A027424 -id:A027424 - cp's OEIS frontend

A062854 First differences of A027424.

1, 2, 3, 3, 5, 4, 7, 5, 6, 6, 11, 6, 13, 8, 9, 8, 17, 9, 19, 10, 12, 12, 23, 10, 16, 14, 15, 13, 29, 12, 31, 15, 18, 18, 20, 13, 37, 20, 21, 16, 41, 17, 43, 20, 21, 24, 47, 17, 31, 22, 27, 23, 53, 22, 31, 22, 30, 30, 59, 19, 61, 32, 28, 26, 36, 26, 67, 30, 36, 26, 71, 23, 73, 38

Offset: 1

Ron Lalonde (ronronronlalonde(AT)hotmail.com), Jun 25 2001

nonn

For prime p, a(p) = p. - Ralf Stephan, Jun 02 2005
a(n) is the number of times n appears in A033677. - Franklin T. Adams-Watters, Nov 18 2005
Conjecture: a(n) > n/log(n) for n > 2. - Thomas Ordowski, Jan 28 2017
a(n) is the number of integers 1<=i<=n such that all divisors of i*n are either <=i or >=n. - Chai Wah Wu, Oct 13 2023

			a(4)=3 because there are 9 unique products in the 4 X 4 multiplication table (1 2 3 4 6 8 9 12 16), which is 3 more than the 6 unique products in the 3 X 3 multiplication table (1 2 3 4 6 9).

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A027421, A062854, A062855, A062856, A062857, A062859.

A062854 := proc(n)
    A027424(n)-A027424(n-1) ;
end proc:
seq(A062854(n),n=1..40) ; # R. J. Mathar, Oct 02 2020

Prepend[Differences@ #, First@ #] &@ Module[{ u = {}}, Table[Length[u = Union[u, n Range@ n]], {n, 100}]] (* Michael De Vlieger, Jan 30 2017 *)

b(n) = #setbinop((x, y)->x*y, vector(n, i, i); );
a(n) = b(n) - b(n-1); \\ Michel Marcus, Jan 28 2017

from itertools import takewhile
from sympy import divisors
def A062854(n): return sum(1 for i in range(1,n+1) if all(d<=i for d in takewhile(lambda d:dChai Wah Wu, Oct 13 2023

More terms from Ralf Stephan, Jun 02 2005

A003991 Multiplication table read by antidiagonals: T(i,j) = i*j, i>=1, j>=1.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 6, 6, 4, 5, 8, 9, 8, 5, 6, 10, 12, 12, 10, 6, 7, 12, 15, 16, 15, 12, 7, 8, 14, 18, 20, 20, 18, 14, 8, 9, 16, 21, 24, 25, 24, 21, 16, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12

Offset: 1

Marc LeBrun

Or, triangle X(n,m) = T(n-m+1,m) read by rows, in which row n gives the numbers n*1, (n-1)*2, (n-2)*3, ..., 2*(n-1), 1*n.
Radius of incircle of Pythagorean triangle with sides a=(n+1)^2-m^2, b=2*(n+1)*m and c=(n+1)^2+m^2. - Floor van Lamoen, Aug 16 2001
A permutation of A061017. - Matthew Vandermast, Feb 28 2003
In the proof of countability of rational numbers they are arranged in a square array. a(n) = p*q where p/q is the corresponding rational number as read from the array. - Amarnath Murthy, May 29 2003
Permanent of upper right n X n corner is A000442. - Marc LeBrun, Dec 11 2003
Row 12 gives total number of partridges, turtle doves, ... and drummers drumming that you have received at the end of the Twelve Days of Christmas song. - Alonso del Arte, Jun 17 2005
Consider a particle with spin S (a half-integer) and 2S+1 quantum states |m>, m = -S,-S+1,...,S-1,S. Then the matrix element  = sqrt((S+m+1)(S-m)) of the spin-raising operator is the square-root of the triangular (tabl) element T(r,o) of this sequence in row r = 2S, and at offset o=2(S+m). T(r,o) is also the intensity || of the transition between the states |m> and |m+1>. For example, the five transitions between the 6 states of a spin S=5/2 particle have relative intensities 5,8,9,8,5. The total intensity of all spin 5/2 transitions (relative to spin 1/2) is 35, which is the tetrahedral number A000292(5). - Stanislav Sykora, May 26 2012
Sum_{k=0..2n-2} (-1)^k*a(A000124(2n-2)+k) = n. See A098359. - Charlie Marion, Apr 22 2013
T(n, k) is also the (k-1)-superdiagonal sum of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 1, ..., n. - Stefano Spezia, Jul 12 2019
From Eric Lengyel, Jun 28 2023: (Start)
X(n, m+1) is the number of degrees of freedom that an m-dimensional flat geometry (point, line, plane, etc.) has when embedded in an n-dimensional Euclidean space.
X(n+1, m+1) is the number of degrees of freedom that an m-ball has when embedded in an n-dimensional Euclidean space. (End)
T(n, k) is also the average number of steps it takes a person to fall off a board of length n+k, if the person starts a random walk at k. - Ruediger Jehn, May 12 2025

