A008595 -id:A008595 - cp's OEIS frontend

A317313 Multiples of 13 and odd numbers interleaved.

0, 1, 13, 3, 26, 5, 39, 7, 52, 9, 65, 11, 78, 13, 91, 15, 104, 17, 117, 19, 130, 21, 143, 23, 156, 25, 169, 27, 182, 29, 195, 31, 208, 33, 221, 35, 234, 37, 247, 39, 260, 41, 273, 43, 286, 45, 299, 47, 312, 49, 325, 51, 338, 53, 351, 55, 364, 57, 377, 59, 390, 61, 403, 63, 416, 65, 429, 67, 442, 69

Offset: 0

Omar E. Pol, Jul 25 2018

Partial sums give the generalized 17-gonal numbers (A303305).
More generally, the partial sums of the sequence formed by the multiples of m and the odd numbers interleaved, give the generalized k-gonal numbers, with m >= 1 and k = m + 4.
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 17-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. A008595 and A005408 interleaved.
Column 13 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), A317311 (k=15), A317312 (k=16).
Cf. A303305.

Table[{13n, 2n + 1}, {n, 0, 35}] // Flatten (* or *)
CoefficientList[Series[(x^3 + 13 x^2 + x)/(x^2 - 1)^2, {x, 0, 69}], x] (* or *)
LinearRecurrence[{0, 2, 0, -1}, {0, 1, 13, 3}, 70] (* Robert G. Wilson v, Jul 26 2018 *)

a(n) = if(n%2==0, return((n/2)*13), return(n)) \\ Felix Fröhlich, Jul 26 2018

concat(0, Vec(x*(1 + 13*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

a(2n) = 13*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 29 2018: (Start)
G.f.: x*(1 + 13*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) = 13*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 + 11/2^s). - Amiram Eldar, Oct 25 2023
a(n) = (15 + 11*(-1)^n)*n/4. - Aaron J Grech, Aug 20 2024

A133371 Triangle read by rows: T(i,j) is the number of i-permutations of 14 objects a,b,c,d,e,f,g,h,i,j,k,l,m,n, with repetition allowed, containing j a's.

Original entry on oeis.org

1, 13, 1, 169, 26, 1, 2197, 507, 39, 1, 28561, 8788, 1014, 52, 1, 371293, 142805, 21970, 1690, 65, 1, 4826809, 2227758, 428415, 43940, 2535, 78, 1, 62748517, 33787663, 7797153, 999635, 76895, 3549, 91, 1

Offset: 0

Zerinvary Lajos, with help from Emeric Deutsch, Dec 21 2007

Mirror image of A123187. - Philippe Deléham, Dec 27 2007

			1
13, 1
169, 26, 1
2197, 507, 39, 1
28561, 8788, 1014, 52, 1
371293, 142805, 21970, 1690, 65, 1
4826809, 2227758, 428415, 43940, 2535, 78, 1
62748517, 33787663, 7797153, 999635, 76895, 3549, 91, 1

Cf. A001022, A008595.

for i from 0 to 7 do seq(binomial(i, j)*13^(i-j), j = 0 .. i) od;

Flatten[Table[Binomial[i,j] 13^(i-j),{i,0,7},{j,0,i}]] (* Harvey P. Dale, Nov 01 2011 *)

A319073 Square array read by antidiagonals upwards: T(n,k) = k*sigma(n), n >= 1, k >= 1.

Original entry on oeis.org

1, 3, 2, 4, 6, 3, 7, 8, 9, 4, 6, 14, 12, 12, 5, 12, 12, 21, 16, 15, 6, 8, 24, 18, 28, 20, 18, 7, 15, 16, 36, 24, 35, 24, 21, 8, 13, 30, 24, 48, 30, 42, 28, 24, 9, 18, 26, 45, 32, 60, 36, 49, 32, 27, 10, 12, 36, 39, 60, 40, 72, 42, 56, 36, 30, 11, 28, 24, 54, 52, 75, 48, 84, 48, 63, 40, 33, 12

Offset: 1

Omar E. Pol, Sep 22 2018

			The corner of the square array begins:
         A000203 A074400 A272027 A239050 A274535 A274536 A319527 A319528
A000027:       1,      2,      3,      4,      5,      6,      7,      8, ...
A008585:       3,      6,      9,     12,     15,     18,     21,     24, ...
A008586:       4,      8,     12,     16,     20,     24,     28,     32, ...
A008589:       7,     14,     21,     28,     35,     42,     49,     56, ...
A008588:       6,     12,     18,     24,     30,     36,     42,     48, ...
A008594:      12,     24,     36,     48,     60,     72,     84,     96, ...
A008590:       8,     16,     24,     32,     40,     48,     56,     64, ...
A008597:      15,     30,     45,     60,     75,     90,    105,    120, ...
A008595:      13,     26,     39,     52,     65,     78,     91,    104, ...
A008600:      18,     36,     54,     72,     90,    108,    126,    144, ...
...

