A169823 -id:A169823 - cp's OEIS frontend

A249674 a(n) = 30*n.

0, 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360, 390, 420, 450, 480, 510, 540, 570, 600, 630, 660, 690, 720, 750, 780, 810, 840, 870, 900, 930, 960, 990, 1020, 1050, 1080, 1110, 1140, 1170, 1200, 1230, 1260, 1290, 1320, 1350, 1380, 1410, 1440

Offset: 0

Kaylan Purisima, Nov 03 2014

Numbers divisible by 2, 3 and 5. - Robert Israel, Nov 19 2014

a(n) is the maximum score of a 10-pin n-frame bowling game and the maximum score of an n-pin 10-frame bowling game, given the rules: a strike is worth the number of pins in each frame plus the number of pins knocked down by the next two balls (except in the last frame), a spare is worth the number of pins in each frame plus the number of pins knocked down by the next ball (except in the last frame), and if a strike or spare is earned in the last frame then the player must continue to throw balls until they have thrown 3 balls in the last frame. - Iain Fox, Mar 02 2018

			a(7) = 7 * 30 = 210.

Iain Fox, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Bowling.
Wikipedia, Ten-pin bowling.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005843, A008585, A008587, A008588, A008592, A008597, A050519, A169823.

[30*n : n in [0..50]]; // Wesley Ivan Hurt, Nov 18 2014

A249674:=n->30*n: seq(A249674(n), n=0..50); # Wesley Ivan Hurt, Nov 18 2014

30*Range[0, 59] (* Alonso del Arte, Nov 18 2014 *)

vector(100,n,30*(n-1)) \\ Derek Orr, Nov 18 2014

first(n) = Vec(30*x/(x-1)^2 + O(x^n), -n) \\ Iain Fox, Mar 02 2018

G.f.: 30*x/(x-1)^2; a(n) = 2*a(n-1) - a(n-2). - Wesley Ivan Hurt, Nov 18 2014
a(n) = 2*A008597(n) = 3*A008592(n) = 5*A008588(n) = 6*A008587(n) = 10*A008585(n) = 15*A005843(n). - Omar E. Pol, Nov 24 2014
From Elmo R. Oliveira, Apr 08 2025: (Start)
E.g.f.: 30*x*exp(x).
a(n) = A169823(n)/2. (End)

A305548 a(n) = 27*n.

Original entry on oeis.org

0, 27, 54, 81, 108, 135, 162, 189, 216, 243, 270, 297, 324, 351, 378, 405, 432, 459, 486, 513, 540, 567, 594, 621, 648, 675, 702, 729, 756, 783, 810, 837, 864, 891, 918, 945, 972, 999, 1026, 1053, 1080, 1107, 1134, 1161, 1188, 1215, 1242, 1269, 1296, 1323, 1350, 1377, 1404, 1431, 1458, 1485, 1512

Offset: 0

Eric Chen, Jun 05 2018

Eric Chen, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

For a(n) = k*n: A001489 (k=-1), A000004 (k=0), A001477 (k=1), A005843 (k=2), A008585 (k=3), A008591 (k=9), A008607 (k=25), A252994 (k=26), this sequence (k=27), A135628 (k=28), A195819 (k=29), A249674 (k=30), A135631 (k=31), A174312 (k=32), A044102 (k=36), A085959 (k=37), A169823 (k=60), A152691 (k=64).

Mathematica
```
Range[0,2000,27]
```
PARI
```
a(n)=27*n
```

a(n) = 27*n.
a(n) = A008585(A008591(n)) = A008591(A008585(n)).
G.f.: 27*x/(x-1)^2.
From Elmo R. Oliveira, Apr 10 2025: (Start)
E.g.f.: 27*x*exp(x).
a(n) = 2*a(n-1) - a(n-2). (End)

A169825 Multiples of 420.

Original entry on oeis.org

0, 420, 840, 1260, 1680, 2100, 2520, 2940, 3360, 3780, 4200, 4620, 5040, 5460, 5880, 6300, 6720, 7140, 7560, 7980, 8400, 8820, 9240, 9660, 10080, 10500, 10920, 11340, 11760, 12180, 12600, 13020, 13440, 13860, 14280, 14700, 15120, 15540, 15960, 16380, 16800

Offset: 0

N. J. A. Sloane, May 30 2010

Numbers that are divisible by all of 1,2,3,4,5,6,7.

