A008602 -id:A008602 - cp's OEIS frontend

A121040 Multiples of 20 containing a 20 in their decimal representation.

20, 120, 200, 220, 320, 420, 520, 620, 720, 820, 920, 1020, 1120, 1200, 1220, 1320, 1420, 1520, 1620, 1720, 1820, 1920, 2000, 2020, 2040, 2060, 2080, 2120, 2200, 2220, 2320, 2420, 2520, 2620, 2720, 2820, 2920, 3020, 3120, 3200, 3220, 3320, 3420, 3520

Offset: 1

Reinhard Zumkeller, Jul 21 2006

Not the same as A044352.

Index entries for 10-automatic sequences.

Cf. A121041, A008602, A011531, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039.

Select[20*Range[200],SequenceCount[IntegerDigits[#],{2,0}]>0&] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, Nov 27 2015 *)

is(n)=if(n%20, return(0)); while(n>19, if(n%100==20, return(1)); n\=10); 0 \\ Charles R Greathouse IV, Feb 12 2017

a(n) ~ 20n. - Charles R Greathouse IV, Feb 12 2017

Corrected by T. D. Noe, Oct 25 2006

A008601 Multiples of 19.

Original entry on oeis.org

0, 19, 38, 57, 76, 95, 114, 133, 152, 171, 190, 209, 228, 247, 266, 285, 304, 323, 342, 361, 380, 399, 418, 437, 456, 475, 494, 513, 532, 551, 570, 589, 608, 627, 646, 665, 684, 703, 722, 741, 760, 779, 798, 817, 836, 855, 874, 893, 912, 931, 950, 969, 988

Offset: 0

N. J. A. Sloane

Ivan Panchenko, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 331.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008599, A008600, A008602.

Range[0, 1500, 19] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)

a(n)=19*n; \\ Michel Marcus, Dec 27 2014

(floor(a(n)/10) + 2*(a(n) mod 10)) == 0 modulo 19, see A076312. - Reinhard Zumkeller, Oct 06 2002
From Vincenzo Librandi, Dec 24 2010: (Start)
a(n) = 19*n.
a(n) = 2*a(n-1) - a(n-2).
G.f.: 19*x/(x-1)^2. (End)
From Elmo R. Oliveira, Apr 10 2025: (Start)
E.g.f.: 19*x*exp(x).
a(n) = (A008600(n) + A008602(n))/2. (End)

A008603 Multiples of 21.

Original entry on oeis.org

0, 21, 42, 63, 84, 105, 126, 147, 168, 189, 210, 231, 252, 273, 294, 315, 336, 357, 378, 399, 420, 441, 462, 483, 504, 525, 546, 567, 588, 609, 630, 651, 672, 693, 714, 735, 756, 777, 798, 819, 840, 861, 882, 903, 924, 945, 966, 987, 1008, 1029, 1050, 1071, 1092

Offset: 0

N. J. A. Sloane

Sum of the numbers from 3*(n-1) to 3*(n+1). - Bruno Berselli, Oct 25 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 333.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008585, A008589, A008601, A008602.

A008603:= func< n | 21*n >; // G. C. Greubel, Apr 16 2025

Range[0, 1500, 21] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)
CoefficientList[Series[21 x/(1-x)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Jun 10 2013 *)

a(n)=21*n \\ Charles R Greathouse IV, Oct 07 2015

def A008603(n): return 21*n # G. C. Greubel, Apr 16 2025

G.f.: 21*x/(1-x)^2. - Vincenzo Librandi, Jun 10 2013
From Elmo R. Oliveira, Apr 08 2025: (Start)
E.g.f.: 21*x*exp(x).
a(n) = A008585(A008589(n)) = A008589(A008585(n)). (End)

A008604 Multiples of 22.

