A074285 -id:A074285 - cp's OEIS frontend

A117949 Index of pentagonal numbers whose sum of divisors is square.

1, 4, 7, 12, 21, 23, 27, 31, 71, 79, 89, 151, 168, 199, 223, 232, 239, 263, 311, 324, 336, 345, 359, 390, 463, 479, 497, 540, 599, 743, 751, 823, 858, 863, 911, 991, 1031, 1063, 1103, 1151, 1302, 1303, 1343, 1399, 1471, 1540, 1583, 1687, 1759, 1802, 1823

Offset: 1

Jonathan Vos Post, Apr 04 2006

n such that A117948(n) is in A000290.

			a(1) = 1 because sigma(1*(3*1-1)/2) = 1 = 1^2.
a(2) = 4 because sigma(4*(3*4-1)/2) = 36 = 6^2.
a(3) = 7 because sigma(7*(3*7-1)/2) = 144 = 12^2.
a(4) = 12 because sigma(12*(3*12-1)/2) = 576 = 24^2.
a(5) = 21 because sigma(21*(3*21-1)/2) = 1024 = 32^2.
a(6) = 23 because sigma(23*(3*23-1)/2) = 1296 = 36^2.
a(7) = 27 because sigma(27*(3*27-1)/2) = 3600 = 60^2.
a(8) = 31 because sigma(31*(3*31-1)/2) = 2304 = 48^2.
a(9) = 71 because sigma(71*(3*71-1)/2) = 11664 = 108^2.

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

Cf. A000203, A000217, A000290, A000326, A074285, A083675, A117948.

with(numtheory): select(n-> sqrt(sigma(n*(3*n-1)/2))::integer, [$1..2200])[]; # Emeric Deutsch, Apr 06 2006

s = {}; Do[If[IntegerQ @ Sqrt @ DivisorSigma[1, (3 n - 1)*n/2], AppendTo[s, n]], {n, 1, 2000}]; s (* Amiram Eldar, Aug 17 2019 *)
Position[DivisorSigma[1,PolygonalNumber[5,Range[2000]]],?(IntegerQ[ Sqrt[ #]]&)]//Flatten (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Oct 23 2020 *)

isok(n) = issquare(sigma(n*(3*n-1)/2)); \\ Michel Marcus, Aug 17 2019

More terms from Emeric Deutsch, Apr 06 2006

a(0) removed by Amiram Eldar, Aug 17 2019

A232355 Numbers k such that sigma(triangular(k)) = sigma(k)^2.

Original entry on oeis.org

1, 11, 695, 991, 2839, 3707, 9347, 10703, 12847, 27089, 42251, 56419, 74671, 115289, 168739, 191051, 219295, 233729, 280111, 300731, 326899, 353651, 430859, 611799, 642991, 661715, 1035827, 1116607, 1181579, 1234519, 1365491, 1485035, 1777099, 1854671, 1905875

Offset: 1

Alex Ratushnyak, Nov 22 2013

nonn

Subsequence of A116990. - Michel Marcus, Jun 13 2015

			11 is in the sequence because sigma(11*12/2) = sigma(66) = 144 = 12^2 = sigma(11)^2.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A000203 (sigma(n): sum of divisors of n), A000217 (triangular(n): = n*(n+1)/2).
Cf. A074285 (sigma(triangular(n))), A072861 (sigma(n)^2).
Cf. A116990 (indices of triangular numbers whose sum of divisors is square).

[n: n in [1..7*10^5] | SumOfDivisors(n*(n+1) div 2) eq SumOfDivisors(n)^2]; // Vincenzo Librandi, Jun 13 2015

Select[Range@1000000, DivisorSigma[1, #]^2==DivisorSigma[1, (# (# + 1)/2)] &] (* Vincenzo Librandi, Jun 13 2015 *)

isok(n) = sigma(n)^2 == sigma(n*(n+1)/2); \\ Michel Marcus, Nov 23 2013

More terms from Michel Marcus, Nov 23 2013

A262613 Sum of divisors of n-th generalized pentagonal number.