Erich Friedman, What's Special About This Number? (See the entry for "420")
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001477, A008589, A008596, A008597, A008602, A008603.

Cf. A135628, A169823, A169827, A249674.

Range[0, 100]*420 (* Paolo Xausa, Apr 17 2024 *)

a(n)=420*n \\ Charles R Greathouse IV, Apr 16 2024

a(n) = 420*n. - Wesley Ivan Hurt, Apr 11 2021
From Elmo R. Oliveira, Apr 16 2024: (Start)
G.f.: 420*x/(x-1)^2.
E.g.f.: 420*x*exp(x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 7*A169823(n) = 14*A249674(n) = 15*A135628(n) = 20*A008603(n) = 21*A008602(n) = 28*A008597(n) = 30*A008596(n) = 60*A008589(n) = 420*A001477(n) = A169827(n)/2. (End)

A169827 Multiples of 840.

Original entry on oeis.org

0, 840, 1680, 2520, 3360, 4200, 5040, 5880, 6720, 7560, 8400, 9240, 10080, 10920, 11760, 12600, 13440, 14280, 15120, 15960, 16800, 17640, 18480, 19320, 20160, 21000, 21840, 22680, 23520, 24360, 25200, 26040, 26880, 27720, 28560, 29400, 30240, 31080, 31920

Offset: 0

N. J. A. Sloane, May 30 2010

Numbers that are divisible by all of 1,2,3,4,5,6,7,8.

Erich Friedman, What's Special About This Number?
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001477, A005843, A008596, A008603, A135628, A169823, A169825.

Cf. A249674, A317095.

840*Range[0,40] (* Harvey P. Dale, Nov 03 2015 *)

a(n)=840*n \\ Charles R Greathouse IV, Apr 16 2024

From Elmo R. Oliveira, Apr 16 2024: (Start)
G.f.: 840*x/(x-1)^2.
E.g.f.: 840*x*exp(x).
a(n) = 840*n = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 2*A169825(n) = 14*A169823(n) = 21*A317095(n) = 28*A249674(n) = 30*A135628(n) = 40*A008603(n) = 60*A008596(n) = 420*A005843(n) = 840*A001477(n). (End)

A334080 Number of Pythagorean triples among the divisors of 60*n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 3, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 6, 4, 6, 2, 8, 2, 6, 4, 5, 4, 9, 2, 4, 6, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 9, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 7, 6, 8, 2, 8, 4, 9, 2, 12, 2, 4, 6, 6, 4, 12, 2, 10, 5, 4, 2, 12, 4

Offset: 1

Michel Lagneau, Apr 14 2020

nonn

The odd numbers of the sequence are rare (see the table below).
The subsequence of odd terms begins with 1, 3, 3, 3, 5, 3, 5, 9, 3, 9, 7, 9, 5, 9, 9, 3, 11, 15, 5, 9, 5, 15, 9, 9, 9, 5, 19, 3, 15, 15, 9, ... (see the table at the link).
It is interesting to note that each set of divisors of A169823(n) contains m primitive Pythagorean triples for some n, m = 1, 2, ...
Examples:
- The set of divisors of A169823(1)= 60 contains only one primitive Pythagorean triple: (3, 4, 5).
- The set of divisors of A169823(136) = 8160 contains two primitive Pythagorean triples: (3, 4, 5) and (8, 15, 17).
- The set of divisors of A169823(910) = 54600 contains three primitive Pythagorean triples: (3, 4, 5), (5, 12, 13) and (7, 24, 25).
There is an interesting property: we observe that a(n) = A000005(n) except for n in the set {13, 26, 34, 39, 52, 65, 68, 70, 78, 91, 102, ...}. This set contains subset of numbers of the form 13*k, 34*k, 70*k, 203*k, 246*k, 259*k, ... for k = 1, 2, ...
We recognize the sequence A081752: {13, 34, 70, 203, 246, 259, 671, ...} (ordered product of the sides of primitive Pythagorean triangles divided by 60).
The following table shows the numbers of odd terms < 10^k for k = 2, 3, 4, 5, 6 and 7. For instance, among the 16 multiples of 60 less than 10^3, the divisors of the five numbers 60, 240, 540, 780 and 960 contain 1, 3, 3, 3 and 5 Pythagorean triples respectively, and that represents 31.25% of odd numbers.
+---------------+-----------------+---------------------+----------+
|   Intervals   |    Number of    | Number of odd terms |          |
|  D(k) < 10^k  | multiples of 60 |      in D(k)        |     %    |
| k = 2,3,...,7 |     in D(k)     |                     |          |
+---------------+-----------------+---------------------+----------+
|   < 10^2      |          1      |          1          | 100%     |
|   < 10^3      |         16      |          5          |  31.250% |
|   < 10^4      |        166      |         18          |  10.843% |
|   < 10^5      |       1666      |         72          |   4.321% |
|   < 10^6      |      16666      |        256          |   1.536% |
|   < 10^7      |     166666      |        879          |   0.527% |
|---------------+-----------------+---------------------+----------+