Original entry on oeis.org

0, 22, 44, 66, 88, 110, 132, 154, 176, 198, 220, 242, 264, 286, 308, 330, 352, 374, 396, 418, 440, 462, 484, 506, 528, 550, 572, 594, 616, 638, 660, 682, 704, 726, 748, 770, 792, 814, 836, 858, 880, 902, 924, 946, 968, 990, 1012, 1034, 1056, 1078, 1100, 1122, 1144

Offset: 0

N. J. A. Sloane

Even numbers for which the sum of "digits" base 100 is divisible by 11. - Daniel Forgues, Feb 22 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 334.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008593, A008602, A008603.

Range[0, 1500, 22] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)
CoefficientList[Series[22 x / (x - 1)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Jun 10 2013 *)
LinearRecurrence[{2,-1},{0,22},50] (* Harvey P. Dale, Aug 06 2018 *)

a(n)=22*n \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 22*x/(x-1)^2. - Vincenzo Librandi, Jun 10 2013
a(n) = A008593(2n). - Daniel Forgues, Feb 22 2016
From Wesley Ivan Hurt, May 19 2024: (Start)
a(n) = 22*n.
a(n) = 2*a(n-1) - a(n-2). (End)
E.g.f.: 22*x*exp(x). - Stefano Spezia, Mar 02 2025

A195322 a(n) = 20*n^2.

Original entry on oeis.org

0, 20, 80, 180, 320, 500, 720, 980, 1280, 1620, 2000, 2420, 2880, 3380, 3920, 4500, 5120, 5780, 6480, 7220, 8000, 8820, 9680, 10580, 11520, 12500, 13520, 14580, 15680, 16820, 18000, 19220, 20480, 21780, 23120, 24500, 25920, 27380, 28880, 30420, 32000, 33620, 35280

Offset: 0

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 0, in the direction 0, 20, ..., in the square spiral whose vertices are the generalized dodecagonal numbers A195162. Semiaxis opposite to A195317 in the same spiral.
a(n) is the sum of all the integers less than 10*n which are not multiple of 2 or 5. a(2) = (1 + 3 + 7 + 9) + (11 + 13 + 17 + 19) = 20 + 60 = 80 = 20 * 2^2. (Link Crux Mathematicorum). - Bernard Schott, May 15 2017
Number of terms less than 10^k (k=0, 1, 2, ...): 1, 1, 3, 8, 23, 71, 224, 708, 2237, 7072, 22361, 70711, ... - Muniru A Asiru, Feb 01 2018

			From _Muniru A Asiru_, Feb 01 2018: (Start)
n=0, a(0) = 20*0^2 = 0.
n=1, a(1) = 20*1^2 = 20.
n=1, a(2) = 20*2^2 = 80.
n=1, a(3) = 20*3^2 = 180.
n=1, a(4) = 20*4^2 = 320.
...
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Léo Sauvé, Problem 53, Crux Mathematicorum, Vol. 1, Nov. 1975, page 88.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A195148.
Cf. A016802, A033581, A033583, A135453, A139098, A144555, A195321, A195323.
Cf. A000290, A001105, A005899, A010014, A016742, A033429, A195162, A195317.
Cf. A008602.

List([0..10^3],n->20*n^2); # Muniru A Asiru, Feb 01 2018

[20*n^2: n in [0..40]]; // Vincenzo Librandi, Sep 20 2011

a := n -> 20*n^2; seq(a(n), n=0..10^3); # Muniru A Asiru, Feb 01 2018

20 Range[0, 40]^2 (* or *) LinearRecurrence[{3, -3, 1}, {0, 20, 80}, 50] (* Harvey P. Dale, Jan 18 2013 *)

a(n) = 20*n^2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 20*A000290(n) = 10*A001105(n) = 5*A016742(n) = 4*A033429(n) = 2*A033583(n).
a(0)=0, a(1)=20, a(2)=80; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jan 18 2013
a(n) = A010014(n) - A005899(n) for n > 0. - R. J. Cano, Sep 29 2015
From Elmo R. Oliveira, Nov 30 2024: (Start)
G.f.: 20*x*(1 + x)/(1-x)^3.
E.g.f.: 20*x*(1 + x)*exp(x).
a(n) = n*A008602(n) = A195148(2*n). (End)

A094053 Triangle read by rows: T(n,k) = k*(n-k), 1 <= k <= n.