			The array T starts in row n=1 with columns m>=1 as:
   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
   2   4   6   8  10  12  14  16  18  20  22  24  26  28  30
   3   6   9  12  15  18  21  24  27  30  33  36  39  42  45
   4   8  12  16  20  24  28  32  36  40  44  48  52  56  60
   5  10  15  20  25  30  35  40  45  50  55  60  65  70  75
   6  12  18  24  30  36  42  48  54  60  66  72  78  84  90
   7  14  21  28  35  42  49  56  63  70  77  84  91  98 105
   8  16  24  32  40  48  56  64  72  80  88  96 104 112 120
   9  18  27  36  45  54  63  72  81  90  99 108 117 126 135
  10  20  30  40  50  60  70  80  90 100 110 120 130 140 150
The triangle X(n, m) begins
   n\m  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
   1:   1
   2:   2  2
   3:   3  4  3
   4:   4  6  6  4
   5:   5  8  9  8  5
   6:   6 10 12 12 10  6
   7:   7 12 15 16 15 12  7
   8:   8 14 18 20 20 18 14  8
   9:   9 16 21 24 25 24 21 16  9
  10:  10 18 24 28 30 30 28 24 18 10
  11:  11 20 27 32 35 36 35 32 27 20 11
  12:  12 22 30 36 40 42 42 40 36 30 22 12
  13:  13 24 33 40 45 48 49 48 45 40 33 24 13
  14:  14 26 36 44 50 54 56 56 54 50 44 36 26 14
  15:  15 28 39 48 55 60 63 64 63 60 55 48 39 28 15
  ... Formatted by _Wolfdieter Lang_, Dec 02 2014

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 46.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 5-6.

T. D. Noe, Rows n = 1..100 of triangle, flattened
Iva Kodrnja and Helena Koncul, Number of Polynomials Vanishing on a Basis of S_m(Gamma_0(N)), arXiv:2405.10747 [math.NT], 2024. See p. 10.
G. W. Leibniz, Dissertatio de arte combinatoria, 1666, Leipzig. (in Latin. This triangle appears on p. 208, page 44 of the PDF file).
Abdelkader Necer, Séries formelles et produit de Hadamard, Journal de théorie des nombres de Bordeaux, 9 no. 2 (1997), p. 319-335.
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.

Main diagonal gives squares A000290. Antidiagonal sums are tetrahedral numbers A000292. See A004247 for another version.
Cf. A003989, A003990, A003056, A049581, A000442, A027424, A002260, A033638, A059895, A059896, A059897, A002620.
Cf. also A051776, A067138, A091257, A325821, A331590, A341520, A350066.

/* As triangle */ [[k*(n-k+1): k in [1..n]]: n in [1..15]]; // Vincenzo Librandi, Jul 12 2019

seq(seq(i*(n-i),i=1..n-1),n=2..10); # Robert Israel, Dec 14 2015

Table[(x + 1 - y) y, {x, 13}, {y, x}] // Flatten (* Robert G. Wilson v, Oct 06 2007 *)
f[n_] := Table[SeriesCoefficient[E^(x + y) (1+ x - y +x*y-y^2), {x, 0, i}, {y, 0, j}]*i!*j!, {i, n, n}, {j, 0, n}]; Flatten[Array[f, 11,0]] (* Stefano Spezia, Jul 12 2019 *)

