A068108 -id:A068108 - cp's OEIS frontend

A038548 Number of divisors of n that are at most sqrt(n).

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

Offset: 1

Tom Verhoeff, N. J. A. Sloane

Number of ways to arrange n identical objects in a rectangle, modulo rotation.
Number of unordered solutions of x*y = n. - Colin Mallows, Jan 26 2002
Number of ways to write n-1 as n-1 = x*y + x + y, 0 <= x <= y <= n. - Benoit Cloitre, Jun 23 2002
Also number of values for x where x+2n and x-2n are both squares (e.g., if n=9, then 18+18 and 18-18 are both squares, as are 82+18 and 82-18 so a(9)=2); this is because a(n) is the number of solutions to n=k(k+r) in which case if x=r^2+2n then x+2n=(r+2k)^2 and x-2n=r^2 (cf. A061408). - Henry Bottomley, May 03 2001
Also number of sums of sequences of consecutive odd numbers or consecutive even numbers including sequences of length 1 (e.g., 12 = 5+7 or 2+4+6 or 12 so a(12)=3). - Naohiro Nomoto, Feb 26 2002
Number of partitions whose consecutive parts differ by exactly two.
a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1). - Christian G. Bower, Jun 06 2005
Also number of partitions of n such that if k is the largest part, then each of the parts 1,2,...,k-1 occurs exactly twice. Example: a(12)=3 because we have [3,3,2,2,1,1],[2,2,2,2,2,1,1] and [1,1,1,1,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Mar 07 2006
a(n) is also the number of nonnegative integer solutions of the Diophantine equation 4*x^2 - y^2 = 16*n. For example, a(24)=4 because there are 4 solutions: (x,y) = (10,4), (11,10), (14,20), (25,46). - N-E. Fahssi, Feb 27 2008
a(n) is the number of even divisors of 2*n that are <= sqrt(2*n). - Joerg Arndt, Mar 04 2010
First differences of A094820. - John W. Layman, Feb 21 2012
a(n) = #{k: A027750(n,k) <= A000196(n)}; a(A008578(n)) = 1; a(A002808(n)) > 1. - Reinhard Zumkeller, Dec 26 2012
Row lengths of the tables in A161906 and A161908. - Reinhard Zumkeller, Mar 08 2013
Number of positive integers in the sequence defined by x_0 = n, x_(k+1) = (k+1)*(x_k-2)/(k+2) or equivalently by x_k = n/(k+1) - k. - Luc Rousseau, Mar 03 2018
Expanding the first comment: Number of rectangles with area n and integer side lengths, modulo rotation. Also number of 2D grids of n congruent squares, in a rectangle, modulo rotation (cf. A000005 for rectangles instead of squares; cf. A034836 for the 3D case). - Manfred Boergens, Jun 08 2021
Number of divisors of n that have an even number of prime divisors (counted with multiplicity), or in other words, number of terms of A028260 that divide n. - Antti Karttunen, Apr 17 2022

			a(4) = 2 since 4 = 2 * 2 = 4 * 1. Also A034178(4*4) = 2 since 16 = 4^2 - 0^2 = 5^2 - 3^2. - _Michael Somos_, May 11 2011
x + x^2 + x^3 + 2*x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + x^11 + ...

George E. Andrews and Kimmo Eriksson, Integer Partitions, Cambridge Univ. Press, 2004, page 18, exer. 21, 22.

T. D. Noe, Table of n, a(n) for n = 1..10000
Cristina Ballantine and Mircea Merca, New convolutions for the number of divisors, Journal of Number Theory, 2016, vol. 170, pp. 17-34.
Christopher Briggs, Y. Hirano, and H. Tsutsui, Positive Solutions to Some Systems of Diophantine Equations, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.4.
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph invariants based on the divides relation and ordered by prime signatures, arXiv:1405.5283 [math.NT], 2014, (2.27).
Madeline Locus Dawsey, Matthew Just and Robert Schneider, A "supernormal" partition statistic, arXiv:2107.14284 [math.NT], 2021. See Table 2 p. 21.
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2 (1999), Article 99.1.6.
Index entries for sequences computed from exponents in factorization of n.

Different from A068108. Records give A038549, A004778, A086921.
Cf. A000005, A001620, A028260, A056924, A072670, A094820, A161841, A108504.
Cf. A066839, A033676, row sums of A303300.
Inverse Möbius transform of A065043.
Cf. A244664 (Dgf at s=2), A244665 (Dgf at s=3).

a038548 n = length $ takeWhile (<= a000196 n) $ a027750_row n
-- Reinhard Zumkeller, Dec 26 2012
(C#)
int A038548(int n)  {
    System.Numerics.BigInteger erg = 0, i;
    for (i = 1; i * i <= n; i++)
        if (n % i == 0) erg++;
    return (int)erg;
} // Frank Hollstein, Jan 08 2023

with(numtheory): A038548 := n->ceil(sigma[0](n)/2);

Table[ Floor[ (DivisorSigma[0, n] + 1)/2], {n, 105}] (* Robert G. Wilson v, Mar 02 2009 *)
Table[Count[Divisors[n],?(#<=Sqrt[n]&)],{n,110}] (* _Harvey P. Dale, Jul 10 2021 *)
Table[Sum[If[n > k*(k-1), 1, 0], {k, Divisors[n]}], {n, 1, 100}] (* Vaclav Kotesovec, Oct 22 2024 *)