Original entry on oeis.org

0, 1, 0, 2, 2, 0, 3, 4, 3, 0, 4, 6, 6, 4, 0, 5, 8, 9, 8, 5, 0, 6, 10, 12, 12, 10, 6, 0, 7, 12, 15, 16, 15, 12, 7, 0, 8, 14, 18, 20, 20, 18, 14, 8, 0, 9, 16, 21, 24, 25, 24, 21, 16, 9, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, 0, 12

Offset: 1

Reinhard Zumkeller, May 31 2004

T(n,k) = A003991(n-1,k) for 1 <= k < n;
T(n,k) = T(n,n-1-k) for k < n;
T(n,1) = n-1; T(n,n) = 0;          T(n,2) = A005843(n-2) for n > 1;
T(n,3) = A008585(n-3) for n>2;     T(n,4) = A008586(n-4) for n > 3;
T(n,5) = A008587(n-5) for n>4;     T(n,6) = A008588(n-6) for n > 5;
T(n,7) = A008589(n-7) for n>6;     T(n,8) = A008590(n-8) for n > 7;
T(n,9) = A008591(n-9) for n>8;     T(n,10) = A008592(n-10) for n > 9;
T(n,11) = A008593(n-11) for n>10;  T(n,12) = A008594(n-12) for n > 11;
T(n,13) = A008595(n-13) for n>12;  T(n,14) = A008596(n-14) for n > 13;
T(n,15) = A008597(n-15) for n>14;  T(n,16) = A008598(n-16) for n > 15;
T(n,17) = A008599(n-17) for n>16;  T(n,18) = A008600(n-18) for n > 17;
T(n,19) = A008601(n-19) for n>18;  T(n,20) = A008602(n-20) for n > 19;
Row sums give A000292; triangle sums give A000332;
All numbers m > 0 occur A000005(m) times;
A002378(n) = T(A005408(n),n+1) = n*(n+1).
k-th columns are arithmetic progressions with step k, starting with 0. If a zero is prefixed to the sequence, then we get a new table where the columns are again arithmetic progressions with step k, but starting with k, k=0,1,2,...: 1st column = (0,0,0,...), 2nd column = (1,2,3,...), 3rd column = (2,4,6,8,...), etc. - M. F. Hasler, Feb 02 2013
Construct the infinite-dimensional matrix representation of angular momentum operators (J_1,J_2,J_3) in the Jordan-Schwinger form (cf. Harter, Klee, Schwinger). The triangle terms T(n,k) = T(2j,j+m) satisfy: (1/2)T(2j,j+m)^(1/2) =  =  = i  = -i . Matrices for J_1 and J_2 are sparse. These equalities determine the only nonzero entries. - Bradley Klee, Jan 29 2016
T(n+1,k+1) is the number of degrees of freedom of a k-dimensional affine subspace within an n-dimensional vector space. This is most readily interpreted geometrically: e.g. in 3 dimensions a line (1-dimensional subspace) has T(4,2) = 4 degrees of freedom and a plane has T(4,3) = 3. T(n+1,1) = n indicates that points in n dimensions have n degrees of freedom. T(n,n) = 0 for any n as all n-dimensional spaces in an n-dimensional space are equivalent. - Daniel Leary, Apr 29 2020

			From _M. F. Hasler_, Feb 02 2013: (Start)
Triangle begins:
  0;
  1, 0;
  2, 2, 0;
  3, 4, 3, 0;
  4, 6, 6, 4, 0;
  5, 8, 9, 8, 5, 0;
  (...)
If an additional 0 was added at the beginning, this would become:
  0;
  0, 1;
  0, 2, 2;
  0, 3, 4; 3;
  0, 4, 6, 6, 4;
  0, 5, 8, 9, 8, 5;
  ... (End)

W. Harter, Principles of Symmetry, Dynamics, Spectroscopy, Wiley, 1993, Ch. 5, page 345-346.
B. Klee, Quantum Angular Momentum Matrices, Wolfram Demonstrations Project, 2016.
J. Schwinger, On Angular Momentum , Cambridge: Harvard University, Nuclear Development Associates, Inc., 1952.

