A063440 -id:A063440 - cp's OEIS frontend

A133949 a(n) = the number of "non-isolated divisors" of n(n+1)/2. A positive divisor k of n is non-isolated if either k-1 or k+1 also divides n.

0, 0, 3, 2, 0, 0, 2, 4, 0, 0, 3, 3, 0, 0, 6, 2, 0, 0, 2, 8, 0, 0, 4, 6, 0, 0, 5, 2, 0, 0, 2, 6, 0, 0, 10, 3, 0, 0, 8, 4, 0, 0, 2, 8, 0, 0, 4, 7, 0, 0, 3, 2, 0, 0, 6, 6, 0, 0, 5, 5, 0, 0, 8, 4, 0, 0, 2, 3, 0, 0, 4, 4, 0, 0, 5, 2, 0, 0, 4, 9, 0, 0, 5, 10, 0, 0, 6, 2, 0, 0, 4, 3, 0, 0, 10, 4, 0, 0, 8, 2, 0, 0, 2, 13

Offset: 1

Leroy Quet, Sep 30 2007

nonn

a(k) = 0 for k mod 4 == {1,2}. - Ray Chandler

Ray Chandler, Table of n, a(n) for n=1..10000

Cf. A133947, A133950, A063440.

Table[Length[Select[Divisors[n*(n + 1)/2], If[ # > 1, Mod[n*(n + 1)/2, #*(# - 1)] == 0] || Mod[n*(n + 1)/2, #*(# + 1)] == 0 &]], {n, 1, 80}] (* Stefan Steinerberger, Nov 01 2007 *)

a(n) = A063440(n) - A133950(n) = A132747(A000217(n)).

More terms from Stefan Steinerberger, Nov 01 2007

Extended by Ray Chandler, Jun 23 2008

A319036 a(n) is the smallest triangular number T(k) such that both it and its successor T(k+1) have exactly 2n divisors, or 0 if no such pair of consecutive triangular numbers exists.

Original entry on oeis.org

0, 6, 153, 66, 0, 3916, 0, 1770, 2556, 327645, 0, 1540, 0, 893862621, 8199225, 17766, 0, 76636, 0, 12720, 662976, 2096128, 0, 10296, 3357936, 416798777159765703, 6221628, 3611328, 0, 1734453, 0, 303810, 111576864636, 1420010137134674578503, 18051523357140153

Offset: 1

Jon E. Schoenfield, Dec 05 2018

nonn

The only primes p for which a(p) > 0 are those for which both 2*3^(p-1) - 1 and 2*3^(p-1) + 1 are prime: 2, 3, and any other primes p such that p-1 appears both in A003307 and A003306. (If such a prime p > 3 exists, then p exceeds 1360105.)

Conjecture: The only primes p for which a(p) > 0 are 2 and 3.

			For n=1, the only triangular number with exactly 2*1 = 2 divisors is T(2) = 2*(2+1)/2 = 3 (the only triangular number that is prime); thus, exists no pair of consecutive triangular numbers having exactly 2 divisors, so a(1)=0.
a(2) is 6 because T(3) = 3*(3+1)/2 = 6 and T(4) = 4*(4+1)/2 = 10 are the first two consecutive triangular numbers having exactly 2*2 = 4 divisors.

Cf. A000005, A000217, A063440, A081978, A319035.

Cf. A003306, A003307.

A330809 Triangular numbers having exactly 8 divisors.

Original entry on oeis.org

66, 78, 105, 136, 190, 231, 351, 406, 435, 465, 561, 595, 741, 861, 903, 946, 1378, 1431, 1653, 2211, 2278, 2485, 3081, 3655, 3741, 4371, 4465, 5151, 5253, 5995, 6441, 7021, 7503, 8515, 8911, 9453, 9591, 10011, 10153, 10585, 11026, 12561, 13366, 14878, 15051

Offset: 1

Jon E. Schoenfield, Jan 11 2020

nonn

Terms may be categorized as belonging to the following types:
type 1: products of 3 distinct primes p,q,r such that 2*p*q + 1 = r: 78, 406, 465, ... (27108 of the first 100000 terms);
type 2: products of 3 distinct primes p,q,r such that 2*p*q - 1 = r: 66, 190, 435, ... (26848 of the first 100000 terms);
type 3: products of 3 distinct primes p,q,r such that p*q + 1 = 2*r: 231, 561, 1653, ... (23050 of the first 100000 terms);
type 4: products of 3 distinct primes p,q,r such that p*q - 1 = 2*r: 105, 595, 741, ... (22983 of the first 100000 terms);
type 5: products of the cube of a prime p and a distinct prime q such that 2*p^3 + 1 = q: 136, 31375, 3544453, ... (6 of the first 100000 terms);
type 6: products of the cube of a prime p and a distinct prime q such that 2*p^3 - 1 = q: 1431, 1774977571, 12642646591, ... (4 of the first 100000 terms);
type 7: products of the cube of a prime p and a distinct prime q such that p^3 - 1 = 2*q: the only term of this type is 351 = 3^3 * 13.
(No term is a product of the cube of a prime p and a distinct prime q such that p^3 + 1 = 2*q.)

