A027750 -id:A027750 - cp's OEIS frontend

A174973 Numbers whose divisors increase by a factor of at most 2.

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252, 256

Offset: 1

T. D. Noe, Apr 02 2010

nonn

That is, if the divisors of a number are listed in increasing order, the ratio of adjacent divisors is at most 2. The only odd number in this sequence is 1. Every term appears to be a practical number (A005153). The first practical number not here is 78.
Let p1^e1 * p2^e2 * ... * pr^er be the prime factorization of a number, with primes p1 < p2 < ... < pr and ek > 0. Then the number is in this sequence if and only if pk <= 2*Product_{j < k} p_j^e_j. This condition is similar to, but more restrictive than, the condition for practical numbers.
The polymath8 project led by Terry Tao refers to these numbers as "2-densely divisible". In general they say that n is y-densely divisible if its divisors increase by a factor of y or less, or equivalently, if for every R with 1 <= R <= n, there is a divisor in the interval [R/y,R]. They use this as a weakening of the condition that n be y-smooth. - David S. Metzler, Jul 02 2013
Is this the same as numbers k with the property that the symmetric representation of sigma(k) has only one part? If not, where is the first place these sequences differ? (cf. A237593). - Omar E. Pol, Mar 06 2014
Yes, the sequence so defined is the same as this sequence; see proof in the links. - Hartmut F. W. Hoft, Nov 26 2014
Saias (1997) called these terms "2-dense numbers" and proved that if N(x) is the number of terms not exceeding x, then there are two positive constants c_1 and c_2 such that c_1 * x/log_2(x) <= N(x) <= c_2 * x/log_2(x) for all x >= 2. - Amiram Eldar, Jul 23 2020
Weingartner (2015, 2019) showed that N(x) = c*x/log(x) + O(x/(log(x))^2), where c = 1.224830... As a result, a(n) = C*n*log(n*log(n)) + O(n), where C = 1/c = 0.816439... - Andreas Weingartner, Jun 22 2021

			The divisors of 12 are 1, 2, 3, 4, 6, 12. The ratios of adjacent divisors is 2, 3/2, 4/3, 3/2, and 2, which are all <= 2. Hence 12 is in this sequence.
Example from _Omar E. Pol_, Mar 06 2014: (Start)
    The symmetric representation of sigma(6) = 12 in the first quadrant looks like this:
   y
   .
   ._ _ _ _
   |_ _ _  |_
   .     |   |_
   .     |_ _  |
   .         | |
   .         | |
   . . . . . |_| . . x
.
6 is in the sequence because the symmetric representation of sigma(6) = 12 has only one part. The 6th row of A237593 is [4, 1, 1, 1, 1, 4] and the 5th row of A237593 is [3, 2, 2, 3] therefore between both symmetric Dyck paths there is only one region (or part) of size 12.
    70 is not in the sequence because the symmetric representation of sigma(70) = 144 has three parts. The 70th row of A237593 is [36, 12, 6, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 6, 12, 36] and the 69th row of A237593 is [35, 12, 7, 4, 2, 3, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 4, 7, 12, 35] therefore between both symmetric Dyck paths there are three regions (or parts) of size [54, 36, 54]. (End)

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
José Manuel Rodríguez Caballero, Jordan's Expansion of the Reciprocal of Theta Functions and 2-densely Divisible Numbers, Integers, Vol. 20 (2020), Article A2.
Hartmut F. W. Hoft, Proof of a conjecture.
Hartmut F. W. Hoft, Proof of a second conjecture.
Eric Saias, Entiers à diviseurs denses 1, Journal of Number Theory, Vol. 62, No. 1 (1997), pp. 163-191.
Terence Tao, A Truncated Elementary Selberg Sieve of Pintz. (blog entry defining y-densely divisible)
Terence Tao et al., Polymath8 home page.
Andreas Weingartner, Practical numbers and the distribution of divisors, The Quarterly Journal of Mathematics, Vol. 66, No. 2 (2015), pp. 743-758; arXiv preprint, arXiv:1405.2585 [math.NT], 2014-2015.
Andreas Weingartner, On the constant factor in several related asymptotic estimates, Math. Comp., Vol. 88, No. 318 (2019), pp. 1883-1902; arXiv preprint, arXiv:1705.06349 [math.NT], 2017-2018.
Andreas Weingartner, The number of prime factors of integers with dense divisors, arXiv:2101.11585 [math.NT], 2021.
Andreas Weingartner, Uniform distribution of alpha*n modulo one for a family of integer sequences, arXiv:2303.16819 [math.NT], 2023.

Subsequence of A196149 and of A071562. A000396 and A000079 are subsequences.
Cf. A027750, A047836, A237593, A365429 (characteristic function).
Column 1 of A240062.
First differs from A103288 and A125225 at a(23). First differs from A005153 at a(24).

a174973 n = a174973_list !! (n-1)
a174973_list = filter f [1..] where
   f n = all (<= 0) $ zipWith (-) (tail divs) (map (* 2) divs)
         where divs = a027750_row' n
-- Reinhard Zumkeller, Jun 25 2015, Sep 28 2011

[k:k in [1..260]|forall{i:i in [1..#Divisors(k)-1]|d[i+1]/d[i] le 2 where d is Divisors(k)}]; // Marius A. Burtea, Jan 09 2020

a:= proc() option remember; local k; for k from 1+`if`(n=1, 0,
      a(n-1)) while (l-> ormap(x-> x, [seq(l[i]>l[i-1]*2, i=2..
      nops(l))]))(sort([(numtheory[divisors](k))[]])) do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 27 2018