			a(4) = 3 because the divisors of A169823(4) = 240 are {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240} with 3 Pythagorean triples: (3, 4, 5), (6, 8, 10) and (12, 16, 20). The first triple is primitive.

Michel Lagneau, Odd terms.

Cf. A000005, A009003, A014442, A081752, A089120, A144255, A169823.

with(numtheory):
for n from 60 by 60 to 5400 do :
   d:=divisors(n):n0:=nops(d):it:=0:
    for i from 1 to n0-1 do:
     for j from i+1 to n0-2 do :
      for m from i+2 to n0 do:
       if d[i]^2 + d[j]^2 = d[m]^2
        then
        it:=it+1:
        else
       fi:
      od:
     od:
    od:
    printf(`%d, `,it):
   od:

ishypo(n) = setsearch(Set(factor(n)[, 1]%4), 1); \\ A009003
a(n) = {n *= 60; my(d=divisors(n), nb=0); for (i=3, #d, if (ishypo(d[i]), for (j=2, i-1, for (k=3, j-1, if (d[j]^2 + d[k]^2 == d[i]^2, nb++););););); nb;} \\ Michel Marcus, Apr 26 2020

A334382 Least k whose set of divisors contains exactly n Pythagorean triples, or 0 if no such k exists.

Original entry on oeis.org

60, 120, 240, 360, 960, 720, 3840, 1440, 2160, 2880, 8160, 3600, 69360, 8400, 8640, 7200, 32640, 9360, 16800, 14400, 34560, 24480, 130560, 18720, 77760, 54600, 28080, 25200, 67200, 37440, 11045580, 61200, 73440, 97920, 294000, 46800, 65520, 50400, 268800, 109200

Offset: 1

Michel Lagneau, Apr 26 2020

This is a subsequence of A169823: a(n) == 0 (mod 60) because one side of every Pythagorean triple is divisible by 3, another by 4, and another by 5. The smallest and best-known Pythagorean triple is (a, b, c) = (3, 4, 5).

			a(3) = 240 because the set of divisors {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240} contains 3 Pythagorean triples: (3, 4, 5), (6, 8, 10) and (12, 16, 20). The first triple is primitive.

Giovanni Resta, Table of n, a(n) for n = 1..450

Cf. A103605, A103606, A169823, A334080.

with(numtheory):
for n from 1 to 52 do :
ii:=0:
for k from 60 by 60 to 10^8 while(ii=0) do:
   d:=divisors(k):n0:=nops(d):it:=0:
    for i from 1 to n0-1 do:
     for j from i+1 to n0-2 do :
      for m from i+2 to n0 do:
       if d[i]^2 + d[j]^2 = d[m]^2
        then
        it:=it+1:
        else
       fi:
      od:
     od:
    od:
    if it = n
     then
     ii:=1: printf (`%d %d \n`,n,k):
     else
    fi:
od:
od:

a(31) from Giovanni Resta, Apr 27 2020

A096472 Numbers containing squares of Pythagorean triples in their divisor set.

