A008589 -id:A008589 - cp's OEIS frontend

A101447 Triangle read by rows: T(n,k) = (2k+1)(n+1-k), 0 <= k < n.

1, 2, 3, 3, 6, 5, 4, 9, 10, 7, 5, 12, 15, 14, 9, 6, 15, 20, 21, 18, 11, 7, 18, 25, 28, 27, 22, 13, 8, 21, 30, 35, 36, 33, 26, 15, 9, 24, 35, 42, 45, 44, 39, 30, 17, 10, 27, 40, 49, 54, 55, 52, 45, 34, 19, 11, 30, 45, 56, 63, 66, 65, 60, 51, 38, 21, 12, 33, 50, 63, 72, 77, 78, 75, 68, 57, 42, 23

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson, Jan 19 2005

The triangle is generated from the product of matrix A and matrix B, i.e., A * B where A = the infinite lower triangular matrix:
  1 0 0 0 0 ...
  1 1 0 0 0 ...
  1 1 1 0 0 ...
  1 1 1 1 0 ...
  1 1 1 1 1 ...
... and B = the infinite lower triangular matrix:
  1 0 0 0 0 ...
  1 3 0 0 0 ...
  1 3 5 0 0 ...
  1 3 5 7 0 ...
  1 3 5 7 9 ...
  ...
Row sums give the square pyramidal numbers A000330.
T(n+0,0)=1*n=A000027(n+1); T(n+1,1)=3*n=A008585(n); T(n+2,2)=5*n=A008587(n); T(n+3,3)=7*n=A008589(n); etc. So T(n,0)*T(n,1)=3*n*(n+1)=A028896(n) (6 times triangular numbers). T(n,1)*T(n,2)/10=3*n*(n+1)/2=A045943(n) for n>0 T(n,2)*T(n,3)/10=7/2*n*(n+1)=A024966(n) for n>1 (7 times triangular numbers), etc.
From Gary W. Adamson, Apr 25 2010: (Start)
Consider the following array, signed as shown:
  ...
  1,   3,  5,   7,  9,  11, ...
  2,  -6, 10, -14, 18, -22, ...
  3,   9, 15,  21, 27,  33, ...
  4, -12, 20, -28, 36, -44, ...
  5,  15, 25,  35, 45,  55, ...
  6, -18, 30, -42, 54, -66, ...
  7,  21, 35,  49, 63,  77, ...
  ...
Let each term (+, -)k = (+, -) phi^(-k).
Consider the inverse terms of the Lucas series (1/1, 1/3, 1/4, 1/7, ...).
By way of example, let q = phi = 1.6180339...; then
...
1/1  = q^(-1) + q^(-3) + q^(-5) + q^(-7) + q^(-9) + ...
1/3  = q^(-2) - q^(-6) + q^(-10) - q^(-14) + q^(-18) + ...
1/4  = q^(-3) + q^(-9) + q^(-15) + q^(-21) + q^(-27) +...
1/7  = q^(-4) - q^(-12) + q^(-20) - q^(-28) + q^(-36) + ...
1/11 = q^(-5) + q^(-15) + q^(-25) + q^(-35) + q^(-45) + ...
...
Relating to the Pell series, the corresponding "Lucas"-like series is (2, 6, 14, 34, 82, 198, ...) such that herein, q = 2.414213... = (1 + sqrt(2)).
Then analogous to the previous set,
...
1/2 = q^(-1) + q^(-3) + q^(-5) + q^(-7) + ...
1/6 = q^(-2) - q^(-6) + q^(-10) - q^(-14) + q^(-18) + ...
... (End)

			From _Bruno Berselli_, Feb 10 2014: (Start)
Triangle begins:
   1;
   2,  3;
   3,  6,  5;
   4,  9, 10,  7;
   5, 12, 15, 14,  9;
   6, 15, 20, 21, 18, 11;
   7, 18, 25, 28, 27, 22, 13;
   8, 21, 30, 35, 36, 33, 26, 15;
   9, 24, 35, 42, 45, 44, 39, 30, 17;
  10, 27, 40, 49, 54, 55, 52, 45, 34, 19;
  11, 30, 45, 56, 63, 66, 65, 60, 51, 38, 21;
  etc.
(End)

Cf. A094728 (triangle generated by B*A), A000330.

t[n_, k_] := If[n < k, 0, (2*k + 1)*(n - k + 1)]; Flatten[ Table[ t[n, k], {n, 0, 11}, {k, 0, n}]] (* Robert G. Wilson v, Jan 20 2005 *)

PARI
```
T(n,k)=if(n
				
```

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)

A209294 a(n) = (7n^2 - 7n + 4)/2.