OK[n_] := Module[{d=Divisors[n]}, And@@(#<=2& /@ (Rest[d]/Most[d]))]; Select[Range[1000], OK]
dif2Q[n_]:=AllTrue[#[[2]]/#[[1]]&/@Partition[Divisors[n],2,1],#<=2&]; Select[Range[300],dif2Q] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 29 2020 *)

is(n)=my(d=divisors(n));for(i=2,#d,if(d[i]>2*d[i-1],return(0)));1 \\ Charles R Greathouse IV, Jul 06 2013

from sympy import divisors
def ok(n):
    d = divisors(n)
    return all(d[i]/d[i-1] <= 2 for i in range(1, len(d)))
print(list(filter(ok, range(1, 257)))) # Michael S. Branicky, Jun 22 2021

a(n) = A047836(n) / 2. - Reinhard Zumkeller, Sep 28 2011

a(n) = C*n*log(n*log(n)) + O(n), where C = 0.816439... (see comments). - Andreas Weingartner, Jun 23 2021

Edited by N. J. A. Sloane, Sep 09 2023

Edited by Peter Munn, Oct 17 2023

A077609 Triangle in which n-th row lists infinitary divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 9, 1, 2, 5, 10, 1, 11, 1, 3, 4, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 16, 1, 17, 1, 2, 9, 18, 1, 19, 1, 4, 5, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 4, 6, 8, 12, 24, 1, 25, 1, 2, 13, 26, 1, 3, 9, 27, 1, 4, 7, 28, 1, 29, 1

Offset: 1

Eric W. Weisstein, Nov 11 2002

The first difference from the triangle A222266 (bi-unitary divisors of n) is in row n = 16; indeed, the 16th row of A222266 is (1, 2, 8, 16) while the 16th of this sequence here is (1, 16). - Bernard Schott, Mar 10 2023

The concept of infinitary divisors was introduced by Cohen (1990). - Amiram Eldar, Mar 09 2024

			The first few rows are:
  1;
  1, 2;
  1, 3;
  1, 4;
  1, 5;
  1, 2, 3, 6;
  1, 7;
  1, 2, 4, 8;
  1, 9;
  1, 2, 5, 10;
  1, 11;
  1, 3, 4, 12;
  1, 13;
  1, 2, 7, 14;
  1, 3, 5, 15;
  1, 16;
  1, 17;

Reinhard Zumkeller, Rows n = 1..1000 of table, flattened
Graeme L. Cohen, On an integer's infinitary divisors, Mathematics of Computation, Vol. 54, No. 189 (1990), pp. 395-411.
Graeme L. Cohen and Peter Hagis, Jr., Arithmetic functions associated with the infinitary divisors of an integer, Internat. J. Math. Math. Sci. 16 (2) (1993) 373-384.
Eric Weisstein's World of Mathematics, Infinitary Divisor.

Cf. A027750, A037445 (row lengths), A049417 (row sums).

Cf. A222266.

import Data.List ((\\))
a077609 n k = a077609_row n !! (k-1)
a077609_row n = filter
   (\d -> d == 1 || null (a213925_row d \\ a213925_row n)) $ a027750_row n
a077609_tabf = map a077609_row [1..]
-- Reinhard Zumkeller, Jul 10 2013

# see the function idivisors() in A049417. # R. J. Mathar, Oct 05 2017

f[x_] := If[x == 1, 1, Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[x] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]] ; Array[f, 30] // Flatten (* Paul Abbott (paul(AT)physics.uwa.edu.au), Apr 29 2005 *) (* edited by Michael De Vlieger, Jun 07 2016 *)

isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k,2]); bde = binary(valuation(d, f[k,1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)););); return (1);}
row(n) = {d = divisors(n); f = factor(n); idiv = []; for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k]));); idiv;} \\ Michel Marcus, Feb 15 2016

A051377 a(1)=1; for n > 1, a(n) = sum of exponential divisors (or e-divisors) of n.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 10, 12, 10, 11, 18, 13, 14, 15, 22, 17, 24, 19, 30, 21, 22, 23, 30, 30, 26, 30, 42, 29, 30, 31, 34, 33, 34, 35, 72, 37, 38, 39, 50, 41, 42, 43, 66, 60, 46, 47, 66, 56, 60, 51, 78, 53, 60, 55, 70, 57, 58, 59, 90, 61, 62, 84, 78, 65, 66, 67, 102, 69, 70, 71

Offset: 1

Yasutoshi Kohmoto

The e-divisors (or exponential divisors) of x=Product p(i)^r(i) are all numbers of the form Product p(i)^s(i) where s(i) divides r(i) for all i.

a(n) = n if and only if n is squarefree. - Jon Perry, Nov 13 2012

			a(8)=10 because 2 and 2^3 are e-divisors of 8 and 2+2^3=10.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
J. Fabrykowski and M. V. Subbarao, The maximal order and the average order of multiplicative function sigma^(e)(n), in Théorie des nombres/Number theory (Quebec, PQ, 1987), 201-206, de Gruyter, Berlin, 1989.
Nicussor Minculete, Concerning some arithmetic functions which use exponential divisors, Acta Universitatis Apulensis, No. 28/2011, pp. 125-133.
Y.-F. S. Pétermann and J. Wu, On the sum of exponential divisors of an integer, Acta Math. Hungar. 77 (1997), 159-175.
Eric Weisstein's World of Mathematics, e-Divisor

Cf. A051378, A049419 (number of e-divisors).
Cf. A027748, A124010, A027750.
Row sums of A322791.
See A307042 and A275480 where the formula and constant appear.