Original entry on oeis.org

3600, 7200, 10800, 14400, 18000, 21600, 25200, 28800, 32400, 36000, 39600, 43200, 46800, 50400, 54000, 57600, 61200, 64800, 68400, 72000, 75600, 79200, 82800, 86400, 90000, 93600, 97200, 100800, 104400, 108000, 111600, 115200, 118800, 122400, 126000, 129600, 133200

Offset: 1

Reinhard Zumkeller, Aug 13 2004

a(n) = m * (A046083(k)*A046084(k)*A009000(k))^2 for appropriate, not necessarily unique m and k.

			5^2 + 12^2 = 13^2: 5^2, 12^2 and 13^2 are divisors of 608400 = (13*5*3*2^2)^2, therefore 608400 is a term.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Tanya Khovanova, Recursive Sequences.
Eric Weisstein's World of Mathematics, Pythagorean Triple.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. Pythagorean triples: A046083, A046084, A009000.

Cf. A044102, A094519, A169823.

Range[50]*3600 (* Paolo Xausa, Jul 01 2025 *)

my(x='x+O('x^38)); Vec(3600*x/(1-x)^2) \\ Elmo R. Oliveira, Jun 30 2025

a(n) = n*60^2.
From Elmo R. Oliveira, Jun 30 2025: (Start)
G.f.: 3600*x/(1-x)^2.
E.g.f.: 3600*x*exp(x).
a(n) = 60*A169823(n) = 100*A044102(n).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

Name clarified by Tanya Khovanova, Jul 05 2021

More terms from Elmo R. Oliveira, Jun 30 2025

A377418 a(n) is the smallest integer k whose set of divisors contains exactly n triples (x,y,z) of distinct divisors considered as integer-sided triangles with integer areas, or 0 if no such k exists.

Original entry on oeis.org

60, 120, 240, 360, 960, 720, 3480, 1440, 1680, 2880, 6600, 2520, 4200, 10440, 5460, 6240, 4680, 5040, 20400, 7800, 18360, 17160, 26520, 10080, 47040, 9360, 15120, 10920, 55080, 20160, 15600, 16380, 34320, 33600, 18720, 27300, 165240, 53040, 37800, 25200, 21840

Offset: 1

Michel Lagneau, Oct 27 2024

nonn

We observe that this sequence is a subsequence of A169823: a(n) == 0 (mod 60).

The area A of a triangle whose sides have lengths x, y, and z is given by Heron's formula: A = sqrt(s*(s-x)*(s-y)*(s-z)), where s = (x+y+z)/2.

			a(3) = 240 because there are 3 triples of divisors (3, 4, 5), (6, 8, 10) and (12, 16, 20) with integer areas 36, 576, 9216 respectively (Pythagorean triples). The first triple is primitive.
a(9)=1680 because there are 9 triples of divisors (3,4,5), (6,8,10), (7,15,20), (12,16,20), (14,30,40), (21,28,35), (28,60,80), (42,56,70), (84,112,140) with 5 Pythagorean triples : (3,4,5), (6,8,10), (21,28,35), (42,56,70), (84,112,70). The other 4 triangles are arbitrary.

Cf. A169823, A188158, A334382.

with(numtheory):
for n from 1 to 41 do:
ii:=0:
for m from 4 to 10^7 while(ii=0) do:it:=0:
 d:=divisors(m):n0:=nops(d):
  for i from 2 to n0-2 do:
   for j from i+1 to n0 do:
     for k from j+1 to n0 do:
       x:=d[i]:y:=d[j]:z:=d[k]:s:=(x+y+z)/2:A:=s*(s-x)*(s-y)*(s-z):
       if A>0 and sqrt(A)=floor(sqrt(A)) then it:=it+1:else
        fi:
       od:
    od:
   od:
    if it=n then printf(`%d %d \n`,it,m):ii:=1:
     else fi:
  od:
od:

cp's OEIS Frontend

A249674 a(n) = 30*n.

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A305548 a(n) = 27*n.

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A169825 Multiples of 420.

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A169827 Multiples of 840.

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A334080 Number of Pythagorean triples among the divisors of 60*n.

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A334382 Least k whose set of divisors contains exactly n Pythagorean triples, or 0 if no such k exists.

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A096472 Numbers containing squares of Pythagorean triples in their divisor set.

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