A130317 -id:A130317 - cp's OEIS frontend

A007862 Number of triangular numbers that divide n.

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

A088726 Smallest numbers having exactly n divisors d>1 such that also d+1 is a divisor.

Original entry on oeis.org

1, 6, 12, 72, 60, 180, 360, 420, 840, 1260, 3780, 2520, 5040, 13860, 36960, 41580, 27720, 55440, 83160, 166320, 277200, 491400, 471240, 360360, 1113840, 720720, 1081080, 3341520, 2162160, 2827440, 5405400, 4324320, 12972960, 6126120

Offset: 0

Reinhard Zumkeller, Oct 12 2003

nonn

A088722(a(n))=n, A088722(k)<>n, for n

Ray Chandler, Table of n, a(n) for n=0..57

Cf. A088722, A088723, A088724, A088725.

a(n) = 2*A130317(n+1) for n>0. - Chandler

Extended by Ray Chandler, May 29 2008

A350756 Integers whose number of divisors that are triangular numbers sets a new record.

Original entry on oeis.org

1, 3, 6, 30, 90, 180, 210, 420, 630, 1260, 2520, 6930, 13860, 27720, 41580, 83160, 138600, 180180, 360360, 540540, 1081080, 1413720, 2162160, 3063060, 6126120, 12252240, 18378360, 36756720, 73513440, 91891800, 116396280, 183783600, 232792560, 349188840

Offset: 1

Bernard Schott, Jan 13 2022

nonn

Terms that are triangular: 1, 3, 6, 210, 630, 2162160, ...

The number of triangular divisors of a(n) is A007862(a(n)): 1, 2, 3, 5, 6, 7, 8, 9, 10, 12, ...

			1260 has 36 divisors of which 12 are triangular numbers {1, 3, 6, 10, 15, 21, 28, 36, 45, 105, 210, 630}. No positive integer smaller than 1260 has as many as twelve triangular divisors; hence 1260 is a term.

Cf. A000217, A007862, A130317.

Similar for A046952 (squares), A053624 (odd), A093036 (palindromes), A181808 (even), A340548 (repdigits), A340549 (repunits) divisors.

max=0;Do[If[(d=Length@Select[Divisors@k,IntegerQ[(Sqrt[8#+1]-1)/2]&])>max,Print@k;max=d],{k,10^10}] (* Giorgos Kalogeropoulos, Jan 13 2022 *)

lista(nn) = {my(r=0); for (n=1, nn, my(m = sumdiv(n, d, ispolygonal(d,3))); if (m>r, r=m; print1(n", ")));} \\ Michel Marcus, Jan 14 2022

A358542 a(n) is the smallest number with exactly n divisors that are tetrahedral numbers.

Original entry on oeis.org

1, 4, 56, 20, 120, 280, 560, 840, 1680, 10920, 9240, 18480, 55440, 120120, 240240, 314160, 628320, 1441440, 2282280, 7225680, 4564560, 9129120, 13693680, 27387360, 54774720, 68468400, 77597520, 136936800, 155195040, 310390080, 465585120, 775975200, 1163962800

Offset: 1

Ilya Gutkovskiy, Nov 21 2022

nonn

			a(3) = 56 because 56 has 3 tetrahedral divisors {1, 4, 56} and this is the smallest such number.

Lucas A. Brown, Table of n, a(n) for n = 1..37
Lucas A. Brown, Python program.
Eric Weisstein's World of Mathematics, Tetrahedral Number
Index entries for sequences related to divisors of numbers

Cf. A000292, A005179, A130317, A279495, A358543, A358544.

istetrah(n) = my(k=sqrtnint(6*n, 3)); k*(k+1)*(k+2)==6*n; \\ A000292
a(n) = my(k=1); while (sumdiv(k, d, istetrah(d)) != n, k++); k; \\ Michel Marcus, Nov 21 2022

a(20)-a(22) from Michel Marcus, Nov 21 2022
a(23)-a(30) from Jinyuan Wang, Nov 28 2022
a(31) from Martin Ehrenstein, Dec 02 2022
a(32) and a(33) from Lucas A. Brown, Dec 14 2022

A358544 a(n) is the smallest number with exactly n divisors that are centered triangular numbers.