A051377:=n->Product(List(Collected(Factors(n)), p -> Sum(DivisorsInt(p[2]),d->p[1]^d))); List([1..10^4], n -> A051377(n)); # Muniru A Asiru, Oct 29 2017

a051377 n = product $ zipWith sum_e (a027748_row n) (a124010_row n) where
   sum_e p e = sum [p ^ d | d <- a027750_row e]
-- Reinhard Zumkeller, Mar 13 2012

A051377 := proc(n)
    local a,pe,p,e;
    a := 1;
    for pe in ifactors(n)[2] do
        p := pe[1] ;
        e := pe[2] ;
        add(p^d,d=numtheory[divisors](e)) ;
        a := a*% ;
    end do:
    a ;
end proc:
seq(A051377(n),n=1..100) ; # R. J. Mathar, Oct 05 2017

a[n_] := Times @@ (Sum[ First[#]^d, {d, Divisors[Last[#]]}] & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 71}] (* Jean-François Alcover, Apr 06 2012 *)

a(n)=my(f=factor(n));prod(i=1,#f[,1],sumdiv(f[i,2],d,f[i,1]^d)) \\ Charles R Greathouse IV, Nov 22 2011

ediv(n,f=factor(n))=my(v=List(),D=apply(divisors,f[,2]~),t=#f~); forvec(u=vector(t,i,[1,#D[i]]), listput(v,prod(j=1,t,f[j,1]^D[j][u[j]]))); Set(v)
a(n)=vecsum(ediv(n)) \\ Charles R Greathouse IV, Oct 29 2018

Multiplicative with a(p^e) = Sum_{d|e} p^d. - Vladeta Jovovic, Apr 23 2002
a(n) = A126164(n)+n. - R. J. Mathar, Oct 05 2017
The average order of a(n) is Dn + O(n^e) for any e > 0, due to Fabrykowski & Subbarao, where D is about 0.568. (D >= 0.5 since a(n) >= n.) - Charles R Greathouse IV, Sep 22 2023

More terms from Jud McCranie, May 29 2000

Definition corrected by Jaroslav Krizek, Feb 27 2009

A001692 Number of irreducible polynomials of degree n over GF(5); dimensions of free Lie algebras.

Original entry on oeis.org

1, 5, 10, 40, 150, 624, 2580, 11160, 48750, 217000, 976248, 4438920, 20343700, 93900240, 435959820, 2034504992, 9536718750, 44878791360, 211927516500, 1003867701480, 4768371093720, 22706531339280

Offset: 0

N. J. A. Sloane

Exponents in expansion of Hardy-Littlewood constant C_5 = 0.409874885.. = A269843 as a product_{n>=2} zeta(n)^(-a(n)).

Number of aperiodic necklaces with n beads of 5 colors. - Herbert Kociemba, Nov 25 2016

E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003
M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..1435 (terms 0..200 from T. D. Noe)
Alin Bostan, Alexander Marynych, and Kilian Raschel, On the least common multiple of several random~integers, arXiv:1901.03002 [math.PR], 2019.
Jeremie Detrey, P. J. Spaenlehauer, and P. Zimmermann, Computing the rho constant, Preprint 2016.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
Gareth A. Jones and Alexander K. Zvonkin, Groups of prime degree and the Bateman-Horn conjecture, 2021.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Dionisel Y. Regalado and Rodel Azura, An Analytic Approximation to the Density of Twin Primes, Recoletos Multidisciplinary Research Journal (2019) Vol. 6, No. 2.
G. J. Simmons, The number of irreducible polynomials of degree n over GF(p), Amer. Math. Monthly, 77 (1970), 743-745.
G. Viennot, Algèbres de Lie Libres et Monoïdes Libres, Lecture Notes in Mathematics 691, Springer Verlag 1978.
Index entries for sequences related to Lyndon words

Cf. A001037, A054720, A002105.
5th column of A074650. - Alois P. Heinz, Aug 08 2008
Cf. A008683, A000351, A027750.

a001692 n = flip div n $ sum $
            zipWith (*) (map a008683 divs) (map a000351 $ reverse divs)
            where divs = a027750_row n
-- Reinhard Zumkeller, Oct 07 2015

a[0] = 1; a[n_] := Sum[MoebiusMu[d]*5^(n/d)/n, {d, Divisors[n]}]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Mar 11 2014 *)
mx=40;f[x_,k_]:=1-Sum[MoebiusMu[i] Log[1-k*x^i]/i,{i,1,mx}];CoefficientList[Series[f[x,5],{x,0,mx}],x] (* Herbert Kociemba, Nov 25 2016 *)

a(n)=if(n,sumdiv(n,d,moebius(d)*5^(n/d))/n,1) \\ Charles R Greathouse IV, Jun 15 2011

a(n) = Sum_{d|n} mu(d)*5^(n/d)/n, for n>0.

