 40 are {1,2,4,5,8,10,20,40}. Of these {1,2,4,5} are inferior and {8,10,20,40} are superior, so a(40) = 8.
(End)

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

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A027424, A056737, A060775, A219695.
The lower central divisor is A033676.
The strictly superior case is A140271.
Leftmost column of A161908 (superior divisors).
Rightmost column of A207375 (central divisors).
A038548 counts superior (or inferior) divisors.
A056924 counts strictly superior (or strictly inferior) divisors.
A063538/A063539 list numbers with/without a superior prime divisor.
A070038 adds up superior divisors.
A341676 selects the unique superior prime divisor.
- Inferior: A063962, A066839, A069288, A161906, A217581, A333749, A333750.
- Superior: A059172, A116882, A116883, A341591, A341592, A341593, A341675.
- Strictly Inferior: A070039, A333805, A333806, A341596, A341674, A341677.
- Strictly Superior: A048098, A064052, A238535, A341594, A341595, A341642, A341643, A341644, A341645, A341646, A341673.
Cf. A000005, A000203, A001248, A001221, A001222, A020639, A051283.

a033677 n = head $
   dropWhile ((< n) . (^ 2)) [d | d <- [1..n], mod n d == 0]
-- Reinhard Zumkeller, Oct 20 2011

A033677 := proc(n)
    n/A033676(n) ;
end proc:

Table[Select[Divisors[n], # >= Sqrt[n] &, 1] // First, {n, 80}]  (* Jean-François Alcover, Apr 01 2011 *)

A033677(n) = {local(d); d=divisors(n); d[length(d)\2+1]} \\ Michael B. Porter, Feb 26 2010

from sympy import divisors
def A033677(n):
    d = divisors(n)
    return d[len(d)//2]  # Chai Wah Wu, Apr 05 2021

a(n) = n/A033676(n).

a(n) = A162348(2n). - Daniel Forgues, Sep 29 2014

A068346 a(n) = n'' = second arithmetic derivative of n.

Original entry on oeis.org

0, 0, 0, 0, 4, 0, 1, 0, 16, 5, 1, 0, 32, 0, 6, 12, 80, 0, 10, 0, 44, 7, 1, 0, 48, 7, 8, 27, 80, 0, 1, 0, 176, 9, 1, 16, 92, 0, 10, 32, 72, 0, 1, 0, 112, 16, 10, 0, 240, 9, 39, 24, 92, 0, 108, 32, 96, 13, 1, 0, 96, 0, 14, 20, 640, 21, 1, 0, 156, 15, 1, 0, 220, 0, 16, 16, 176, 21, 1, 0, 368, 216

Offset: 0

Reinhard Zumkeller, Feb 28 2002

a(2p) = 1 for any prime p implies p,p+2 form a twin prime pair. - Kevin J. Gomez, Aug 29 2017

Indices of records > 0 appear to all belong to A116882. - Bill McEachen, Oct 16 2023

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 2000 terms from T. D. Noe)
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.

Cf. A003415 (arithmetic derivative of n), A099306 (third arithmetic derivative of n).

Column k=2 of A258651.

a068346 = a003415 . a003415  -- Reinhard Zumkeller, Nov 10 2013

d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
a:= n-> d(d(n));
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 29 2017

dn[0]=0; dn[1]=0; dn[n_]:=Module[{f=Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus@@(n*f[[2]]/f[[1]])]]; Table[dn[dn[n]], {n, 100}] (T. D. Noe)
f[n_] := If[ Abs@ n < 2, 0, n*Total[#2/#1 & @@@ FactorInteger[Abs@ n]]]; Table[ f[ f[ n]], {n, 81}] (* Robert G. Wilson v, May 12 2012 *)

a(n) = A003415(A003415(n)).

a(A000040(n)) = 0; a(A157037(n)) = 1. - Reinhard Zumkeller, Feb 22 2009

More terms from T. D. Noe, Oct 12 2004

A063539 Numbers n that are sqrt(n-1)-smooth: largest prime factor of n (=A006530(n)) < sqrt(n).