J_3: A114327; J_1^2, J_2^2: A141387, A268759.
Cf. A000005, A002378, A003991, A005408.
Cf. A000292 (row sums), A000332 (triangle sums).
T(n,k) for values of k:
  A005843 (k=2), A008585 (k=3), A008586 (k=4), A008587 (k=5), A008588 (k=6), A008589 (k=7), A008590 (k=8), A008591 (k=9), A008592 (k=10), A008593 (k=11), A008594 (k=12), A008595 (k=13), A008596 (k=14), A008597 (k=15), A008598 (k=16), A008599 (k=17), A008600 (k=18), A008601 (k=19), A008602 (k=20).

/* As triangle */ [[k*(n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jan 30 2016

Flatten[Table[(j - m) (j + m + 1), {j, 0, 10, 1/2}, {m, -j, j}]] (* Bradley Klee, Jan 29 2016 *)

{for(n=1, 13, for(k=1, n, print1(k*(n - k)," ");); print(););} \\ Indranil Ghosh, Mar 12 2017

A169823 Multiples of 60.

Original entry on oeis.org

0, 60, 120, 180, 240, 300, 360, 420, 480, 540, 600, 660, 720, 780, 840, 900, 960, 1020, 1080, 1140, 1200, 1260, 1320, 1380, 1440, 1500, 1560, 1620, 1680, 1740, 1800, 1860, 1920, 1980, 2040, 2100, 2160, 2220, 2280, 2340, 2400, 2460, 2520, 2580, 2640, 2700

Offset: 0

N. J. A. Sloane, May 29 2010

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

Erich Friedman, What's Special About This Number? (See the entry for "60")
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001477, A005843, A008585, A008586, A008587, A008588, A008592, A008594.

Cf. A008597, A008602, A169825, A169827, A249674.

Range[0, 2700, 60] (* Vladimir Joseph Stephan Orlovsky, Jul 12 2011 *)

a(n)=60*n \\ Charles R Greathouse IV, Mar 19 2017

From Elmo R. Oliveira, Apr 16 2024: (Start)
G.f.: 60*x/(x-1)^2.
E.g.f.: 60*x*exp(x).
a(n) = 60*n = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 2*A249674(n) = 3*A008602(n) = 4*A008597(n) = 5*A008594(n) = 6*A008592(n) = 10*A008588(n) = 12*A008587(n) = 15*A008586(n) = 20*A008585(n) = 30*A005843(n) = 60*A001477(n) = A169827(n)/14 = A169825(n)/7. (End)

A110303 Alternators.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

Walter Nissen, Jul 18 2005

An alternating integer is a positive integer for which, in base-10, the parity of its digits alternates. E.g., 121 is alternating because its consecutive digits are odd-even-odd, 1 being odd and 2 even. Of course, 1234567890 is also alternating. An alternator is a positive integer which has a multiple which is alternating.

This sequence is the answer to the 6th problem proposed the 2nd day by Iran during the 45th International Mathematical Olympiad, in Athens (Greece), 2004 (see links). - Bernard Schott, Apr 12 2021

			11 is an alternator and in the sequence because it has a multiple which is alternating. The least of these multiples is 121.

Michael De Vlieger, Table of n, a(n) for n = 1..10001 (adapted to offset by Michel Marcus)
45th International Mathematical Olympiad (45th IMO), Problem #6 and Solution, Mathematics Magazine, 2005, Vol. 78, No. 3, pp. 247, 250-251.
The IMO Compendium, Problem 6, 45th IMO 2004.
Index to sequences related to Olympiads.

Cf. A030141, A030142, A110304, A110305, A008602 (complement), A343335, A343336.