G.f.: k=5, 1 - Sum_{i>=1} mu(i)*log(1 - k*x^i)/i. - Herbert Kociemba, Nov 25 2016

A034699 Largest prime power factor of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 4, 13, 7, 5, 16, 17, 9, 19, 5, 7, 11, 23, 8, 25, 13, 27, 7, 29, 5, 31, 32, 11, 17, 7, 9, 37, 19, 13, 8, 41, 7, 43, 11, 9, 23, 47, 16, 49, 25, 17, 13, 53, 27, 11, 8, 19, 29, 59, 5, 61, 31, 9, 64, 13, 11, 67, 17, 23, 7, 71, 9, 73, 37, 25, 19, 11, 13, 79

Offset: 1

David W. Wilson

n divides lcm(1, 2, ..., a(n)).
a(n) = A210208(n,A073093(n)) = largest term of n-th row in A210208. - Reinhard Zumkeller, Mar 18 2012
a(n) = smallest m > 0 such that n divides A003418(m). - Thomas Ordowski, Nov 15 2013
a(n) = n when n is a prime power (A000961). - Michel Marcus, Dec 03 2013
Conjecture: For all n between two consecutive prime numbers, all a(n) are different. - I. V. Serov, Jun 19 2019
Disproved with between p=prime(574) = 4177 and prime(575) = 4201, a(4180) = a(4199) = 19. See A308752. - Michel Marcus, Jun 19 2019
Conjecture: For any N > 0, there exist numbers n and m, N < n < n+a(n) <= m, such that all n..m are composite and a(n) = a(m). - I. V. Serov, Jun 21 2019
Conjecture: For all n between two consecutive prime numbers, all (-1)^n*a(n) are different. Checked up to 5*10^7. - I. V. Serov, Jun 23 2019
Disproved: between p = prime(460269635) = 10120168277 and p = prime(460269636) = 10120168507 the numbers n = 10120168284 and m = 10120168498 form a pair such that (-1)^n*a(n) = (-1)^m*a(m) = 107. - L. Joris Perrenet, Jan 05 2020
a(n) = cardinality of smallest set on which idempotence of order n+1 (f^{n+1} = f) differs from idempotence of order e for 2 <= e <= n (see von Eitzen link for proof); derivable from A245501. - Mark Bowron, May 22 2025

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from T. D. Noe)
Hagen von Eitzen, Least k to distinguish f^n = f on {1,...,k} from f^e = f for every 1 < e < n is largest prime power factor of n-1?, Math Stack Exchange.

Cf. A006530, A010055, A020639, A027750, A034684, A028233, A051283, A052128, A053585, A057110, A060818, A038610, A081805, A088387, A088388, A100574, A210208, A245501, A284600, A308752.

Numbers n ordered by a(n): A305325.

a034699 = last . a210208_row
-- Reinhard Zumkeller, Mar 18 2012, Feb 14 2012

f[n_] := If[n == 1, 1, Max[ #[[1]]^#[[2]] & /@ FactorInteger@n]]; Array[f, 79] (* Robert G. Wilson v, Sep 02 2006 *)
Array[Max[Power @@@ FactorInteger@ #] &, 79] (* Michael De Vlieger, Jul 26 2018 *)

a(n) = if(1==n,n,my(f=factor(n)); vecmax(vector(#f[, 1], i, f[i, 1]^f[i, 2]))); \\ Charles R Greathouse IV, Nov 20 2012, check for a(1) added by Antti Karttunen, Aug 06 2018

A034699(n) = if(1==n,n,fordiv(n, d, if(isprimepower(n/d), return(n/d)))); \\ Antti Karttunen, Aug 06 2018

from sympy import factorint
def A034699(n): return max((p**e for p, e in factorint(n).items()), default=1) # Chai Wah Wu, Apr 17 2023

If n = p_1^e_1 *...* p_k^e_k, p_1 < ... < p_k primes, then a(n) = Max_i p_i^e_i.
a(n) = A088387(n)^A088388(n). - Antti Karttunen, Jul 22 2018
a(n) = n/A284600(n) = n - A081805(n) = A034684(n) + A100574(n). - Antti Karttunen, Aug 06 2018
a(n) = a(m) iff m = d*a(n), where d is a divisor of A038610(a(n)). - I. V. Serov, Jun 19 2019

A162306 Irregular triangle in which row n contains the numbers <= n whose prime factors are a subset of prime factors of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 4, 6, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 4, 5, 8, 10, 1, 11, 1, 2, 3, 4, 6, 8, 9, 12, 1, 13, 1, 2, 4, 7, 8, 14, 1, 3, 5, 9, 15, 1, 2, 4, 8, 16, 1, 17, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 1, 19, 1, 2, 4, 5, 8, 10, 16, 20, 1, 3, 7, 9, 21, 1, 2, 4, 8, 11, 16, 22, 1, 23

Offset: 1

T. D. Noe, Jun 30 2009

Row n begins with 1, ends with n, and has A010846(n) terms.
From Michael De Vlieger, Jul 08 2014: (Start)
Prime p has {1, p} and A010846(p) = 2.
Prime power p^e has {1, p, ..., p^e} and A010846(p^e) = A000005(p^e) = e + 1.
Composite c that are not prime powers have A010846(c) = A000005(c) + A243822(c), where A243822(c) is nonzero positive, since the minimum prime divisor p of c produces at least one semidivisor (e.g., p^2 < c). Thus these have the set of divisors of c and at least one semidivisor p^2. For squareful c that are not prime powers, p^2 may divide c, but p^3 does not. The minimum squareful c = 12, 2^3 does not divide 12 yet is less than 12 and is a product of the minimum prime divisor of 12. All other even squareful c admit a power of 2 that does not divide c, since there must be another prime divisor q > 2. (End)
Numbers 1 <= k <= n such that (floor(n^k/k) - floor((n^k - 1)/k)) = 1. - Michael De Vlieger, May 26 2016
Numbers 1 <= k <= n such that k | n^e with e >= 0. - Michael De Vlieger, May 29 2018

			n =  6: {1, 2, 3, 4, 6}.
n =  7: {1, 7}.
n =  8: {1, 2, 4, 8}.
n =  9: {1, 3, 9}.
n = 10: {1, 2, 4, 5, 8, 10}.
n = 11: {1, 11}.
n = 12: {1, 2, 3, 4, 6, 8, 9, 12}.