Original entry on oeis.org

1, 8, 12, 16, 18, 24, 27, 30, 32, 36, 40, 45, 48, 50, 54, 56, 60, 63, 64, 70, 72, 75, 80, 81, 84, 90, 96, 98, 100, 105, 108, 112, 120, 125, 126, 128, 132, 135, 140, 144, 147, 150, 154, 160, 162, 165, 168, 175, 176, 180, 182, 189, 192, 195, 196

Offset: 1

N. J. A. Sloane, Aug 14 2001

nonn

Sometimes (Weisstein) called the "usual numbers" as opposed to what Greene and Knuth define as "unusual numbers" (A063538), which turn out to not be so unusual after all (Greene and Knuth 1990, Finch 2001). - Jonathan Vos Post, Sep 11 2010
If we define a divisor d|n to be superior if d >= n/d, then superior divisors are counted by A038548 and listed by A161908. This sequence lists numbers without a superior prime divisor, which is unique (A341676) when it exists. For example, the set of superior prime divisors of each n starts: {},{2},{3},{2},{5},{3},{7},{},{3},{5},{11},{},{13},{7}. The positions of empty sets give the sequence. - Gus Wiseman, Feb 24 2021
As Jonathan Vos Post's comment suggests, the sqrt(n-1)-smooth numbers are asymptotically less dense than their "unusual" complement. This is part of a larger picture of "typical" relative sizes of a number's prime factors: see, for example, the medians of the n-th smallest prime factors of the positive integers in A281889. - Peter Munn, Mar 03 2021

			a(100) = 360; a(1000) = 3744; a(10000) = 37665; a(100000)=375084;
a(10^6) = 3697669; a(10^7) = 36519633; a(10^8) = 360856296;
a(10^9) = 3571942311; a(10^10) = 35410325861; a(10^11) = 351498917129. - _Giovanni Resta_, Apr 12 2020

Greene, D. H. and Knuth, D. E., Mathematics for the Analysis of Algorithms, 3rd ed. Boston, MA: Birkhäuser, pp. 95-98, 1990.

N. J. A. Sloane, Table of a(n) for n = 1..10622 [Extending and correcting earlier b-files from T. D. Noe and Marius A. Burtea]
M. Beeler, R. W. Gosper and R. Schroeppel, HAKMEM, ITEM 29
Steven Finch, "RE: Unusual Numbers.", Aug 27 2001.
Hugo Pfoertner, Illustration of deviation from linear fit 3.7642*n, (2020).
Hugo Pfoertner, Illustration of relative error of asymptotic approximation, (2020).
Project Euler, Problem 668: Square root smooth numbers
V. Ramaswami, On the number of positive integers less than x and free of prime divisors greater than x^c, Bull. Amer. Math. Soc. 55 (1949), 1122-1127.
Eric W. Weisstein, Rough Number. [From _Jonathan Vos Post_, Sep 11 2010]
Wikipedia, Dickman function

Set difference of A048098 and A001248.
Complement of A063538.
Cf. A006530.
The following are all different versions of sqrt(n)-smooth numbers: A048098, A063539, A064775, A295084, A333535, A333536.
Positions of zeros in A341591.
A001221 counts prime divisors, with sum A001414.
A001222 counts prime-power divisors.
A033677 selects the smallest superior divisor.
A038548 counts superior (or inferior) divisors.
A051283 lists numbers without a superior prime-power divisor.
A056924 counts strictly superior (or strictly inferior) divisors.
A059172 lists numbers without a superior squarefree divisor.
A063962 counts inferior prime divisors.
A116882/A116883 list numbers with/without a superior odd divisor.
A161908 lists superior divisors.
A207375 lists central divisors.
A217581 selects the greatest inferior prime divisor.
A341642 counts strictly superior prime divisors.
A341676 gives unique superior prime divisors, with strict case A341643.
- Inferior: A033676, A066839, A069288, A161906, A333749, A333750.
- Superior: A070038, A072500, A341592, A341593, A341675.
- Strictly inferior: A060775, A070039, A333805, A333806, A341596, A341674.
- Strictly superior: A064052, A140271, A238535, A341594, A341595, A341644, A341645, A341646, A341673.
Cf. A000005, A000203, A020639, A112798, A281889.