PARI
```
A003991(n,k) = if(k<1 || n<1,0,k*n)
```

Rectangular array: T(n, m) = n*m, n>=1, m>= 1.
Triangle X(n, m) = T(n-m+1, m) = (n-m+1)*m.
Sum_{i=1..n} Sum_{j=1..n} a(n) = A000537(n) [Sum of first n cubes; or n-th triangular number squared.] Determinant of all n X n contiguous subarrays of A003991 is 0. - Gerald McGarvey, Sep 26 2004
G.f. as rectangular array: x*y/((1 - x)^2*(1 - y)^2).
a(n) = i*j, where i=floor((1+sqrt(8n-7))/2), j=n-i*(i-1)/2. - Hieronymus Fischer, Aug 08 2007
As an infinite lower triangular matrix equals A000012 * A002260; where A000012 = (1; 1,1; 1,1,1; ...) and A002260 = (1; 1,2; 1,2,3; ...). - Gary W. Adamson, Oct 23 2007
As a linear array, the sequence is a(n) = A002260(n)*A004736(n) or a(n) = ((t*t+3*t+4)/2-n)*(n-(t*(t+1)/2)), where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 17 2012
G.f. as linear array: (x - 3*x^2 + Sum_{k >= 0} ((k+2-x-(k+1)*x^2)*x^((k^2+3*k+4)/2)))/(1-x)^3. - Robert Israel, Dec 14 2015
E.g.f. as triangle: exp(x+y)*(1 + x - y + x*y - y^2). - Stefano Spezia, Jul 12 2019
a(n) = (1/2)*t + (n - 1/4)*t^2 - (1/4)*t^4 - n^2 + n, where t = floor(sqrt(2*n) + 1/2). - Ridouane Oudra, Nov 21 2020
a(n) = A003989(n) * A003990(n) = A059895(n) * A059896(n) = A059895(n)^2 * A059897(n). - Antti Karttunen, Dec 13 2021
T(n,k) = A002620(n+k) - A002620(n-k). - Michel Marcus, Jan 06 2023
T(n,k) = number of sums |x-y|+|y-z| = k, where x,y,z are in {1,2,...,n} and x < y < z. - Clark Kimberling, Jan 22 2024
E.g.f. as rectangular array: x*y*exp(x+y). - Stefano Spezia, Jun 27 2025

More terms from Michael Somos

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

A027425 Number of distinct products ijk with 1 <= i,j,k <= n.

Original entry on oeis.org

1, 4, 10, 16, 30, 40, 65, 80, 100, 120, 173, 194, 266, 301, 343, 378, 492, 536, 678, 732, 804, 876, 1075, 1130, 1247, 1343, 1450, 1537, 1833, 1909, 2248, 2362, 2515, 2668, 2850, 2940, 3400, 3587, 3789, 3919, 4477, 4624, 5242, 5440, 5654, 5916

Offset: 1

N. J. A. Sloane

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 200 terms from T. D. Noe)

Cf. A027424, A027426, A027430.

import Data.List (nub)
a027425 n = length $ nub [i*j*k | i <- [1..n], j <- [1..n], k <- [1..n]]
-- Reinhard Zumkeller, Jan 01 2012

f:=proc(n) local i,j,k,t1,t2; t1:={}; for i from 1 to n do for j from i to n do for k from j to n do t1:={op(t1),i*j*k}; od: od: od: t1:=convert(t1,list); nops(t1); end;

a[n_] := Reap[Do[Sow[i*j*k], {i, 1, n}, {j, i, n}, {k, j, n}]][[2, 1]] // Union // Length; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Jan 30 2018 *)

pr(n)=my(v=List());for(i=1,n, for(j=i,n, listput(v, i*j))); Set(v)
a(n)=my(v=pr(n),u=v); for(i=2,n,u=Set(concat(u,v*i))); #u \\ Charles R Greathouse IV, Mar 04 2014

def A027425(n): return len({i*j*k for i in range(1,n+1) for j in range(1,i+1) for k in range(1,j+1)}) # Chai Wah Wu, Oct 16 2023

a(n) = A027426(n)-1. - T. D. Noe, Jan 16 2007

A027384 Number of distinct products i*j with 0 <= i, j <= n.

