A008601 -id:A008601 - cp's OEIS frontend

A121039 Multiples of 19 containing a 19 in their decimal representation.

19, 190, 1197, 1900, 1919, 1938, 1957, 1976, 1995, 3192, 3819, 4199, 5719, 6194, 7619, 9196, 9519, 11191, 11419, 11913, 11932, 11951, 11970, 11989, 12198, 13319, 14193, 15219, 17119, 17195, 19000, 19019, 19038, 19057, 19076, 19095, 19114

Offset: 1

Reinhard Zumkeller, Jul 21 2006

Index entries for 10-automatic sequences.

Cf. A121041, A008601, A011531, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121040.

Select[19*Range[1500],MemberQ[Partition[IntegerDigits[#],2,1],{1,9}]&] (* Harvey P. Dale, Jun 08 2014 *)

is(n)=if(n%19, return(0)); while(n>18, if(n%100==19, return(1)); n\=10); 0 \\ Charles R Greathouse IV, Feb 12 2017

a(n) ~ 19n. - Charles R Greathouse IV, Feb 12 2017

Corrected by T. D. Noe, Oct 25 2006

A008602 Multiples of 20.

Original entry on oeis.org

0, 20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220, 240, 260, 280, 300, 320, 340, 360, 380, 400, 420, 440, 460, 480, 500, 520, 540, 560, 580, 600, 620, 640, 660, 680, 700, 720, 740, 760, 780, 800, 820, 840, 860, 880, 900, 920, 940, 960, 980, 1000

Offset: 0

N. J. A. Sloane

The multiples of 20 are exactly those integers which do not have a multiple whose decimal digits are of alternating parity. (International Mathematical Olympiad 2004, problem 6, see A110303) - Joseph Myers, Jul 13 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 332.
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. A005843, A008592, A008600, A008601, A110303 (complement), A215145, A317095.

Range[0, 1500, 20] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)
CoefficientList[Series[20 x / (x - 1)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Jun 10 2013 *)

a(n)=20*n \\ Charles R Greathouse IV, Sep 28 2015

(0 to 1000 by 20).toList // Alonso del Arte, Feb 20 2020

G.f.: 20*x/(x - 1)^2. - Vincenzo Librandi, Jun 10 2013
E.g.f.: 20*x*exp(x). - Stefano Spezia, Feb 20 2020
From Elmo R. Oliveira, Apr 10 2025: (Start)
a(n) = 20*n = 2*A008592(n) = 10*A005843(n) = A317095(n)/2.
a(n) = 2*a(n-1) - a(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)

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

A076312 a(n) = floor(n/10) + 2*(n mod 10).

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 7, 9, 11, 13, 15, 17, 19, 21

Offset: 0

Reinhard Zumkeller, Oct 06 2002

nonn

Delete the last digit from n and add twice this digit to the shortened number. - N. J. A. Sloane, May 25 2019

(n==0 modulo 19) iff (a(n)==0 modulo 19); applied recursively, this property provides a useful test for divisibility by 19.

			26468 is not a multiple of 19, as 26468 -> 2646+2*8=2662 -> 266+2*2=270 -> 27+2*0=27=19*1+8, therefore the answer is NO.
Is 12882 divisible by 19? 12882 -> 1288+2*2=1292 -> 129+2*2=133 -> 13+2*3=19, therefore the answer is YES.

Erdős, Paul, and János Surányi. Topics in the Theory of Numbers. New York: Springer, 2003. Problem 6, page 3.
Karl Menninger, Rechenkniffe, Vandenhoeck & Ruprecht in Goettingen (1961), 79A.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Divisibility Tests.
Wikipedia, Divisibility rule
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A008601, A076309, A076310, A076311.

a076312 n =  n' + 2 * m where (n', m) = divMod n 10
-- Reinhard Zumkeller, Jun 01 2013

[Floor(n/10) + 2*(n mod 10): n in [0..100]]; // Vincenzo Librandi, Mar 05 2020

f[n_]:=Module[{idn=IntegerDigits[n]},FromDigits[Most[idn]]+2idn[[-1]]]; Array[ f,80,0] (* Harvey P. Dale, Mar 01 2020 *)

G.f.: -x(17x^9-2-2x-2x^2-2x^3-2x^4-2x^5-2x^6-2x^7-2x^8)/((1-x)^2(1+x)(x^4+x^3+x^2+x+1)(x^4-x^3+x^2-x+1)). a(n)=A059995(n)+2*A010879(n). [R. J. Mathar, Jan 24 2009]

A139620 a(n) = 190*n + 20.