Original entry on oeis.org

2, 9, 23, 44, 72, 107, 149, 198, 254, 317, 387, 464, 548, 639, 737, 842, 954, 1073, 1199, 1332, 1472, 1619, 1773, 1934, 2102, 2277, 2459, 2648, 2844, 3047, 3257, 3474, 3698, 3929, 4167, 4412, 4664, 4923, 5189, 5462

Offset: 1

Marco Piazzalunga, Jan 17 2013

a(n) is the sum of the n-th centered triangular number and n-th centered square number.

Difference of consecutive terms gives A008589 (multiples of 7).

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A005448, A001844, A008589.

[(7*n^2 - 7*n + 4)/2: n in [1..30]]; // G. C. Greubel, Jan 04 2018

Table[(7*n^2 - 7*n + 4)/2, {n, 1, 50}] (* G. C. Greubel, Jan 04 2018 *)
LinearRecurrence[{3,-3,1},{2,9,23},40] (* Harvey P. Dale, Nov 02 2020 *)

a(n)=7*n*(n-1)/2+2 \\ Charles R Greathouse IV, Jan 17 2013

a(n) = (7*n^2 - 7*n + 4) = 7*T(n) + 2 with T = A000217.
G.f.: x*(2+3*x+2*x^2)/(1-x)^3. - Bruno Berselli, Jan 18 2013
a(n) = a(-n+1) = 3*a(n-1)-3*a(n-2)+a(n-3).  - Bruno Berselli, Jan 18 2013
a(n) = 1 + A069099(n). - Omar E. Pol, Apr 27 2017
E.g.f.: ((7*x^2 + 4)*exp(x) - 4)/2. - G. C. Greubel, Jan 04 2018

A279895 a(n) = n(5n + 11)/2.

Original entry on oeis.org

0, 8, 21, 39, 62, 90, 123, 161, 204, 252, 305, 363, 426, 494, 567, 645, 728, 816, 909, 1007, 1110, 1218, 1331, 1449, 1572, 1700, 1833, 1971, 2114, 2262, 2415, 2573, 2736, 2904, 3077, 3255, 3438, 3626, 3819, 4017, 4220, 4428, 4641, 4859, 5082, 5310, 5543, 5781, 6024, 6272, 6525

Offset: 0

Bruno Berselli, Dec 22 2016

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000566, A008589, A017281.
Second bisection of A165720.
The first differences are in A016885.
Cf. similar sequences provided by P(s,m)+s*m, where P(s,m) = ((s-2)*m^2-(s-4)*m)/2 is the m-th s-gonal number: A008585 (s=2), A055999 (s=3), A028347 (s=4), A140091 (s=5), A033537 (s=6), this sequence (s=7), A067725 (s=8).

Magma
```
[n*(5*n+11)/2: n in [0..60]];
```

Table[n (5 n + 11)/2, {n, 0, 60}]
LinearRecurrence[{3,-3,1},{0,8,21},60] (* Harvey P. Dale, Nov 14 2022 *)

PARI
```
vector(60, n, n--; n*(5*n+11)/2)
```
Python
```
[n*(5*n+11)/2 for n in range(60)]
```
Sage
```
[n*(5*n+11)/2 for n in range(60)]
```

O.g.f.: x*(8 - 3*x)/(1 - x)^3.
E.g.f.: x*(16 + 5*x)*exp(x)/2.
a(n+h) - a(n-h) = h*A017281(n+1), with h>=0. A particular case:
a(n) - a(-n) = 11*n = A008593(n).
a(n+h) + a(n-h) = 2*a(n) + A033429(h), with h>=0. A particular case:
a(n) + a(-n) = A033429(n).
a(n) - a(n-2) = A017281(n) for n>1. Also:
40*a(n) + 121 = A017281(n+1)^2.
a(n) = A000566(n) + 7*n, also a(n) = A000566(n) + A008589(n). - Michel Marcus, Dec 22 2016

A336596 Numbers whose number of divisors is divisible by 7.