Original entry on oeis.org

1, 2, 4, 7, 10, 15, 19, 26, 31, 37, 43, 54, 60, 73, 81, 90, 98, 115, 124, 143, 153, 165, 177, 200, 210, 226, 240, 255, 268, 297, 309, 340, 355, 373, 391, 411, 424, 461, 481, 502, 518, 559, 576, 619, 639, 660, 684, 731, 748, 779, 801, 828, 851, 904, 926, 957, 979, 1009, 1039

Offset: 0

Fred Schwab (fschwab(AT)nrao.edu)

a(n) = A027420(n,0) = A027420(n,n). - Reinhard Zumkeller, May 02 2014

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
R. P. Brent, C. Pomerance, D. Purdum, and J. Webster, Algorithms for the multiplication table, arXiv:1908.04251 [math.NT], 2019-2021.

Equals A027424 + 1, n>0.

Cf. A027417, A027427, A027425.

import Data.List (nub)
a027384 n = length $ nub [i*j | i <- [0..n], j <- [0..n]]
-- Reinhard Zumkeller, Jan 01 2012

A027384 := proc(n)
    local L,i,j ;
    L := {};
    for i from 0 to n do
        for j from i to n do
            L := L union {i*j};
        end do:
    end do:
    nops(L);
end proc: # R. J. Mathar, May 06 2016

u = {}; Table[u = Union[u, n*Range[0, n]]; Length[u], {n, 0, 100}] (* T. D. Noe, Jan 07 2012 *)

a(n) = {my(s=Set()); for (i=0, n, s = setunion(s, Set(vector(n+1, k, i*(k-1))));); #s;} \\ Michel Marcus, Jan 01 2019

def A027384(n): return len({i*j for i in range(1,n+1) for j in range(1,i+1)})+1 # Chai Wah Wu, Oct 13 2023

For prime p, a(p) = a(p - 1) + p. - David A. Corneth, Jan 01 2019

A062851 Number of k such that 1 < k < n X n and k not of the form ij for 1 <= {i, j} <= n.

Original entry on oeis.org

0, 1, 3, 7, 11, 18, 24, 34, 45, 58, 68, 85, 97, 116, 136, 159, 175, 201, 219, 248, 277, 308, 330, 367, 400, 437, 475, 517, 545, 592, 622, 670, 717, 766, 815, 873, 909, 964, 1020, 1083, 1123, 1189, 1231, 1298, 1366, 1433, 1479, 1557, 1623, 1700, 1774, 1854

Offset: 1

Ron A. Lalonde (ronronronlalonde(AT)hotmail.com), Jun 25 2001

nonn

Smallest k for given n is given by A007918, largest by A005563 (except for some initial terms).

			a(4)=7 because there are 9 unique products in the 4 X 4 multiplication table (1 2 3 4 6 8 9 12 16), which excludes 7 non-product integers within the range 1 to 16 (5 7 10 11 13 14 15).

a(n) = n^2 - A027424.

Cf. A027424, A062854, A062855, A062856, A062857, A062859.

def A062851(n): return n**2-len({i*j for i in range(1,n+1) for j in range(1,i+1)}) # Chai Wah Wu, Oct 13 2023

A062855 First differences of A062851.

Original entry on oeis.org

0, 1, 2, 4, 4, 7, 6, 10, 11, 13, 10, 17, 12, 19, 20, 23, 16, 26, 18, 29, 29, 31, 22, 37, 33, 37, 38, 42, 28, 47, 30, 48, 47, 49, 49, 58, 36, 55, 56, 63, 40, 66, 42, 67, 68, 67, 46, 78, 66, 77, 74, 80, 52, 85, 78, 89, 83

Offset: 1

Ron A. Lalonde (ronronronlalonde(AT)hotmail.com), Jun 25 2001

nonn

			a(4)=4 because there are 7 integer non-products for the 4 X 4 multiplication table (5 7 10 11 13 14 15), which is 4 more than the 3 non-products for the 3 X 3 multiplication table (5 7 8).

Cf. A027424, A062851, A062854, A062856, A062857, A062858, A062859.

from itertools import takewhile
from sympy import divisors
def A062855(n): return (n<<1)-1-sum(1 for i in range(1,n+1) if all(d<=i for d in takewhile(lambda d:dChai Wah Wu, Oct 13 2023

a(n) = 2*n-1-A062854(n). - Chai Wah Wu, Oct 13 2023

A062856 In the square multiplication table of size A062857(n), the smallest number which appears n times.