[1] cat [m:m in [2..200]| Max(PrimeFactors(m)) lt Sqrt(m) ]; // Marius A. Burtea, May 08 2019

N:= 1000: # to get all terms <= N
Primes:= select(isprime, [2, seq(2*i+1, i=1..floor((N-1)/2))]):
S:= {$1..N} minus {seq(seq(m*p, m = 1 .. min(p, N/p)), p=Primes)}:
sort(convert(S, list)); # Robert Israel, Sep 02 2015

Prepend[Select[Range[192], FactorInteger[#][[-1, 1]] < Sqrt[#] &], 1] (* Ivan Neretin, Sep 02 2015 *)

from math import isqrt
from sympy import primepi
def A063539(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+primepi(x//(y:=isqrt(x)))+sum(primepi(x//i)-primepi(i) for i in range(1,y)))
    return bisection(f,n,n) # Chai Wah Wu, Oct 05 2024

From Hugo Pfoertner, Apr 02 - Apr 12 2020: (Start)
For small n (e.g. n < 10000) a(n) can apparently be approximated by 3.7642*n.
Asymptotically, the number of sqrt(n)-smooth numbers < x is known to be (1-log(2))*x + O(x/log(x)), see Ramaswami (1949).
n = (1-log(2))*a(n) - 0.59436*a(n)/log(a(n)) is a fitted approximation. (End)
However, it is known that this fit only leads to an increase of accuracy in the range up to a(10^11). The improvement in accuracy suggested by the plot of the relative error for even larger n does not occur. For larger n the behavior of the error term O(x/log(x)) is not known. - Hugo Pfoertner, Nov 12 2023

A161906 Triangle read by rows in which row n lists the divisors of n that are <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 3, 1, 2, 4, 1, 1, 2, 3, 1, 1, 2, 4, 1, 3, 1, 2, 1, 1, 2, 3, 4, 1, 5, 1, 2, 1, 3, 1, 2, 4, 1, 1, 2, 3, 5, 1, 1, 2, 4, 1, 3, 1, 2, 1, 5, 1, 2, 3, 4, 6, 1, 1, 2, 1, 3, 1, 2, 4, 5, 1, 1, 2, 3, 6, 1, 1, 2, 4, 1, 3, 5, 1, 2, 1, 1, 2, 3

Offset: 1

Omar E. Pol, Jun 27 2009

If we define a divisor d|n to be inferior if d <= n/d, then inferior divisors are counted by A038548 and listed by this sequence. - Gus Wiseman, Mar 08 2021

			Triangle begins:
   1....... 1;
   2....... 1;
   3....... 1;
   4..... 1,2;
   5....... 1;
   6..... 1,2;
   7....... 1;
   8..... 1,2;
   9..... 1,3;
  10..... 1,2;
  11....... 1;
  12... 1,2,3;
  13....... 1;
  14..... 1,2;
  15..... 1,3;
  16... 1,2,4;

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened

Initial terms are A000012.
Final terms are A033676.
Row lengths are A038548 (number of inferior divisors).
Row sums are A066839 (sum of inferior divisors).
The prime terms are counted by A063962.
The odd terms are counted by A069288.
Row products are A072499.
Row LCMs are A072504.
The superior version is A161908.
The squarefree terms are counted by A333749.
The prime-power terms are counted by A333750.
The strictly superior version is A341673.
The strictly inferior version is A341674.
A001221 counts prime divisors, with sum A001414.
A000005 counts divisors, listed by A027750 with sum A000203.
A056924 count strictly superior (or strictly inferior divisors).
A207375 lists central divisors.
- Inferior: A217581.
- Superior: A033677, A051283, A059172, A063538, A063539, A070038, A116882, A116883, A341591, A341592, A341593, A341675, A341676.
- Strictly Inferior: A060775, A070039, A333805, A333806, A341596, A341677.
- Strictly Superior: A048098, A064052, A140271, A238535, A341594, A341595, A341642, A341643, A341644, A341645, A341646.
Cf. A000196, A001055, A001248, A006530, A020639, A050320, A068101, A161901.