Select[Range[75], Mod[#, 20] != 0 &] (* Michael De Vlieger, Apr 13 2021 *)

Positive n, not congruent to 0 mod 20.

a(n + 19) = a(n) + 20. - David A. Corneth, Apr 13 2021

Offset 1 from Michel Marcus, May 12 2021

A008605 Multiples of 23.

Original entry on oeis.org

0, 23, 46, 69, 92, 115, 138, 161, 184, 207, 230, 253, 276, 299, 322, 345, 368, 391, 414, 437, 460, 483, 506, 529, 552, 575, 598, 621, 644, 667, 690, 713, 736, 759, 782, 805, 828, 851, 874, 897, 920, 943, 966, 989, 1012, 1035, 1058, 1081, 1104, 1127, 1150

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 335.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008602, A008603, A008604, A008606.

Range[0, 1500, 23] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)
CoefficientList[Series[23 x / (x - 1)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Jun 10 2013 *)

a(n)=23*n \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 23*x/(x-1)^2. - Vincenzo Librandi, Jun 10 2013
E.g.f.: 23*x*exp(x). - Stefano Spezia, Mar 02 2025
From Elmo R. Oliveira, Apr 10 2025: (Start)
a(n) = 23*n = (A008604(n) + A008606(n))/2.
a(n) = 2*a(n-1) - a(n-2). (End)

A094519 Numbers having at least one pair (x,y) of divisors with x

Original entry on oeis.org

6, 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 90, 96, 100, 102, 108, 110, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 180, 182, 186, 192, 198, 200, 204, 210, 216, 220, 222, 224, 228, 234, 240, 246

Offset: 1

Reinhard Zumkeller, May 06 2004

nonn

If m is in the sequence then so is k*m for k > 0. Furthermore, all terms are even. - David A. Corneth, Aug 31 2019
If (x,y) = (1,m) with m > 1, then oblong numbers m*(m+1) >= 6 belong to this sequence, and each oblong number >= 6 is a primitive term of the subsequence {k*m*(m+1), k >= 1}. Examples: with pair (1,2), we get multiples of 6 (see A008588); with (1,3) we get multiples of 12 (see A008594); with (1,4) we get multiples of 20 (see A008602); with (1,7) we get multiples of 56. - Bernard Schott, Aug 31 2019
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 22, 230, 2317, 23201, 232209, 2322920, 23232166, 232332309, 2323370184, ... . Apparently, the asymptotic density of this sequence exists and equals 0.23233... . - Amiram Eldar, Apr 20 2025

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

Cf. A094518.
Complement of A094520.
A superset of A088723. - R. J. Mathar, Sep 16 2007
Subsequences: A002378 \ {0, 2}, A008588 \ {0}, A008602 \ {0}.

aQ[n_] := AnyTrue[Total /@ Subsets[Divisors[n], {2}], Divisible[n, #] &]; Select[Range[250], aQ] (* Amiram Eldar, Aug 31 2019 *)

is(n) = {my(d = divisors(n)); for(i = 1, #d - 2, for(j = i + 1, #d - 1, if(n % (d[i] + d[j]) == 0, return(1) ) ) ); 0 } \\ David A. Corneth, Aug 31 2019

from itertools import count, islice
from sympy import divisors
def A094519_gen(): # generator of terms
    for n in count(1):
        for i in range(1,len(d:=divisors(n))):
            di = d[i]
            for j in range(i):
                if n % (di+d[j]) == 0:
                    yield n
                    break
            else:
                continue
            break
A094519_list = list(islice(A094519_gen(),20)) # Chai Wah Wu, Dec 26 2021

A094518(a(n)) > 0.

cp's OEIS Frontend

A121040 Multiples of 20 containing a 20 in their decimal representation.

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A008601 Multiples of 19.

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A008603 Multiples of 21.

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A008604 Multiples of 22.

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A195322 a(n) = 20*n^2.

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A094053 Triangle read by rows: T(n,k) = k*(n-k), 1 <= k <= n.

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A169823 Multiples of 60.

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