Original entry on oeis.org

1, 3, 6, 8, 28, 24, 36, 42, 48, 90, 72, 80, 144, 96, 168, 217, 182, 312, 180, 192, 372, 216, 576, 456, 280, 588, 336, 352, 864, 576, 720, 855, 558, 756, 702, 936, 1120, 600, 1080, 1116, 1024, 2016, 1008, 816, 1296, 1152, 2016, 2072, 1178, 1860, 1344, 1120, 3600

Offset: 1

Omar E. Pol, Nov 24 2015

For a remarkable connection between the sum-of-divisors function (A000203) and the generalized pentagonal numbers (A001318) see A238442.

Antti Karttunen, Table of n, a(n) for n = 1..5000

Cf. A000203, A001318, A074285, A117948, A196020, A238442.

[DivisorSigma(1,(3*n^2+2*n+(n mod 2)*(2*n+1)) div 8): n in [1..70]]; // Vincenzo Librandi, Dec 21 2015

DivisorSigma[1, Select[Accumulate[Range[200]]/3, IntegerQ]] (* G. C. Greubel, Jun 06 2017 *)

a(n) = sigma((3*n^2 + 2*n + (n%2) * (2*n + 1)) / 8); \\ Michel Marcus, Dec 21 2015

(define (A262613 n) (A000203 (A001318 n))) ;; Scheme-program for A000203 given in that entry.
;; This uses memoization-macro definec:
(definec (A001318 n) (if (zero? n) 0 (+ (if (even? n) (/ n 2) n) (A001318 (- n 1)))))
;; Antti Karttunen, Dec 20 2015

a(n) = A000203(A001318(n)).

Sum_{k=1..n} a(k) ~ (9/40) * n^3. - Amiram Eldar, Dec 14 2024

A325838 a(n) is the product of divisors of the n-th triangular number.

Original entry on oeis.org

1, 3, 36, 100, 225, 441, 21952, 10077696, 91125, 3025, 18974736, 37015056, 8281, 121550625, 42998169600000000, 342102016, 3581577, 5000211, 1303210000, 3782285936100000000, 2847396321, 64009, 442032795979776, 19683000000000000000000, 34328125, 15178486401

Offset: 1

Jaroslav Krizek, Sep 07 2019

nonn

			The 5th triangular number is 15, whose divisors are {1, 3, 5, 15}; their product is 225.

Cf. A000005, A000203, A000217, A007955.

See A063440 and A074285 for number and sum of such divisors.

[&*[d: d in Divisors(n * (n+1) div 2)] : n in [1..1000]];

pd[n_] := n^(DivisorSigma[0, n]/2); t[n_] := n (n + 1)/2; pd /@ t /@ Range[26] (* Amiram Eldar, Sep 07 2019 *)

a(n) = vecprod(divisors(n*(n+1)/2)); \\ Michel Marcus, Oct 14 2019

from math import isqrt
from sympy import divisor_count
def A325838(n): return (lambda m:(isqrt(m) if (c:=divisor_count(m)) & 1 else 1)*m**(c//2))(n*(n+1)//2) # Chai Wah Wu, Jun 25 2022

a(n) = A007955(A000217(n)).

A347155 Sum of divisors of nontriangular numbers.