a161906 n k = a161906_tabf !! (n-1) !! (k-1)
a161906_row n = a161906_tabf !! (n-1)
a161906_tabf = zipWith (\m ds -> takeWhile ((<= m) . (^ 2)) ds)
                       [1..] a027750_tabf'
-- Reinhard Zumkeller, Jun 24 2015, Mar 08 2013

div[n_] := Select[Divisors[n], # <= Sqrt[n] &]; div /@ Range[48] // Flatten (* Amiram Eldar, Nov 13 2020 *)

row(n) = select(x->(x<=sqrt(n)), divisors(n)); \\ Michel Marcus, Nov 13 2020

More terms from Sean A. Irvine, Nov 29 2010

A161908 Array read by rows in which row n lists the divisors of n that are >= sqrt(n).

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 3, 6, 7, 4, 8, 3, 9, 5, 10, 11, 4, 6, 12, 13, 7, 14, 5, 15, 4, 8, 16, 17, 6, 9, 18, 19, 5, 10, 20, 7, 21, 11, 22, 23, 6, 8, 12, 24, 5, 25, 13, 26, 9, 27, 7, 14, 28, 29, 6, 10, 15, 30, 31, 8, 16, 32, 11, 33, 17, 34, 7, 35, 6, 9, 12, 18, 36, 37, 19, 38, 13, 39, 8, 10, 20, 40, 41, 7, 14, 21, 42, 43, 11, 22, 44, 9, 15, 45, 23, 46, 47, 8, 12, 16

Offset: 1

Omar E. Pol, Jun 27 2009

T(n,A038548(n)) = n. - Reinhard Zumkeller, Mar 08 2013

If we define a divisor d|n to be superior if d >= n/d, then superior divisors are counted by A038548 and listed by this sequence. - Gus Wiseman, Mar 08 2021

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

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened

Final terms are A000027.
Initial terms are A033677.
Row lengths are A038548 (number of superior divisors).
Row sums are A070038 (sum of superior divisors).
The inferior version is A161906.
The prime terms are counted by A341591.
The squarefree terms are counted by A341592.
The prime-power terms are counted by A341593.
The strictly superior version is A341673.
The strictly inferior version is A341674.
The odd terms are counted by A341675.
A001221 counts prime divisors, with sum A001414.
A056924 counts strictly superior (or strictly inferior divisors).
A207375 lists central divisors.
- Inferior: A033676, A063962, A066839, A069288, A217581, A333749, A333750.
- Superior: A051283, A059172, A063538, A063539, A072500, A116882, A116883, A341592, A341676.
- Strictly Inferior: A060775, A070039, A333805, A333806, A341596, A341677.
- Strictly Superior: A048098, A064052, A140271, A238535, A341594, A341595, A341642, A341643, A341644, A341645, A341646.
Cf. A000005, A000203, A001055, A001248, A006530, A020639, A050320, A161901.

a161908 n k = a161908_tabf !! (n-1) !! (k-1)
a161908_row n = a161908_tabf !! (n-1)
a161908_tabf = zipWith
               (\x ds -> reverse $ map (div x) ds) [1..] a161906_tabf
-- Reinhard Zumkeller, Mar 08 2013

Table[Select[Divisors[n],#>=Sqrt[n]&],{n,100}]//Flatten (* Harvey P. Dale, Jan 01 2021 *)

More terms from Sean A. Irvine, Nov 29 2010

A063538 Numbers n that are not sqrt(n-1)-smooth: largest prime factor of n (=A006530(n)) >= sqrt(n).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91