Original entry on oeis.org

20, 210, 400, 590, 780, 970, 1160, 1350, 1540, 1730, 1920, 2110, 2300, 2490, 2680, 2870, 3060, 3250, 3440, 3630, 3820, 4010, 4200, 4390, 4580, 4770, 4960, 5150, 5340, 5530, 5720, 5910, 6100, 6290, 6480, 6670, 6860, 7050, 7240, 7430

Offset: 0

Omar E. Pol, May 21 2008

Numbers of the 20th column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 20th column in the square array A057145.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008601, A057145, A139600.

[10*(19*n + 2): n in [0..50]]; // Vincenzo Librandi, Jul 23 2011

190*Range[0,40]+20 (* or *) LinearRecurrence[{2,-1},{20,210},40] (* Harvey P. Dale, Feb 04 2019 *)

a(n)=190*n+20 \\ Charles R Greathouse IV, Oct 05 2011

From Elmo R. Oliveira, Apr 08 2024: (Start)
G.f.: 10*(2+17*x)/(x-1)^2.
E.g.f.: 10*exp(x)*(2 + 19*x).
a(n) = 10*(A008601(n) + 2).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

A244631 a(n) = 19*n^2.

Original entry on oeis.org

0, 19, 76, 171, 304, 475, 684, 931, 1216, 1539, 1900, 2299, 2736, 3211, 3724, 4275, 4864, 5491, 6156, 6859, 7600, 8379, 9196, 10051, 10944, 11875, 12844, 13851, 14896, 15979, 17100, 18259, 19456, 20691, 21964, 23275, 24624, 26011, 27436, 28899, 30400, 31939, 33516

Offset: 0

Vincenzo Librandi, Jul 03 2014

First bisection of A195048. - Bruno Berselli, Jul 03 2014

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

Cf. A000290, A008601, A195048.

Cf. similar sequences listed in A244630.

Magma
```
[19*n^2: n in [0..40]];
```
Mathematica
```
Table[19 n^2, {n, 0, 40}]
```

a(n)=19*n^2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: 19*x*(1 + x)/(1 - x)^3. [corrected by Bruno Berselli, Jul 03 2014]
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = 19*A000290(n). - Omar E. Pol, Jul 03 2014
From Elmo R. Oliveira, Dec 01 2024: (Start)
E.g.f.: 19*x*(1 + x)*exp(x).
a(n) = n*A008601(n) = A195048(2*n). (End)

A317319 Multiples of 19 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 19, 3, 38, 5, 57, 7, 76, 9, 95, 11, 114, 13, 133, 15, 152, 17, 171, 19, 190, 21, 209, 23, 228, 25, 247, 27, 266, 29, 285, 31, 304, 33, 323, 35, 342, 37, 361, 39, 380, 41, 399, 43, 418, 45, 437, 47, 456, 49, 475, 51, 494, 53, 513, 55, 532, 57, 551, 59, 570, 61, 589, 63, 608, 65, 627, 67, 646, 69

Offset: 0

Omar E. Pol, Jul 25 2018

Partial sums give the generalized 23-gonal numbers (A303303).

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 23-gonal numbers.

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

Cf. A008601 and A005408 interleaved.
Column 19 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14).
Cf. A303303.

a[n_] := If[OddQ[n], n, 19*n/2]; Array[a, 70, 0] (* Amiram Eldar, Oct 14 2023 *)

concat(0, Vec(x*(1 + 19*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

a(2n) = 19*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 29 2018: (Start)
G.f.: x*(1 + 19*x + x^2) / ((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
Multiplicative with a(2^e) = 19*2^(e-1), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 17/2^s). - Amiram Eldar, Oct 26 2023

A309131 Triangle read by rows: T(n, k) = (n - k)*prime(1 + k), with 0 <= k < n.