Original entry on oeis.org

3, 7, 6, 8, 15, 13, 12, 28, 14, 24, 31, 18, 39, 20, 42, 36, 24, 60, 31, 42, 40, 30, 72, 32, 63, 48, 54, 48, 38, 60, 56, 90, 42, 96, 44, 84, 72, 48, 124, 57, 93, 72, 98, 54, 120, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 68, 126, 96, 144, 72, 195, 74, 114, 124, 140

Offset: 1

Omar E. Pol, Aug 20 2021

nonn

The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram both Dyck paths have the same orientation, that is both Dyck paths have peaks or both Dyck paths have valleys.
So knowing this characteristic shape we can know if a number is a nontriangular number (or not) just by looking at the diagram, even ignoring the concept of nontriangular number.
Therefore we can see a geometric pattern of the distribution of the nontriangular numbers in the stepped pyramid described in A245092.
If both Dyck paths have peaks on the main diagonal then the related subsequence of nontriangular numbers A014132 is A317303.
If both Dyck paths have valleys on the main diagonal then the related subsequence of nontriangular numbers A014132 is A317304.

			a(6) = 13 because the sum of divisors of the 6th nontriangular (i.e., 9) is 1 + 3 + 9 = 13.
On the other we can see that in the main diagonal of the diagrams both Dyck paths have the same orientation, that is both Dyck paths have peaks or both Dyck paths have valleys as shown below.
Illustration of initial terms:
m(n) = A014132(n).
.
   n   m(n) a(n)   Diagram
.                    _   _ _   _ _ _   _ _ _ _   _ _ _ _ _   _ _ _ _ _ _
                   _| | | | | | | | | | | | | | | | | | | | | | | | | | |
   1    2    3    |_ _|_| | | | | | | | | | | | | | | | | | | | | | | | |
                   _ _|  _|_| | | | | | | | | | | | | | | | | | | | | | |
   2    4    7    |_ _ _|    _|_| | | | | | | | | | | | | | | | | | | | |
   3    5    6    |_ _ _|  _|  _ _|_| | | | | | | | | | | | | | | | | | |
                   _ _ _ _|  _| |  _ _|_| | | | | | | | | | | | | | | | |
   4    7    8    |_ _ _ _| |_ _|_|    _ _|_| | | | | | | | | | | | | | |
   5    8   15    |_ _ _ _ _|  _|     |  _ _ _|_| | | | | | | | | | | | |
   6    9   13    |_ _ _ _ _| |      _|_| |  _ _ _|_| | | | | | | | | | |
                   _ _ _ _ _ _|  _ _|    _| |    _ _ _|_| | | | | | | | |
   7   11   12    |_ _ _ _ _ _| |  _|  _|  _|   |  _ _ _ _|_| | | | | | |
   8   12   28    |_ _ _ _ _ _ _| |_ _|  _|  _ _| | |  _ _ _ _|_| | | | |
   9   13   14    |_ _ _ _ _ _ _| |  _ _|  _|    _| | |    _ _ _ _|_| | |
  10   14   24    |_ _ _ _ _ _ _ _| |     |     |  _|_|   |  _ _ _ _ _|_|
                   _ _ _ _ _ _ _ _| |  _ _|  _ _|_|       | | |
  11   16   31    |_ _ _ _ _ _ _ _ _| |  _ _|  _|      _ _|_| |
  12   17   18    |_ _ _ _ _ _ _ _ _| | |_ _ _|      _| |  _ _|
  13   18   39    |_ _ _ _ _ _ _ _ _ _| |  _ _|    _|  _|_|
  14   19   20    |_ _ _ _ _ _ _ _ _ _| | |       |_ _|
  15   20   42    |_ _ _ _ _ _ _ _ _ _ _| |  _ _ _|  _|
                   _ _ _ _ _ _ _ _ _ _ _| | |  _ _| |
  16   22   36    |_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _|
  17   23   24    |_ _ _ _ _ _ _ _ _ _ _ _| | |
  18   24   60    |_ _ _ _ _ _ _ _ _ _ _ _ _| |
  19   25   31    |_ _ _ _ _ _ _ _ _ _ _ _ _| |
  20   26   42    |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
  21   27   40    |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
Column m gives the nontriangular numbers.
Also the diagrams have on the main diagonal the following property: diagram [1] has peaks, diagrams [2, 3] have valleys, diagrams [4, 5, 6] have peaks, diagrams [7, 8, 9, 10] have valleys, and so on.
a(n) is also the area (and the number of cells) of the n-th diagram.
For n = 3 the sum of the regions (or parts) of the third diagram is 3 + 3 = 6, so a(3) = 6.
For more information see A237593.