Offset: 1

N. J. A. Sloane, Aug 14 2001

If we define a divisor d|n to be superior if d >= n/d, then superior divisors are counted by A038548 and listed by A161908. This sequence lists all numbers with a superior prime divisor, which is unique (A341676) when it exists. For example, 42 is in the sequence because it has a prime divisor 7 which is greater than the quotient 42/7 = 6. - Gus Wiseman, Feb 19 2021

D. H. Greene and D. E. Knuth, Mathematics for the Analysis of Algorithms; see pp. 95-98.

Robert Israel, Table of n, a(n) for n = 1..10000
Beeler, M., Gosper, R. W. and Schroeppel, R., HAKMEM, ITEM 29

Cf. A006530, A063762.
Complement of A063539. Supersequence of A001358 (semiprimes).
The strictly superior version is A064052 (complement: A048098), with associated unique prime divisor A341643.
The case of odd instead of prime divisors is A116883 (complement: A116882).
Also nonzeros of A341591 (number of superior prime divisors).
The unique superior prime divisors of the terms are A341676.
A001221 counts prime divisors, with sum A001414.
A033677 selects the smallest superior divisor.
A038548 counts superior (also inferior) divisors.
A161908 lists superior divisors.
- Inferior: A033676, A063962, A066839, A161906, A217581, A333749, A333750.
- Superior: A051283, A059172, A070038, A072500, A341592, A341593, A341675.
- Strictly Inferior: A056924, A060775, A070039, A333805, A333806, A341596, A341674, A341677.
- Strictly Superior: A140271, A238535, A341594, A341595, A341642, A341644, A341645/A341646, A341673.
Cf. A000005, A001055, A001222, A001248, A020639, A056239, A112798.

N:= 1000: # to get all terms <= N
Primes:= select(isprime, [2,seq(2*i+1, i=1..floor((N-1)/2))]):
S:= {seq(seq(m*p, m = 1 .. min(p, N/p)),p=Primes)}:
sort(convert(S,list)); # Robert Israel, Sep 01 2015

Select[Range[2, 91], FactorInteger[#][[-1, 1]] >= Sqrt[#] &] (* Ivan Neretin, Aug 30 2015 *)

from math import isqrt
from sympy import primepi
def A063538(n):
    def f(x): return int(n+x-primepi(x//(y:=isqrt(x)))-sum(primepi(x//i)-primepi(i) for i in range(1,y)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Oct 05 2024

Union of A001248 and A064052. - Gus Wiseman, Feb 24 2021

A140271 Least divisor of n that is > sqrt(n), with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 4, 9, 5, 11, 4, 13, 7, 5, 8, 17, 6, 19, 5, 7, 11, 23, 6, 25, 13, 9, 7, 29, 6, 31, 8, 11, 17, 7, 9, 37, 19, 13, 8, 41, 7, 43, 11, 9, 23, 47, 8, 49, 10, 17, 13, 53, 9, 11, 8, 19, 29, 59, 10, 61, 31, 9, 16, 13, 11, 67, 17, 23, 10, 71, 9, 73, 37, 15, 19, 11, 13, 79, 10, 27

Offset: 1

Leroy Quet, May 16 2008

nonn

If n is not a square, then a(n) = A033677(n).

If we define a divisor d|n to be strictly superior if d > n/d, then strictly superior divisors are counted by A056924 and listed by A341673. This sequence selects the smallest strictly superior divisor of n. - Gus Wiseman, Apr 06 2021