Original entry on oeis.org

2, 4, 3, 6, 6, 5, 8, 9, 10, 7, 10, 12, 15, 14, 11, 12, 15, 20, 21, 22, 13, 14, 18, 25, 28, 33, 26, 17, 16, 21, 30, 35, 44, 39, 34, 19, 18, 24, 35, 42, 55, 52, 51, 38, 23, 20, 27, 40, 49, 66, 65, 68, 57, 46, 29, 22, 30, 45, 56, 77, 78, 85, 76, 69, 58, 31

Offset: 1

Stefano Spezia, Jul 14 2019

T(n, k) is the k-superdiagonal sum of an n X n Toeplitz matrix M(n) whose first row consists of successive prime numbers prime(1), ..., prime(n).
The h-th subdiagonal of the triangle T gives the primes multiplied by (h + 1).
The k-th column of the triangle T gives the multiples of prime(1 + k).
Also array A(n, k) = n*prime(1 + k) read by ascending antidiagonals, with 0 <= k < n. - Michel Marcus, Jul 15 2019

			The triangle T(n, k) begins:
---+-----------------------------------------------------
n\k|    0     1     2     3     4     5     6     7     8
---+-----------------------------------------------------
1  |    2
2  |    4     3
3  |    6     6     5
4  |    8     9    10     7
5  |   10    12    15    14    11
6  |   12    15    20    21    22    13
7  |   14    18    25    28    33    26    17
8  |   16    21    30    35    44    39    34    19
9  |   18    24    35    42    55    52    51    38    23
...
For n = 3 the matrix M(3) is
          2,         3,         5
    M_{2,1},         2,         3
    M_{3,1},   M_{3,2},         2
and therefore T(3, 0) = 2 + 2 + 2 = 6, T(3, 1) = 3 + 3 = 6, and T(3, 2) = 5.

Wikipedia, Toeplitz matrix

Cf. A025581, A318173, A325516.

Cf. A000040: diagonal; A001747: 1st subdiagonal; A001748: 2nd subdiagonal; A001749: 3rd subdiagonal; A001750: 4th subdiagonal; A005843: 0th column; A008585: 1st column; A008587: 2nd column; A008589: 3rd column; A008593: 4th column; A008595: 5th column; A008599: 6th column; A008601: 7th column; A014148: row sums; A138636: 5th subdiagonal; A272470: 6th subdiagonal.

[[(n-k)*NthPrime(1+k): k in [0..n-1]]: n in [1..11]]; // triangle output

a:=(n, k)->(n-k)*ithprime(1+k): seq(seq(a(n, k), k=0..n-1), n=1..11);

Flatten[Table[(n-k)*Prime[1+k],{n,1,11},{k,0,n-1}]]

T(n, k) = (n - k)*prime(1 + k);
tabl(nn) = for(n=1, nn, for(k=0, n-1, print1(T(n, k), ", ")); print); \\ triangle output

[[(n-k)*Primes().unrank(k) for k in (0..n-1)] for n in (1..11)] # triangle output

T(n, k) = A025581(n, k)*A000040(1 + k).

A121669 Numbers with sum of digits = 19, divisible by 19 and containing the string "19".

Original entry on oeis.org

17119, 19171, 19342, 19513, 20197, 21907, 33193, 34219, 41914, 51319, 61921, 101935, 102619, 112195, 119035, 119206, 121942, 125191, 171019, 171190, 190171, 190342, 190513, 191026, 191710, 192052, 192223, 193420, 194104, 195130, 195301, 197011, 201970, 204193

Offset: 1

Hassan Taifour (hytaifour(AT)yahoo.co.uk), Sep 10 2006

Conjecture: There are approximately k(n-1)(n-2)^(n-2) terms of this sequence up to 10^n, where k is about e/(19e-19). - Charles R Greathouse IV, Oct 13 2022

Chai Wah Wu, Table of n, a(n) for n = 1..12794 (all terms < 10^11)

Intersection of A008601, A166459 and A293879.

d19Q[n_]:=Module[{idn=IntegerDigits[n]},Total[idn]==19&&MemberQ[ Partition[ idn,2,1],{1,9}]]; Select[19*Range[20000],d19Q] (* Harvey P. Dale, Jun 10 2014 *)

def ok(n): s = str(n); return n%19==0 and '19' in s and sum(map(int, s))==19
print(list(filter(ok, range(205000)))) # Michael S. Branicky, Aug 06 2021

More terms from Zak Seidov, Sep 12 2006

cp's OEIS Frontend

A121039 Multiples of 19 containing a 19 in their decimal representation.

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A008602 Multiples of 20.

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A008603 Multiples of 21.

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

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A076312 a(n) = floor(n/10) + 2*(n mod 10).

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A139620 a(n) = 190*n + 20.

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A244631 a(n) = 19*n^2.

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