T. D. Noe and Michael De Vlieger, Rows n = 1..1000 of triangle, flattened (first rows n=1..200 from T. D. Noe)
Michael De Vlieger, Plot k in row n at (x,y) = (k,-n) for n = 1..2^10.

Cf. A007947, A010846 (number of terms in row n), A027750 (terms k that divide n), A243103 (product of terms in row n), A244974 (sum of terms in row n), A272618 (terms k that do not divide n).

A:= proc(n) local F, S, s, j, p;
  F:= numtheory:-factorset(n);
  S:= {1};
  for p in F do
    S:= {seq(seq(s*p^j, j=0..floor(log[p](n/s))), s=S)}
  od;
  S
end proc; map(op,[seq(A(n), n=1..100)]); # Robert Israel, Jul 15 2014

pf[n_] := If[n==1, {}, Transpose[FactorInteger[n]][[1]]]; SubsetQ[lst1_, lst2_] := Intersection[lst1,lst2]==lst1; Flatten[Table[pfn=pf[n]; Select[Range[n], SubsetQ[pf[ # ],pfn] &], {n,27}]]
(* Second program: *)
f[x_, y_ : 0] :=
  Block[{m, n, nn, j, k, p, t, v, z},
    n = Abs[x]; nn = If[y == 0, n, y];
    If[n == 1, {1},
      z = Length@
        MapIndexed[Set[{p[#2], m[#2]}, {#1, 0}] & @@
        {#1, First[#2]} &, FactorInteger[n][[All, 1]] ];
    k = Times @@ Array[p[#]^m[#] &, z]; Set[{v, t}, {1, False}];
    Union@ Reap[Do[Set[t, k > nn];
      If[t, k /= p[v]^m[v]; m[v] = 0; v++; If[v > z, Break[]],
      v = 1; Sow[k] ]; m[v]++; k *= p[v], {i, Infinity}] ][[-1, 1]] ] ];
Array[f, 120] (* Michael De Vlieger, Jun 18 2024 *)

Union of A027750 and nonzero terms of A272618.

Row n of this sequence is {k <= n : rad(k) | n }, where rad = A007947. - Michael De Vlieger, Jun 18 2024

A183063 Number of even divisors of n.

Original entry on oeis.org

0, 1, 0, 2, 0, 2, 0, 3, 0, 2, 0, 4, 0, 2, 0, 4, 0, 3, 0, 4, 0, 2, 0, 6, 0, 2, 0, 4, 0, 4, 0, 5, 0, 2, 0, 6, 0, 2, 0, 6, 0, 4, 0, 4, 0, 2, 0, 8, 0, 3, 0, 4, 0, 4, 0, 6, 0, 2, 0, 8, 0, 2, 0, 6, 0, 4, 0, 4, 0, 4, 0, 9, 0, 2, 0, 4, 0, 4, 0, 8, 0, 2, 0, 8, 0, 2

Offset: 1

Jaroslav Krizek, Dec 22 2010

Number of divisors of n that are divisible by 2. More generally, it appears that the sequence formed by starting with an initial set of k-1 zeros followed by the members of A000005, with k-1 zeros between every one of them, can be defined as "the number of divisors of n that are divisible by k", (k >= 1). For example if k = 1 we have A000005 by definition, if k = 2 we have this sequence. Note that if k >= 3 the sequences are not included in the OEIS because the usual OEIS policy is not to include sequences like this where alternate terms are zero; this is an exception. - Omar E. Pol, Oct 18 2011
Number of zeros in n-th row of triangle A247795. - Reinhard Zumkeller, Sep 28 2014
a(n) is also the number of partitions of n into equal parts, minus the number of partitions of n into consecutive parts. - Omar E. Pol, May 04 2017
a(n) is also the number of partitions of n into an even number of equal parts. - Omar E. Pol, May 14 2017

			For n = 12, set of even divisors is {2, 4, 6, 12}, so a(12) = 4.
On the other hand, there are six partitions of 12 into equal parts: [12], [6, 6], [4, 4, 4], [3, 3, 3, 3], [2, 2, 2, 2, 2, 2] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]. And there are two partitions of 12 into consecutive parts: [12] and [5, 4, 3], so a(12) = 6 - 2 = 4, equaling the number of even divisors of 12. - _Omar E. Pol_, May 04 2017

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function tau_e(n).