			From _Gus Wiseman_, Apr 06 2021: (Start)
a(n) is the smallest element in the following sets of strictly superior divisors:
   1: {1}       16: {8,16}        31: {31}
   2: {2}       17: {17}          32: {8,16,32}
   3: {3}       18: {6,9,18}      33: {11,33}
   4: {4}       19: {19}          34: {17,34}
   5: {5}       20: {5,10,20}     35: {7,35}
   6: {3,6}     21: {7,21}        36: {9,12,18,36}
   7: {7}       22: {11,22}       37: {37}
   8: {4,8}     23: {23}          38: {19,38}
   9: {9}       24: {6,8,12,24}   39: {13,39}
  10: {5,10}    25: {25}          40: {8,10,20,40}
  11: {11}      26: {13,26}       41: {41}
  12: {4,6,12}  27: {9,27}        42: {7,14,21,42}
  13: {13}      28: {7,14,28}     43: {43}
  14: {7,14}    29: {29}          44: {11,22,44}
  15: {5,15}    30: {6,10,15,30}  45: {9,15,45}
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A060775, A033676, A033677.
These divisors are counted by A056924.
These divisors add up to A238535.
These divisors that are odd are counted by A341594.
These divisors that are squarefree are counted by A341595
These divisors that are prime are counted by A341642.
These divisors are listed by A341673.
A038548 counts superior (or inferior) divisors.
A161906 lists inferior divisors.
A161908 lists superior divisors.
A207375 list central divisors.
A341674 lists strictly inferior divisors.
- Inferior: A063962, A066839, A069288, A217581, A333749, A333750.
- Superior: A051283, A059172, A063538, A063539, A070038, A072500, A116882, A116883, A341591, A341592, A341593, A341675, A341676.
- Strictly Inferior: A070039, A333805, A333806, A341596, A341677.
- Strictly Superior: A048098, A064052, A341643, A341644, A341646.
Cf. A000005, A000203, A001221, A001222, A001248, A006530, A020639, A112798.

with(numtheory):
a:= n-> min(select(d-> is(d=n or d>sqrt(n)), divisors(n))):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 29 2018

Table[Select[Divisors[n], # > Sqrt[n] &][[1]], {n, 2, 70}] (* Stefan Steinerberger, May 18 2008 *)

A140271(n)={local(d,a);d=divisors(n);a=n;for(i=1,length(d),if(d[i]>sqrt(n),a=min (d[i],a)));a} \\ Michael B. Porter, Apr 06 2010

More terms from Stefan Steinerberger, May 18 2008
a(70)-a(80) from Ray Chandler, Jun 25 2009
Franklin T. Adams-Watters, Jan 26 2018, added a(1) = 1 to preserve the relation a(n) | n.

A333805 Number of odd divisors of n that are < sqrt(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 3, 1, 1, 3, 1, 2, 2, 1, 1, 2, 3, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 2, 1, 1, 3, 2, 1, 2, 1, 1, 4, 2, 1, 2, 1, 2, 2, 1, 2, 3, 2, 1, 2, 1, 1, 4

Offset: 1

Ilya Gutkovskiy, Apr 05 2020

nonn

If we define a divisor d|n to be strictly inferior if d < n/d, then strictly inferior divisors are counted by A056924 and listed by A341674. This sequence counts strictly inferior odd divisors. - Gus Wiseman, Feb 26 2021

			The strictly inferior odd divisors of 945 are 1, 3, 5, 7, 9, 15, 21, 27, so a(945) = 8. - _Gus Wiseman_, Feb 27 2021

Antti Karttunen, Table of n, a(n) for n = 1..65537

Dominated by A001227 (number of odd divisors).
Strictly inferior divisors (not just odd) are counted by A056924.
The non-strict version is A069288.
These divisors add up to A070039.
The case of prime divisors is A333806.
The strictly superior version is A341594.
The case of squarefree divisors is A341596.
The superior version is A341675.
The case of prime-power divisors is A341677.
A006530 selects the greatest prime factor.
A020639 selects the smallest prime factor.
- Odd -
A000009 counts partitions into odd parts, ranked by A066208.
A026424 lists numbers with odd Omega.
A027193 counts odd-length partitions.
A067659 counts strict partitions of odd length, ranked by A030059.
- Inferior divisors -
A033676 selects the greatest inferior divisor.
A033677 selects the smallest superior divisor.
A038548 counts superior (or inferior) divisors.
A060775 selects the greatest strictly inferior divisor.
A341674 lists strictly inferior divisors.
Cf. A001248, A051283, A063538/A063539, A063962, A116883/A116882, A300272, A333807, A333809, A340832.