{a(n) = if( n<1, 0, sumdiv(n, d, d*d <= n))} /* Michael Somos, Jan 25 2005 */

a(n)=ceil(numdiv(n)/2) \\ Charles R Greathouse IV, Sep 28 2012

from sympy import divisor_count
def A038548(n): return divisor_count(n)+1>>1 # Chai Wah Wu, Dec 19 2023

a(n) = ceiling(d(n)/2), where d(n) = number of divisors of n (A000005).
a(2k) = A034178(2k) + A001227(k). a(2k+1) = A034178(2k+1). - Naohiro Nomoto, Feb 26 2002
G.f.: Sum_{k>=1} x^(k^2)/(1-x^k). - Jon Perry, Sep 10 2004
Dirichlet g.f.: (zeta(s)^2 + zeta(2*s))/2. - Christian G. Bower, Jun 06 2005 [corrected by Vaclav Kotesovec, Aug 19 2019]
a(n) = (A000005(n) + A010052(n))/2. - Omar E. Pol, Jun 23 2009
a(n) = A034178(4*n). - Michael Somos, May 11 2011
2*a(n) = A161841(n). - R. J. Mathar, Mar 07 2021
a(n) = A000005(n) - A056924(n) = A056924(n) + A010052(n) = Sum_{d|n} A065043(d). - Antti Karttunen, Apr 17 2022
Sum_{k=1..n} a(k) ~ n*log(n)/2 + (gamma - 1/2)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 27 2022

A337135 a(1) = 1; for n > 1, a(n) = Sum_{d|n, d <= sqrt(n)} a(d).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Nov 21 2020

nonn

From Gus Wiseman, Mar 05 2021: (Start)
This sequence counts all of the following essentially equivalent things:
1. Chains of distinct inferior divisors from n to 1, where a divisor d|n is inferior if d <= n/d. Inferior divisors are counted by A038548 and listed by A161906.
2. Chains of divisors from n to 1 whose first-quotients (in analogy with first-differences) are term-wise greater than or equal to their decapitation (maximum element removed). For example, the divisor chain q = 60/4/2/1 has first-quotients (15,2,2), which are >= (4,2,1), so q is counted under a(60).
3. Chains of divisors from n to 1 such that x >= y^2 for all adjacent x, y.
4. Factorizations of n where each factor is greater than or equal to the product of all previous factors.
(End)

			From _Gus Wiseman_, Mar 05 2021: (Start)
The a(n) chains for n = 1, 2, 4, 12, 16, 24, 36, 60:
  1  2/1  4/1    12/1    16/1      24/1      36/1      60/1
          4/2/1  12/2/1  16/2/1    24/2/1    36/2/1    60/2/1
                 12/3/1  16/4/1    24/3/1    36/3/1    60/3/1
                         16/4/2/1  24/4/1    36/4/1    60/4/1
                                   24/4/2/1  36/6/1    60/5/1
                                             36/4/2/1  60/6/1
                                             36/6/2/1  60/4/2/1
                                                       60/6/2/1
The a(n) factorizations for n = 2, 4, 12, 16, 24, 36, 60:
    2  4    12   16     24     36     60
       2*2  2*6  2*8    3*8    4*9    2*30
            3*4  4*4    4*6    6*6    3*20
                 2*2*4  2*12   2*18   4*15
                        2*2*6  3*12   5*12
                               2*2*9  6*10
                               2*3*6  2*2*15
                                      2*3*10
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..65536

Cf. A002033, A008578 (positions of 1's), A068108.
The restriction to powers of 2 is A018819.
Not requiring inferiority gives A074206 (ordered factorizations).
The strictly inferior version is A342083.
The strictly superior version is A342084.
The weakly superior version is A342085.
The additive version is A000929, or A342098 forbidding equality.
A000005 counts divisors, with sum A000203.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n-1, with strict case A122651.
A038548 counts inferior (or superior) divisors.
A056924 counts strictly inferior (or strictly superior) divisors.
A067824 counts strict chains of divisors starting with n.
A167865 counts strict chains of divisors > 1 summing to n.
A207375 lists central divisors.
A253249 counts strict chains of divisors.
A334996 counts ordered factorizations by product and length.
A334997 counts chains of divisors of n by length.
A342086 counts strict factorizations of divisors.
- Inferior: A033676, A066839, A072499, A161906.
- Superior: A033677, A070038, A161908.
- Strictly Inferior: A060775, A070039, A333806, A341674.
- Strictly Superior: A048098, A064052, A140271, A238535, A341673.
Cf. A001248, A006530, A020639, A040039, A045690, A337105, A342087, A342094, A342095, A342096, A342097.

a:= proc(n) option remember; `if`(n=1, 1, add(
      `if`(d<=n/d, a(d), 0), d=numtheory[divisors](n)))
    end:
seq(a(n), n=1..128);  # Alois P. Heinz, Jun 24 2021

a[1] = 1; a[n_] := a[n] = DivisorSum[n, a[#] &, # <= Sqrt[n] &]; Table[a[n], {n, 95}]
(* second program *)
asc[n_]:=Prepend[#,n]&/@Prepend[Join@@Table[asc[d],{d,Select[Divisors[n],#Gus Wiseman, Mar 05 2021 *)

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

a(2^n) = A018819(n). - Gus Wiseman, Mar 08 2021

cp's OEIS Frontend

A038548 Number of divisors of n that are at most sqrt(n).

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A337135 a(1) = 1; for n > 1, a(n) = Sum_{d|n, d <= sqrt(n)} a(d).

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A348955 a(1) = 1; a(n) = Sum_{d|n, d <= sqrt(n)} a(d)^2.

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