Cf. A000005, A000265, A001227, A007814, A000593, A183064, A136655, A125911, A320111.
Column 2 of A195050. - Omar E. Pol, Oct 19 2011
Cf. A001620, A027750, A083910, A244009.
Cf. A247795.

a183063 = sum . map (1 -) . a247795_row
-- Reinhard Zumkeller, Sep 28 2014, Jan 15 2013, Jan 10 2012

[IsOdd(n) select 0 else #[d:d in Divisors(n)|IsEven(d)]:n in [1..100]]; // Marius A. Burtea, Dec 16 2019

A183063 := proc(n)
    if type(n,'even') then
        numtheory[tau](n/2) ;
    else
        0;
    end if;
end proc: # R. J. Mathar, Jun 18 2015

Table[Length[Select[Divisors[n], EvenQ]], {n, 90}] (* Alonso del Arte, Jan 10 2012 *)
a[n_] := (e = IntegerExponent[n, 2]) * DivisorSigma[0, n / 2^e]; Array[a, 100] (* Amiram Eldar, Jul 06 2022 *)

a(n)=if(n%2,0,numdiv(n/2)) \\ Charles R Greathouse IV, Jul 29 2011

from sympy import divisor_count
def A183063(n): return divisor_count(n>>(m:=(~n&n-1).bit_length()))*m # Chai Wah Wu, Jul 16 2022

def A183063(n): return len([1 for d in divisors(n) if is_even(d)])
[A183063(n) for n in (1..80)]  # Peter Luschny, Feb 01 2012

a(n) = A000005(n) - A001227(n).
a(2n-1) = 0; a(2n) = A000005(n).
G.f.: Sum_{d>=1} x^(2*d)/(1 - x^(2*d)) and generally for the number of divisors that are divisible by k: Sum_{d>=1} x^(k*d)/(1 - x^(k*d)). - Geoffrey Critzer, Apr 15 2014
Dirichlet g.f.: zeta(s)^2/2^s and generally for the number of divisors that are divisible by k: zeta(s)^2/k^s. - Geoffrey Critzer, Mar 28 2015
From Ridouane Oudra, Sep 02 2019: (Start)
a(n) = Sum_{i=1..n} (floor(n/(2*i)) - floor((n-1)/(2*i))).
a(n) = 2*A000005(n) - A000005(2n). (End)
Conjecture: a(n) = lim_{x->n} f(Pi*x), where f(x) = sin(x)*Sum_{k>0} (cot(x/(2*k))/(2*k) - 1/x). - Velin Yanev, Dec 16 2019
a(n) = A000005(A000265(n))*A007814(n) - Chai Wah Wu, Jul 16 2022
Sum_{k=1..n} a(k) ~ n*log(n)/2 + (gamma - 1/2 - log(2)/2)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 27 2022
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A000005(k) = 1-log(2) (A244009). - Amiram Eldar, Mar 01 2023

Formula corrected by Charles R Greathouse IV, Jul 29 2011

A007862 Number of triangular numbers that divide n.

Original entry on oeis.org

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

Offset: 1

Richard Stanley

nonn

Also a(n) is the total number of ways to represent n+1 as a centered polygonal number of the form km(m+1)/2+1 for k>1. - Alexander Adamchuk, Apr 26 2007
Number of oblong numbers that divide 2n. - Ray Chandler, Jun 24 2008
The number of divisors d of 2n such that d+1 is also a divisor of 2n, see first formula. - Michel Marcus, Jun 18 2015
From Gus Wiseman, May 03 2019: (Start)
Also the number of integer partitions of n forming a finite arithmetic progression with offset 0, i.e. the differences are all equal (with the last part taken to be 0). The Heinz numbers of these partitions are given by A325327. For example, the a(1) = 1 through a(12) = 3 partitions are (A = 10, B = 11, C = 12):
  1    2    3     4    5    6      7    8    9     A       B     C
            21              42               63    4321          84
                            321                                  642
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Centered Polygonal Number.
Wikipedia, Arithmetic progression.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A046951, A130317.
Cf. A010054, A027750, A000005, A239930.
Cf. A000217, A007294, A049988, A325324, A325327, A325407.

a007862 = sum . map a010054 . a027750_row
-- Reinhard Zumkeller, Jul 05 2014

sup=90; TriN=Array[ (#+1)(#+2)/2&, Floor[ N[ Sqrt[ sup*2 ] ] ]-1 ]; Array[ Function[n, 1+Count[ Map[ Mod[ n, # ]&, TriN ], 0 ] ], sup ]
Table[Count[Divisors[k], ?(IntegerQ[Sqrt[8 # + 1]] &)], {k, 105}] (* _Jayanta Basu, Aug 12 2013 *)
Table[Length[Select[IntegerPartitions[n],SameQ@@Differences[Append[#,0]]&]],{n,0,30}] (* Gus Wiseman, May 03 2019 *)

a(n) = sumdiv(n, d, ispolygonal(d, 3)); \\ Michel Marcus, Jun 18 2015

from itertools import pairwise
from sympy import divisors
def A007862(n): return sum(1 for a, b in pairwise(divisors(n<<1)) if a+1==b)  # Chai Wah Wu, Jun 09 2025

a(n) = Sum_{d|2*n,d+1|2*n} 1.
G.f.: Sum_{k>=1} x^A000217(k)/(1-x^A000217(k)). - Jon Perry, Jul 03 2004
a(A130317(n)) = n and a(m) <> n for m < A130317(n). - Reinhard Zumkeller, May 23 2007
a(n) = A129308(2n). - Ray Chandler, Jun 24 2008
a(n) = Sum_{k=1..A000005(n)} A010054(A027750(n,k)). - Reinhard Zumkeller, Jul 05 2014
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - Amiram Eldar, Dec 31 2023

Extended by Ray Chandler, Jun 24 2008

A027751 Irregular triangle read by rows in which row n lists the proper divisors of n (those divisors of n which are < n), with the first row {1} by convention.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 1, 3, 1, 2, 5, 1, 1, 2, 3, 4, 6, 1, 1, 2, 7, 1, 3, 5, 1, 2, 4, 8, 1, 1, 2, 3, 6, 9, 1, 1, 2, 4, 5, 10, 1, 3, 7, 1, 2, 11, 1, 1, 2, 3, 4, 6, 8, 12, 1, 5, 1, 2, 13, 1, 3, 9, 1, 2, 4, 7, 14, 1, 1, 2, 3, 5, 6, 10, 15, 1, 1, 2, 4, 8, 16, 1, 3, 11, 1, 2, 17, 1, 5, 7, 1, 2, 3, 4, 6, 9, 12, 18

Offset: 1

N. J. A. Sloane

Or, take the list 1,2,3,4,... of natural numbers (A000027) and replace each number by its proper divisors.