Original entry on oeis.org

1, 2, 4, 6, 12, 12, 36, 60, 60, 60, 120, 120, 120, 120, 360, 360, 360, 360, 360, 360, 840, 840, 840, 840, 2520, 2520, 2520, 2520, 2520, 2520, 2520, 2520, 2520, 2520, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 10080, 10080

Offset: 1

Ron Lalonde (ronronronlalonde(AT)hotmail.com), Jun 25 2001

nonn

Smallest number to appear n times in any m X m multiplication table.

			a(7)=36 because 36 is the first product to appear in the m X m multiplication tables 7 times as m increases from 1 to infinity.

Cf. A027424, A062851, A062854, A062855, A062857, A062858, A062859.

b[1] = {1, 1}; b[n_] := b[n] = For[m = b[n-1][[1]], True, m++, T = Table[i j, {i, m}, {j, m}] // Flatten // Tally; sel = SelectFirst[T, #[[2]] >= n&]; If[sel != {}, Print[n, " ", m, " ", sel[[1]]]; Return[{m, sel[[1]]}] ]];
a[n_] := b[n][[2]];
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Mar 25 2019 *)

from itertools import count
from collections import Counter
def A062856(n):
    if n == 1: return 1
    c = Counter()
    for m in count(1):
        for i in range(1,m):
            ij = i*m
            c[ij] += 2
            if c[ij]>=n:
                return ij
        c[m*m] = 1 # Chai Wah Wu, Oct 16 2023

More terms from Don Reble, Nov 08 2001

A062857 Size of smallest square multiplication table which contains some number at least n times.

Original entry on oeis.org

1, 2, 4, 6, 12, 12, 18, 20, 30, 30, 40, 40, 60, 60, 72, 72, 90, 90, 120, 120, 140, 140, 168, 168, 180, 180, 210, 210, 252, 252, 280, 280, 315, 315, 336, 336, 360, 360, 420, 420, 504, 504, 560, 560, 630, 630, 672, 672, 720, 720, 792, 792, 840, 840, 924, 924, 990

Offset: 1

Ron Lalonde (ronronronlalonde(AT)hotmail.com), Jun 25 2001

nonn

a(n) is the least number m such that there exists k with 1 <= k <= m^2 such that k has at least n divisors t with k/m <= t <= m. - Robert Israel, Jan 30 2017

			a(7)=18 because the 18 X 18 multiplication table is the smallest to contain a product of frequency 7 (namely the number A062856(7)=36).

The least such number is A062856(n).

Cf. A027424, A062851, A062854, A062855, A062856, A062858, A062859.

N = 1000; % to get all terms with a(n) <= N
M = sparse(1,N^2);
A(1) = 1;
imax = 1;
for k = 2:N
  M(k*[1:k-1]) = M(k*[1:k-1])+2;
  M(k^2) = 1;
  newimax = max(M);
  A(imax+1:newimax) = k;
  imax = newimax;
end
A  % Robert Israel, Jan 30 2017

a[1] = 1; a[n_] := a[n] = For[m = a[n-1], True, m++, T = Table[i j, {i, m}, {j, m}] // Flatten // Tally; sel = SelectFirst[T, #[[2]] >= n&]; If[sel != {}, Print[n, " ", m, " ", sel[[1]]]; Return[m]]];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 25 2019 *)

from itertools import count
from collections import Counter
def A062857(n):
    if n == 1: return 1
    c = Counter()
    for m in count(1):
        for i in range(1,m):
            ij = i*m
            c[ij] += 2
            if c[ij]>=n:
                return m
        c[m*m] = 1 # Chai Wah Wu, Oct 16 2023

More terms from Don Reble, Nov 08 2001

Name clarified by Robert Israel, Jan 30 2017

cp's OEIS Frontend

A062854 First differences of A027424.

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A003991 Multiplication table read by antidiagonals: T(i,j) = i*j, i>=1, j>=1.

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

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A027425 Number of distinct products ijk with 1 <= i,j,k <= n.

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A027384 Number of distinct products i*j with 0 <= i, j <= n.

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A062851 Number of k such that 1 < k < n X n and k not of the form ij for 1 <= {i, j} <= n.

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