Original entry on oeis.org

64, 192, 320, 448, 576, 704, 729, 832, 960, 1088, 1216, 1344, 1458, 1472, 1600, 1728, 1856, 1984, 2112, 2240, 2368, 2496, 2624, 2752, 2880, 2916, 3008, 3136, 3264, 3392, 3520, 3645, 3648, 3776, 3904, 4032, 4160, 4288, 4416, 4544, 4672, 4800, 4928, 5056, 5103

Offset: 1

Amiram Eldar, Jul 26 2020

nonn

The asymptotic density of this sequence is 1 - zeta(7)/zeta(6) = 0.0088404638... (Sathe, 1945).

			64 is a term since A000005(64) = 7 is divisible by 7.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical Notes, XIII. A Sequal to Note IV, Elemente der Mathematik, Vol. 18 (1963), pp. 8-11.
S. S. Pillai, On a congruence property of the divisor function, J. Indian Math. Soc. (N. S.), Vol. 6, (1942), pp. 118-119.
L. G. Sathe, On a congruence property of the divisor function, American Journal of Mathematics, Vol. 67, No. 3 (1945), pp. 397-406.

Cf. A000005, A008589, A059269, A336595.

Cf. A030516, A113851 and A138031 are subsequences.

q:= n-> is(irem(numtheory[tau](n), 7)=0):
select(q, [$1..5500])[];  # Alois P. Heinz, Jul 26 2020

Select[Range[5000], Divisible[DivisorSigma[0, #], 7] &]

A030516 UNION A030632 UNION A137484 UNION A137491 UNION A175745 UNION A175750 UNION ... - R. J. Mathar, May 05 2023

A016985 a(n) = (7n)^5.

Original entry on oeis.org

0, 16807, 537824, 4084101, 17210368, 52521875, 130691232, 282475249, 550731776, 992436543, 1680700000, 2706784157, 4182119424, 6240321451, 9039207968, 12762815625, 17623416832, 23863536599, 31757969376, 41615795893, 53782400000, 68641485507, 86617093024

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000584, A008589.

[(7*n)^5: n in [0..25]]; // Vincenzo Librandi, May 24 2011

A016985:=n->(7*n)^5: seq(A016985(n), n=0..30); # Wesley Ivan Hurt, Aug 27 2015

(7Range[0,20])^5  (* Harvey P. Dale, Feb 13 2011 *)
CoefficientList[Series[16807 (x + 26 x^2 + 66 x^3 + 26 x^4 + x^5)/(x - 1)^6, {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 27 2015 *)

From Wesley Ivan Hurt, Aug 27 2015: (Start)
a(n) = (7n)^5 = 16807*n^5 = A000584(A008589(n)).
G.f.: 16807*(x+26*x^2+66*x^3+26*x^4+x^5)/(x-1)^6.
a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6), n>5. (End)

A101468 Triangle read by rows: T(n,k)=(n+1-k)(3k+1).

Original entry on oeis.org

1, 2, 4, 3, 8, 7, 4, 12, 14, 10, 5, 16, 21, 20, 13, 6, 20, 28, 30, 26, 16, 7, 24, 35, 40, 39, 32, 19, 8, 28, 42, 50, 52, 48, 38, 22, 9, 32, 49, 60, 65, 64, 57, 44, 25, 10, 36, 56, 70, 78, 80, 76, 66, 50, 28, 11, 40, 63, 80, 91, 96, 95, 88, 75, 56, 31, 12, 44, 70, 90, 104, 112, 114

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson, Jan 21 2005

The triangle is generated from the product A*B
of the infinite lower triangular matrices A =
1 0 0 0...
1 1 0 0...
1 1 1 0...
1 1 1 1...
... and B =
1 0 0 0...
1 4 0 0...
1 4 7 0...
1 4 7 10...
...
Row sums give pentagonal pyramidal numbers A002411 T(n+0,0)= 1*n=A000027(n) T(n+0,1)= 4*n=A008586(n) T(n+1,2)= 7*n=A008589(n) T(n+2,3)=10*n=A008592(n) ...
so for example T(n+1,n-0)=6*n+2=A016933(n) T(n+1,n-1)=9*n+3=A017197(n) T(n+2,n-1)=12*n+4=A017569(n)
T(n,0)*T(n,1) = A033996(n) (8 times triangular numbers)
T(n,n)*T(n,0) = A000567(n+1) (Octagonal numbers) etc.