			Type
(see
cmts)  Initial terms             Notes
-----  ------------------------  -----------------------------
  1    78, 406, 465, ...         p*q*r such that 2*p*q + 1 = r
  2    66, 190, 435, ...         p*q*r such that 2*p*q - 1 = r
  3    231, 561, 1653, ...       p*q*r such that p*q + 1 = 2*r
  4    105, 595, 741, ...        p*q*r such that p*q - 1 = 2*r
  5    136, 31375, 3544453, ...  p^3*q such that 2*p^3 + 1 = q
  6    1431, 1774977571, ...     p^3*q such that 2*p^3 - 1 = q
  7    351 (only)                p^3*q such that p^3 - 1 = 2*q

Robert Israel, Table of n, a(n) for n = 1..10000

Intersection of A000217 (triangular numbers) and A030626 (8 divisors).

Cf. A063440 (number of divisors of n-th triangular number), A292989 (triangular numbers having exactly 6 divisors).

[k:k in [1..16000]| IsSquare(8*k+1) and NumberOfDivisors(k) eq 8]; // Marius A. Burtea, Jan 12 2020

select(t -> numtheory:-tau(t) = 8, [seq(i*(i+1)/2, i=1..1000)]); # Robert Israel, Jan 13 2020

Select[PolygonalNumber@ Range[180], DivisorSigma[0, #] == 8 &] (* Michael De Vlieger, Jan 11 2020 *)

isok(k) = ispolygonal(k, 3) && (numdiv(k) == 8); \\ Michel Marcus, Jan 11 2020

A272524 Refactorable triangular numbers.

Original entry on oeis.org

1, 36, 136, 276, 1176, 2016, 2556, 2628, 3240, 4560, 11628, 12720, 12880, 18336, 18528, 25200, 32640, 32896, 51360, 64620, 73920, 86320, 89676, 100128, 114960, 115440, 126756, 131328, 148240, 166176, 248160, 253116, 265356, 270480, 294528, 295296, 320400, 345696, 373680, 380628, 400960, 401856, 438516

Offset: 1

Waldemar Puszkarz, May 01 2016

nonn

Intersection of A000217 and A033950.

			36 is a term as the number of divisors of 36 (see A000005) is 9 which divides 36.

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

Cf. A000217 (triangular numbers), A033950 (refactorable numbers), A063440 (number of divisors of triangular numbers), A000005 (number of divisors), A036907 (refactorable squares).

Select[PolygonalNumber@Range@1000, Divisible[#, DivisorSigma[0,#]]&]

for (n=1, 1000, t=n*(n+1)/2; t%numdiv(t)==0 && print1(t ", " ))

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)).

A331234 Triangular numbers having exactly 9 divisors.

Original entry on oeis.org

36, 1225, 1413721, 7885505171090778556470578126753302097454601, 67594562493730400324395236678194231988753004665644702944024074038452164931450549832074201

Offset: 1

Jon E. Schoenfield, Jan 12 2020

nonn

Any number having an odd number of divisors is a square, so each term in this sequence is a term of A001110 (numbers that are both triangular and square). Since A001110(k) = (A000129(k)*A001333(k))^2, A001110(k) will have exactly 9 divisors iff A000129(k) and A001333(k) are both prime (i.e., k is in both A096650 and A099088); the first 5 values of k at which this occurs are 2, 3, 5, 29, and 59.

Conjecture: a(5) is the final term of this sequence.

			Writing the k-th triangular number A000217(k) as T(k):
a(1) = T(8) = 8*9/2 = 36 = 2^2 * 3^2;
a(2) = T(49) = 49*50/2 = 1225 = 5^2 * 7^2;
a(3) = T(1681) = 1681*1682/2 = 1413721 = 29^2 * 41^2.
Factorization of larger known terms:
a(4) = 44560482149^2 * 63018038201^2;
a(5) = 13558774610046711780701^2 * 19175002942688032928599^2.

Intersection of A000217 (triangular numbers) and A030627 (numbers with exactly 9 divisors).
Triangular numbers having exactly k divisors: A068443 (k=4), A292989 (k=6), A330809 (k=8).
Cf. A063440 (number of divisors of n-th triangular number), A242585 (number of divisors of the n-th positive number that is both triangular and square).
Cf. A001110, A000129, A001333, A096650, A099088.

A375640 The smallest triangular number that begins a run of at least n consecutive triangular numbers with the same number of divisors.

Original entry on oeis.org

1, 6, 6, 6, 3181503, 213118335, 21363375160, 83495980981, 595814595900153, 2153650189195351, 49340595918273751, 3380756271799042653, 13175529217290940503

Offset: 1

Shyam Sunder Gupta, Aug 22 2024

			a(4) = 6 because 6 is the smallest triangular number that begins a run of 4 consecutive triangular numbers (6, 10, 15, 21) with the same number of divisors, i.e. 4.
a(5) = 3181503 because 3181503 is the smallest triangular number that begins a run of 5 consecutive triangular numbers (3181503, 3184026, 3186550, 3189075, 3191601) with the same number of divisors, i.e. 24.

Cf. A000217, A063440, A319037, A319038.

cp's OEIS Frontend

A133949 a(n) = the number of "non-isolated divisors" of n(n+1)/2. A positive divisor k of n is non-isolated if either k-1 or k+1 also divides n.

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A319036 a(n) is the smallest triangular number T(k) such that both it and its successor T(k+1) have exactly 2n divisors, or 0 if no such pair of consecutive triangular numbers exists.

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A330809 Triangular numbers having exactly 8 divisors.

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A272524 Refactorable triangular numbers.

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

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A331234 Triangular numbers having exactly 9 divisors.

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A375640 The smallest triangular number that begins a run of at least n consecutive triangular numbers with the same number of divisors.

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