Table[DivisorSum[n, 1 &, # < Sqrt[n] && OddQ[#] &], {n, 1, 90}]
nmax = 90; CoefficientList[Series[Sum[x^(2 k (2 k - 1))/(1 - x^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

A333805(n) = sumdiv(n,d,(d%2)&&((d*d)Antti Karttunen, Nov 02 2022

G.f.: Sum_{k>=1} x^(2*k*(2*k - 1)) / (1 - x^(2*k - 1)).

Data section extended up to a(105) by Antti Karttunen, Nov 02 2022

A341674 Irregular triangle read by rows giving the strictly inferior divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 3, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 1, 3, 1, 2, 1, 1, 2, 3, 4, 1, 1, 2, 1, 3, 1, 2, 4, 1, 1, 2, 3, 5, 1, 1, 2, 4, 1, 3, 1, 2, 1, 5, 1, 2, 3, 4, 1, 1, 2, 1, 3, 1, 2, 4, 5, 1, 1, 2, 3, 6, 1, 1, 2, 4, 1, 3

Offset: 1

Gus Wiseman, Feb 23 2021

We define a divisor d|n to be strictly inferior if d < n/d. The number of strictly inferior divisors of n is A056924(n).

			Triangle begins:
     1: {}        16: 1,2        31: 1
     2: 1         17: 1          32: 1,2,4
     3: 1         18: 1,2,3      33: 1,3
     4: 1         19: 1          34: 1,2
     5: 1         20: 1,2,4      35: 1,5
     6: 1,2       21: 1,3        36: 1,2,3,4
     7: 1         22: 1,2        37: 1
     8: 1,2       23: 1          38: 1,2
     9: 1         24: 1,2,3,4    39: 1,3
    10: 1,2       25: 1          40: 1,2,4,5
    11: 1         26: 1,2        41: 1
    12: 1,2,3     27: 1,3        42: 1,2,3,6
    13: 1         28: 1,2,4      43: 1
    14: 1,2       29: 1          44: 1,2,4
    15: 1,3       30: 1,2,3,5    45: 1,3,5

Initial terms are A000012.
Row lengths are A056924 (number of strictly inferior divisors).
Final terms are A060775.
Row sums are A070039 (sum of strictly inferior divisors).
The weakly inferior version is A161906.
The weakly superior version is A161908.
The odd terms are counted by A333805.
The prime terms are counted by A333806.
The squarefree terms are counted by A341596.
The strictly superior version is A341673.
The prime-power terms are counted by A341677.
A001221 counts prime divisors, with sum A001414.
A001222 counts prime-power divisors.
A005117 lists squarefree numbers.
A038548 counts superior (or inferior) divisors.
A207375 lists central divisors.
- Inferior: A033676, A063962, A066839, A069288, A217581, A333749, A333750.
- Superior: A033677, A051283, A059172, A063538, A063539, A070038, A116882, A116883, A341591, A341592, A341593, A341675, A341676.
- Strictly Superior: A048098, A064052, A140271, A238535, A341594, A341595, A341642, A341643, A341644, A341645, A341646.
Cf. A000005, A000203, A001055, A001248, A006530, A020639, A050320.

Mathematica
```
Table[Select[Divisors[n],#
				
```

cp's OEIS Frontend

A363121 Primitive terms of A116882: terms k of A116882 such that k/2 is not a term of A116882.

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A033677 Smallest divisor of n >= sqrt(n).

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A068346 a(n) = n'' = second arithmetic derivative of n.

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A063539 Numbers n that are sqrt(n-1)-smooth: largest prime factor of n (=A006530(n)) < sqrt(n).

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A161906 Triangle read by rows in which row n lists the divisors of n that are <= sqrt(n).

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A161908 Array read by rows in which row n lists the divisors of n that are >= sqrt(n).

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