Original entry on oeis.org

1, 4, 20, 320, 460, 5440, 14260, 12920, 168640, 103360, 594320, 3878720, 2377280, 9211960, 18423920, 36847840, 125995840, 73695680, 865924240, 976467760, 1952935520, 3463696960, 3905871040, 31246968320, 22946992360

Offset: 1

Ilya Gutkovskiy, Nov 21 2022

Any subsequent terms are > 10^10. - Lucas A. Brown, Dec 24 2022

			a(3) = 20 because 20 has 3 centered triangular divisors {1, 4, 10} and this is the smallest such number.

Lucas A. Brown, Python program.
Eric Weisstein's World of Mathematics, Centered Triangular Number
Index entries for sequences related to divisors of numbers

Cf. A005179, A005448, A130317, A300409, A358542, A358545.

isct(n) = my(k=(2*n-2)/3, m); (n==1) || ((denominator(k)==1) && (m=sqrtint(k)) && (m*(m+1)==k)); \\ A005448
a(n) = my(k=1); while (sumdiv(k, d, isct(d)) != n, k++); k; \\ Michel Marcus, Nov 21 2022

a(14) from Michel Marcus, Nov 21 2022

a(15)-a(25) from Jinyuan Wang, Nov 29 2022

A287142 Least k such that the number of pairs of consecutive divisors of k equals n.

Original entry on oeis.org

1, 2, 6, 12, 72, 60, 180, 360, 420, 840, 1260, 3780, 2520, 5040, 13860, 36960, 41580, 27720, 55440, 83160, 166320, 277200, 491400, 471240, 360360, 1113840, 720720, 1081080, 3341520, 2162160, 2827440, 5405400, 4324320, 12972960, 6126120, 16576560, 28274400

Offset: 0

Michel Lagneau, May 20 2017

nonn

a(n) is even for n > 0.
We observe numbers of the decimal form (abcabc) = 360360, 720720 and numbers of the decimal form (abcabc0) = 1081080, 2162160, 5405400, 4324320, 6126120.
Observation and questions: many terms are products of powers of a contiguous set of the smallest primes. Many early terms of a(n) are in A002182; e.g., a(35) - A002182(44). The smallest exception outside of the empty product a(0) = 1 is a(22) = 491400 = 2^3 * 3^3 * 5^2 * 7 * 13. In other words, many terms have A006530(a(n)) < A053669(a(n)); a(22) is the smallest exception. Other exceptions include {471240, 1113840, 3341520, 2827440, 16576560, 28274400, ...}. A000720(A053669(a(22))) - A000720(A006530(a(22))) = 1, but the first instance of 2 for this function is a(35) = 16576560. This is evident by mapping A054841 across a(n). Are there a finite number of terms of a(n) that are also in A002182? Are there a finite number of terms of a(n) that have A006530(a(n)) < A053669(a(n)); are they becoming less frequent as n increases? - Michael De Vlieger, May 20 2017
In other words, a(n) is the least integer with exactly n divisors that are oblong (A002378). - Bernard Schott, Jul 30 2022

			a(3) = 12 because the divisors of 12 are {1, 2, 3, 4, 6, 12} with three pairs of consecutive divisors (1, 2), (2, 3) and (3, 4).

Cf. A000005, A002378, A027750, A129308, A130317.

Essentially the same as A088726.

with(numtheory):
for n from 0 to 60 do:
ii:=0:
  for k from 1 to 10^8 while(ii=0) do:
    d0:=divisors(k):n0:=nops(d0):c0:=0:
      for i from 1 to n0-1 do:
        if d0[i+1]=d0[i]+1
         then
          c0:=c0+1:
          else
         fi:
       od:
       if c0=n
       then
     ii:=1:printf(“%d %d \n”,n,k):
     else
     fi:
   od:
  od:

Function[s, Function[t, ReplacePart[t, Map[#1 -> #2 & @@ # &, Transpose@{1 + Keys@ s, Values[s][[All, 1]]}]]]@ ConstantArray[0, Max@ Keys@ s]]@ KeySort@ PositionIndex@ Table[DivisorSum[n, 1 &, Divisible[n, # + 1] &], {n, 2 * 10^6}] (* Michael De Vlieger, May 20 2017, Version 10 *)

isok(n,k) = {dk = divisors(k); ddk = vector(#dk-1, j, dk[j+1] - dk[j]); #select(x->x==1, ddk) == n;}
a(n) = {my(k=1); while (!isok(n, k), k++); k;} \\ Michel Marcus, May 20 2017

a(n) = 2*A130317(n) for n >= 1. - Bernard Schott, Jul 30 2022

A356132 Least integer with n pentagonal divisors.