			Triangle begins:
1,
2,  4,
3,  8,  7,
4,  12, 14, 10,
5,  16, 21, 20, 13,
6,  20, 28, 30, 26, 16,
7,  24, 35, 40, 39, 32, 19,
8,  28, 42, 50, 52, 48, 38, 22,
9,  32, 49, 60, 65, 64, 57, 44, 25,
10, 36, 56, 70, 78, 80, 76, 66, 50, 28,
11, 40, 63, 80, 91, 96, 95, 88, 75, 56, 31, etc.
[_Bruno Berselli_, Feb 10 2014]

Cf. A095871 (product B*A), A002411.

t[n_, k_] := If[n < k, 0, (3*k + 1)*(n - k + 1)]; Flatten[ Table[ t[n, k], {n, 0, 11}, {k, 0, n}]] (* Robert G. Wilson v, Jan 21 2005 *)

T(n,k)=if(k>n,0,(n-k+1)*(3*k+1)) for(i=0,10, for(j=0,i,print1(T(i,j),", "));print())

A117795 Heptagonal numbers divisible by 7.

Original entry on oeis.org

0, 7, 112, 189, 469, 616, 1071, 1288, 1918, 2205, 3010, 3367, 4347, 4774, 5929, 6426, 7756, 8323, 9828, 10465, 12145, 12852, 14707, 15484, 17514, 18361, 20566, 21483, 23863, 24850, 27405, 28462, 31192, 32319, 35224, 36421, 39501, 40768, 44023

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), Apr 29 2006

Intersection of A000566 and A008589. Their indices are given by A047352. - Michel Marcus, Feb 27 2014

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000566, A008589, A047352.

Select[PolygonalNumber[7,Range[0,200]],Divisible[#,7]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 02 2019 *)

isok(n) = ispolygonal(n, 7) && !(n % 7); \\ Michel Marcus, Feb 27 2014

A137182 Lucky numbers (A000959) which are congruent to 0 mod 7.

Original entry on oeis.org

7, 21, 49, 63, 105, 133, 189, 231, 259, 273, 357, 385, 399, 427, 483, 511, 553, 651, 679, 693, 735, 777, 805, 819, 903, 931, 1029, 1057, 1155, 1183, 1197, 1281, 1309, 1323, 1365, 1435, 1491, 1519, 1533, 1575, 1645, 1659, 1701, 1771, 1827, 1869, 1995, 2023, 2065

Offset: 1

N. J. A. Sloane, Mar 07 2008

nonn

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

Intersection of A000959 and A008589.

A166389 Multiples of 7 whose reversal + 1 is also a multiple of 7.

Original entry on oeis.org

14, 84, 140, 147, 231, 238, 322, 329, 392, 399, 413, 483, 504, 574, 665, 756, 840, 847, 931, 938, 1043, 1134, 1225, 1295, 1316, 1386, 1400, 1407, 1470, 1477, 1561, 1568, 1652, 1659, 1743, 1834, 1925, 1995, 2044, 2135, 2226, 2296, 2310, 2317, 2380, 2387

Offset: 1

Claudio Meller, Oct 13 2009

G. C. Greubel, Table of n, a(n) for n = 1..10000

Subsequence of A008589.

Select[7 Range[6!], Divisible[FromDigits[Reverse[IntegerDigits[#]]] + 1, 7] &] (* G. C. Greubel, May 12 2016 *)
Select[7Range[400],Mod[IntegerReverse[#]+1,7]==0&] (* Harvey P. Dale, Aug 16 2024 *)

isok(n) = !(n%7) && !((subst(Polrev(digits(n)),x,10)+1) % 7); \\ Michel Marcus, May 12 2016

cp's OEIS Frontend

A101447 Triangle read by rows: T(n,k) = (2*k+1)*(n+1-k), 0 <= k < n.

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Mathematica

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A169825 Multiples of 420.

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A209294 a(n) = (7*n^2 - 7*n + 4)/2.

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Magma

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A279895 a(n) = n*(5*n + 11)/2.

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A336596 Numbers whose number of divisors is divisible by 7.

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Maple

Mathematica

Formula

A016985 a(n) = (7n)^5.

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Maple

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A101468 Triangle read by rows: T(n,k)=(n+1-k)*(3*k+1).

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PARI

A101447 Triangle read by rows: T(n,k) = (2k+1)(n+1-k), 0 <= k < n.

A209294 a(n) = (7n^2 - 7n + 4)/2.

A279895 a(n) = n(5n + 11)/2.

A101468 Triangle read by rows: T(n,k)=(n+1-k)(3k+1).