The row length is 1 for n = 1 and A032741(n) for n >= 2. - Wolfdieter Lang, Jan 16 2016

			The irregular triangle T(n,k) begins:
n\k  1 2 3 4  5 ...
1:   1  (by convention)
2:   1
3:   1
4:   1 2
5:   1
6:   1 2 3
7:   1
8:   1 2 4
9:   1 3
10:  1 2 5
11:  1
12:  1 2 3 4  6
13:  1
14:  1 2 7
15:  1 3 5
16:  1 2 4 8
17:  1
18:  1 2 3 6  9
19:  1
20:  1 2 4 5 10
.... reformatted - _Wolfdieter Lang_, Jan 16 2016

Alois P. Heinz, Rows n = 1..1540, flattened

Cf. A027750, A032741 (row lengths), A001065, A000005.

Row sums give A173455. - Omar E. Pol, Nov 23 2010

a027751 n k = a027751_tabf !! (n-1) !! (k-1)
a027751_row n = a027751_tabf !! (n-1)
a027751_tabf = [1] : map init (tail a027750_tabf)
-- Reinhard Zumkeller, Apr 18 2012

with(numtheory):
T:= n-> sort([(divisors(n) minus {n})[]])[]: T(1):=1:
seq(T(n), n=1..50); # Alois P. Heinz, Apr 11 2012

Table[ Divisors[n] // Most, {n, 1, 36}] // Flatten // Prepend[#, 1] & (* Jean-François Alcover, Jun 10 2013 *)

row(n) = if (n==1, [1], my(d = divisors(n)); vector(#d-1,k, d[k])); \\ Michel Marcus, Apr 30 2017

from sympy import divisors
def a(n): return [1] if n==1 else divisors(n)[:-1]
for n in range(21): print(a(n)) # Indranil Ghosh, Apr 30 2017

More terms from Patrick De Geest, May 15 1998

Example edited by Omar E. Pol, Nov 23 2010

A127093 Triangle read by rows: T(n,k)=k if k is a divisor of n; otherwise, T(n,k)=0 (1 <= k <= n).

Original entry on oeis.org

1, 1, 2, 1, 0, 3, 1, 2, 0, 4, 1, 0, 0, 0, 5, 1, 2, 3, 0, 0, 6, 1, 0, 0, 0, 0, 0, 7, 1, 2, 0, 4, 0, 0, 0, 8, 1, 0, 3, 0, 0, 0, 0, 0, 9, 1, 2, 0, 0, 5, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 1, 2, 3, 4, 0, 6, 0, 0, 0, 0, 0, 12, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 1, 2, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 14

Offset: 1

Gary W. Adamson, Jan 05 2007, Apr 04 2007

Sum of terms in row n = sigma(n) (sum of divisors of n).
Euler's derivation of A127093 in polynomial form is in his proof of the formula for Sigma(n): (let S=Sigma, then Euler proved that S(n) = S(n-1) + S(n-2) - S(n-5) - S(n-7) + S(n-12) + S(n-15) - S(n-22) - S(n-26), ...).
[Young, pp. 365-366], Euler begins, s = (1-x)*(1-x^2)*(1-x^3)*... = 1 - x - x^2 + x^5 + x^7 - x^12 ...; log s = log(1-x) + log(1-x^2) + log(1-x^3) ...; differentiating and then changing signs, Euler has t = x/(1-x) + 2x^2/(1-x^2) + 3x^3/(1-x^3) + 4x^4/(1-x^4) + 5x^5/(1-x^5) + ...
Finally, Euler expands each term of t into a geometric series, getting A127093 in polynomial form: t =
  x +  x^2 +  x^3 +  x^4 +  x^5 +  x^6 +  x^7 +  x^8 + ...
    + 2x^2        + 2x^4        + 2x^6        + 2x^8 + ...
           + 3x^3               + 3x^6               + ...
                  + 4x^4                      + 4x^8 + ...
                         + 5x^5                      + ...
                                + 6x^6               + ...
                                       + 7x^7        + ...
                                              + 8x^8 + ...
T(n,k) is the sum of all the k-th roots of unity each raised to the n-th power. - Geoffrey Critzer, Jan 02 2016
From Davis Smith, Mar 11 2019: (Start)
For n > 1, A020639(n) is the leftmost term, other than 0 or 1, in the n-th row of this array. As mentioned in the Formula section, the k-th column is period k: repeat [k, 0, 0, ..., 0], but this also means that it's the characteristic function of the multiples of k multiplied by k. T(n,1) = A000012(n), T(n,2) = 2*A059841(n), T(n,3) = 3*A079978(n), T(n,4) = 4*A121262(n), T(n,5) = 5*A079998(n), and so on.
The terms in the n-th row, other than 0, are the factors of n. If n > 1 and for every k, 1 <= k < n, T(n,k) = 0 or 1, then n is prime. (End)
From Gary W. Adamson, Aug 07 2019: (Start)
Row terms of the triangle can be used to calculate E(n) in A002654): (1, 1, 0, 1, 2, 0, 0, 1, 1, 2, ...), and A004018, the number of points in a square lattice on the circle of radius sqrt(n), A004018: (1, 4, 4, 0, 4, 8, 0, 0, 4, ...).
As to row terms in the triangle, let E(n) of even terms = 0,
  E(integers of the form 4*k - 1 = (-1), and E(integers of the form 4*k + 1 = 1.
Then E(n) is the sum of the E(n)'s of the factors of n in the triangle rows.  Example: E(10) = Sum: ((E(1) + E(2) + E(5) + E(10)) = ((1 + 0 + 1 + 0) = 2, matching A002654(10).
To get A004018, multiply the result by 4, getting A004018(10) = 8.
The total numbers of lattice points = 4r^2 = E(1) + ((E(2))/2 + ((E(3))/3 + ((E(4))/4 + ((E(5))/5 + .... Since E(even integers) are zero, E(integers of the form (4*k - 1)) = (-1), and E(integers of the form (4*k + 1)) = (+1); we are left with 4r^2 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ..., which is approximately equal to Pi(r^2). (End)
T(n,k) is also the number of parts in the partition of n into k equal parts. - Omar E. Pol, May 05 2020