Original entry on oeis.org

1, 5, 35, 70, 210, 420, 2310, 4620, 18480, 32340, 60060, 120120, 240240, 720720, 1261260, 1141140, 2042040, 4084080, 4564560, 13693680, 19399380, 58198140, 95855760, 38798760, 116396280, 193993800, 77597520, 232792560, 543182640, 387987600, 1125164040

Offset: 1

Michel Marcus, Jul 27 2022

nonn

Rémy Sigrist, C program

Cf. A000326, A130317, A279497.

C
```
See Links section.
```

a(n) = my(k=1); while (sumdiv(k, d, ispolygonal(d, 5)) != n, k++); k;

More terms from Rémy Sigrist, Jul 27 2022

A358539 a(n) is the smallest number with exactly n divisors that are n-gonal numbers.

Original entry on oeis.org

6, 36, 210, 1260, 6426, 3360, 351000, 207900, 3749460, 1153152, 15036840, 204204000, 213825150, 11737440, 91797866160, 1006485480, 2310808500, 4966241280, 22651328700, 325269404460, 14266470332400, 11203920000, 256653797400, 45843256859400, 59207908359600, 46822406400

Offset: 3

Ilya Gutkovskiy, Nov 21 2022

nonn

			a(5) = 210 because 210 has 5 pentagonal divisors {1, 5, 35, 70, 210} and this is the smallest such number.

Lucas A. Brown, Python program.
Eric Weisstein's World of Mathematics, Polygonal Number
Index entries for sequences related to divisors of numbers

Cf. A005179, A130279, A130317, A356132, A358540, A358541.

a(n) = my(k=1); while (sumdiv(k, d, ispolygonal(d, n)) != n, k++); k; \\ Michel Marcus, Nov 21 2022

a(12)-a(13) from Michel Marcus, Nov 21 2022
a(14)-a(16) from Daniel Suteu, Dec 04 2022
a(17)-a(28) from Martin Ehrenstein, Dec 05 2022

A357842 a(n) is the smallest number k for which k and the arithmetic derivative k' (A003415) have exactly n triangular divisors (A000217).

Original entry on oeis.org

2, 27, 18, 72, 612, 1764, 756, 8100, 27000, 97200, 66528, 175500, 93600, 280800, 1731600, 661500, 680400, 3704400, 34177500, 11107800, 16581600, 20065500, 108486000, 102910500, 108353700, 181912500, 314874000, 462672000, 4408236000, 229975200, 2297786400, 672348600, 925041600, 1344697200, 158230800

Offset: 1

Marius A. Burtea, Oct 20 2022

nonn

			2 has only the divisor 1 = A000217(1) and 2' = 1 = A000217(1), so a(1) = 2.
27 and 27' = 27 have the divisors 1 = A000217(1), 3 = A000217(2) triangular numbers, so a(2) = 27.

Cf. A000217, A003415, A007862, A130317.

tr:=func; f:=func;  a:=[]; for n in [1..30] do k:=2 ; while tr(k) ne n or tr(Floor(f(k))) ne n do k:=k+1; end while; Append(~a,k); end for; a;

d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); tridiv[n_] := DivisorSum[n, 1 &, IntegerQ[Sqrt[8*# + 1]] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 2, i}, While[c < len && n < nmax, i = tridiv[n]; If[i <= len && s[[i]] == 0 && tridiv[d[n]] == i, c++; s[[i]] = n]; n++]; s]; seq[10, 10^6] (* Amiram Eldar, Oct 21 2022 *)

f(n) = sumdiv(n, d, ispolygonal(d, 3)); \\ A007862
ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
a(n) = my(k=2); while((f(k)!=n) || (f(ad(k))!=n), k++); k; \\ Michel Marcus, Oct 23 2022

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A007862 Number of triangular numbers that divide n.

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A088726 Smallest numbers having exactly n divisors d>1 such that also d+1 is a divisor.

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A350756 Integers whose number of divisors that are triangular numbers sets a new record.

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A358542 a(n) is the smallest number with exactly n divisors that are tetrahedral numbers.

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A358544 a(n) is the smallest number with exactly n divisors that are centered triangular numbers.

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A287142 Least k such that the number of pairs of consecutive divisors of k equals n.

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A356132 Least integer with n pentagonal divisors.

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A358539 a(n) is the smallest number with exactly n divisors that are n-gonal numbers.

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