Cf. A000203, A000217, A014132, A237591, A237593, A245092, A262626, A317303, A317304, A346873.

Some sequences that gives sum of divisors: A000225 (of powers of 2), A008864 (of prime numbers), A065764 (of squares), A073255 (of composites), A074285 (of triangular numbers, also of generalized hexagonal numbers), A139256 (of perfect numbers), A175926 (of cubes), A224613 (of multiples of 6), A346865 (of hexagonal numbers), A346866 (of second hexagonal numbers), A346867 (of numbers with middle divisors), A346868 (of numbers with no middle divisors).

Array[DivisorSigma[1,#+Round@Sqrt[2#]]&,100] (* Giorgos Kalogeropoulos, Aug 20 2021 *)

a(n) = sigma(n + round(sqrt(2*n))); \\ Michel Marcus, Aug 21 2021

a(n) = A000203(A014132(n)).

A166162 Twin prime averages which are also the sum of the divisors of a triangular number.

Original entry on oeis.org

4, 12, 18, 72, 192, 270, 1488, 1872, 1482, 1872, 1152, 2268, 6552, 3672, 6552, 10890, 3528, 5280, 7560, 7488, 13680, 23040, 17388, 29760, 21600, 23040, 65520, 87120, 51480, 34848, 65268, 127680, 122400, 134400, 114660, 137088, 206640, 134400

Offset: 1

Vladimir Joseph Stephan Orlovsky, Oct 08 2009

nonn

			The triangular number A000217(3)= 6 has a sum of divisors 1+2+3+6 = 12 = A014574(3) which is also the average of the twin primes 11 and 13.

Cf. A074285, A014574 [R. J. Mathar, Oct 14 2008]

t[n_]:=n*(n+1)/2; f[n_]:=Plus@@Divisors[t[n]]; lst={};Do[p=f[n];If[PrimeQ[p-1]&&PrimeQ[p+1], AppendTo[lst,p]],{n,7!}];lst

Definition and example rewritten by R. J. Mathar, Oct 14 2009

A275374 Numbers k that divide sigma(k*(k+1)/2).

Original entry on oeis.org

1, 2, 3, 7, 12, 15, 31, 56, 63, 127, 135, 168, 234, 240, 255, 260, 384, 504, 511, 720, 819, 896, 992, 1023, 1080, 1344, 1512, 1638, 2047, 2240, 2352, 3276, 3564, 3584, 3744, 3840, 4095, 4320, 4655, 7280, 7448, 8191, 9360, 10304, 12825, 12896, 13104, 14256, 14725, 15795, 16256

Offset: 1

Altug Alkan, Jul 25 2016

nonn

			7 is a term because 7 divides sigma(28) = 56.

Cf. A000203, A000217, A074285.

Select[Range@ 17000, Divisible[DivisorSigma[1, # (# + 1)/2], #] &] (* Michael De Vlieger, Jul 25 2016 *)

PARI
```
isok(n) = sigma(n*(n+1)/2) % n == 0
```

cp's OEIS Frontend

A117949 Index of pentagonal numbers whose sum of divisors is square.

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A232355 Numbers k such that sigma(triangular(k)) = sigma(k)^2.

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Magma

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A262613 Sum of divisors of n-th generalized pentagonal number.

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Magma

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A325838 a(n) is the product of divisors of the n-th triangular number.

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Magma

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A347155 Sum of divisors of nontriangular numbers.

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Mathematica

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A166162 Twin prime averages which are also the sum of the divisors of a triangular number.

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Mathematica

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A275374 Numbers k that divide sigma(k*(k+1)/2).

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Mathematica