			T(8,4) = 4 since 4 divides 8.
T(9,3) = 3 since 3 divides 9.
First few rows of the triangle:
  1;
  1, 2;
  1, 0, 3;
  1, 2, 0, 4;
  1, 0, 0, 0, 5;
  1, 2, 3, 0, 0, 6;
  1, 0, 0, 0, 0, 0, 7;
  1, 2, 0, 4, 0, 0, 0, 8;
  1, 0, 3, 0, 0, 0, 0, 0, 9;
  ...

David Wells, "Prime Numbers, the Most Mysterious Figures in Math", John Wiley & Sons, 2005, appendix.
L. Euler, "Discovery of a Most Extraordinary Law of the Numbers Concerning the Sum of Their Divisors"; pp. 358-367 of Robert M. Young, "Excursions in Calculus, An Interplay of the Continuous and the Discrete", MAA, 1992. See p. 366.

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened
Grant Sanderson, Pi hiding in prime regularities
Leonhard Euler, Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs, 1747, The Euler Archive, (Eneström Index) E175.
Leonhard Euler, Observatio de summis divisorum
Eric Weisstein's World of Mathematics, Divisor

Reversal = A127094
Cf. A127094, A123229, A127096, A127097, A127098, A127099, A000203, A126988, A127013, A127057, A038040, A024916, A060640, A001001.
Cf. A000005, A060640.
Cf. A027750.
Cf. A000012 (the first column), A020639, A059841 (the second column when multiplied by 2), A079978 (the third column when multiplied by 2), A079998 (the fifth column when multiplied by 5), A121262 (the fourth column when multiplied by 4).
Cf. A002654, A004018, A008683, A050873.

mod(row()-1;column()) - mod(row();column()) + 1 - Mats Granvik, Aug 31 2007

a127093 n k = a127093_row n !! (k-1)
a127093_row n = zipWith (*) [1..n] $ map ((0 ^) . (mod n)) [1..n]
a127093_tabl = map a127093_row [1..]
-- Reinhard Zumkeller, Jan 15 2011

A127093:=proc(n,k) if type(n/k, integer)=true then k else 0 fi end:
for n from 1 to 16 do seq(A127093(n,k),k=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Jan 20 2007

t[n_, k_] := k*Boole[Divisible[n, k]]; Table[t[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 17 2014 *)
Table[ SeriesCoefficient[k*x^k/(1 - x^k), {x, 0, n}], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 14 2015 *)

trianglerows(n) = for(x=1, n, for(k=1, x, if(x%k==0, print1(k, ", "), print1("0, "))); print(""))
/* Print initial 9 rows of triangle as follows: */
trianglerows(9) \\ Felix Fröhlich, Mar 26 2019

k-th column is composed of "k" interspersed with (k-1) zeros.
Let M = A127093 as an infinite lower triangular matrix and V = the harmonic series as a vector: [1/1, 1/2, 1/3, ...]. then M*V = d(n), A000005: [1, 2, 2, 3, 2, 4, 2, 4, 3, 4, ...]. M^2 * V = A060640: [1, 5, 7, 17, 11, 35, 15, 49, 34, 55, ...]. - Gary W. Adamson, May 10 2007
T(n,k) = ((n-1) mod k) - (n mod k) + 1 (1 <= k <= n). - Mats Granvik, Aug 31 2007
T(n,k) = k * 0^(n mod k). - Reinhard Zumkeller, Jan 15 2011
G.f.: Sum_{k>=1} k * x^k * y^k/(1-x^k) = Sum_{m>=1} x^m * y/(1 - x^m*y)^2. - Robert Israel, Aug 08 2016
T(n,k) = Sum_{d|k} mu(k/d)*sigma(gcd(n,d)). - Ridouane Oudra, Apr 05 2025

cp's OEIS Frontend

A174973 Numbers whose divisors increase by a factor of at most 2.

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A077609 Triangle in which n-th row lists infinitary divisors of n.

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A051377 a(1)=1; for n > 1, a(n) = sum of exponential divisors (or e-divisors) of n.

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A001692 Number of irreducible polynomials of degree n over GF(5); dimensions of free Lie algebras.

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A034699 Largest prime power factor of n.

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A162306 Irregular triangle in which row n contains the numbers <= n whose prime factors are a subset of prime factors of n.

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