Index entries for sequences related to sigma(n)

Another version of A274824.
Antidiagonal sums give A175254.
Main diagonal gives A064987.
Row n lists the multiples of A000203(n).
Row 1 is A000027.
Initial zeros should be omitted in the following sequences related to the rows of the array:
Row 2-5: A008585, A008586, A008589, A008588.
Rows 6 and 11: A008594.
Rows 7-9: A008590, A008597, A008595.
Rows 10 and 17: A008600.
Rows 12-13: A135628, A008596.
Rows 14, 15 and 23: A008606.
Rows 16 and 25: A135631.
(Note that in the OEIS there are many other sequences that are also rows of this square array.)
Cf. A000203, A237593, A319526.

T:=Flat(List([1..12],n->List([1..n],k->k*Sigma(n-k+1))));; Print(T); # Muniru A Asiru, Jan 01 2019

with(numtheory): T:=(n,k)->k*sigma(n-k+1): seq(seq(T(n,k),k=1..n),n=1..12); # Muniru A Asiru, Jan 01 2019

Table[k DivisorSigma[1, #] &[m - k + 1], {m, 12}, {k, m}] // Flatten (* Michael De Vlieger, Dec 31 2018 *)

A062903 Numbers n such that n and its reversal are both multiples of 13.

Original entry on oeis.org

0, 494, 585, 676, 767, 858, 949, 1001, 1495, 1586, 1677, 1768, 1859, 2002, 2496, 2587, 2678, 2769, 3003, 3497, 3588, 3679, 4004, 4498, 4589, 4940, 5005, 5499, 5850, 5941, 6006, 6760, 6851, 6942, 7007, 7670, 7761, 7852, 7943, 8008, 8580, 8671

Offset: 1

Amarnath Murthy, Jul 01 2001

			1495 and 5941 are both multiples of 13.

Oren Meisner, Table of n, a(n) for n = 1..10000

Subsequence of A008595.

Select[13*Range[0,700],Divisible[FromDigits[Reverse[IntegerDigits[ #]]],13]&] (* Harvey P. Dale, Nov 30 2014 *)

isok(n) = !(n % 13) && !(fromdigits(Vecrev(digits(n))) % 13); \\ Michel Marcus, Aug 14 2018

Corrected and extended by Dean Hickerson, Jul 06 2001

Zero added by Zak Seidov, May 31 2010

A283737 Numbers with digit sum 13 that are multiples of 13.

Original entry on oeis.org

247, 364, 481, 715, 832, 1066, 1183, 1417, 1534, 1651, 2119, 2236, 2353, 2470, 2704, 2821, 3055, 3172, 3406, 3523, 3640, 4108, 4225, 4342, 4810, 5044, 5161, 5512, 6214, 6331, 7033, 7150, 7501, 8203, 8320, 9022, 10075, 10192, 10309, 10426, 10543, 10660, 11128, 11245

Offset: 1

Zak Seidov, Mar 15 2017

			a(1) = A143164(16) = 247 = 19*13,
a(90) = A143164(1186) = 25402 = 1954*13.

Alois P. Heinz, Table of n, a(n) for n = 1..38073

Intersection of A008595 and A143164.

Cf. A007953.

With[{k = 13}, Select[Range@ 12000, Total@ IntegerDigits@ # == k && Divisible[#, k] &]] (* Michael De Vlieger, Mar 16 2017 *)

for(k=1,1000,if(13==sumdigits(m=13*k),print1(m",")))

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).

A098961 Sums of two squares and divisible by 13.

Original entry on oeis.org

13, 26, 52, 65, 104, 117, 130, 169, 208, 221, 234, 260, 325, 338, 377, 416, 442, 468, 481, 520, 533, 585, 637, 650, 676, 689, 754, 793, 832, 845, 884, 936, 949, 962, 1040, 1053, 1066, 1105, 1157, 1170, 1261, 1274, 1300, 1313

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Oct 22 2004

			Sums not divisible by 13 are shown in asterisks:
.
      |   1    4    9   16   25   36   49   64   81  100  121
  ----+------------------------------------------------------
    1 |   *    *    *    *   26    *    *   65    *    *    *
    4 |   *    *   13    *    *    *    *    *    *  104    *
    9 |   *   13    *    *    *    *    *    *    *    *  130
   16 |   *    *    *    *    *   52   65    *    *    *    *
   25 |  26    *    *    *    *    *    *    *    *    *    *
   36 |   *    *    *   52    *    *    *    *  117    *    *
   49 |   *    *    *   65    *    *    *    *  130    *    *
   64 |  65    *    *    *    *    *    *    *    *    *    *
   81 |   *    *    *    *    *  117  130    *    *    *    *
  100 |   *  104    *    *    *    *    *    *    *    *  221
  121 |   *    *  130    *    *    *    *    *    *  221    *

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

Intersection of A001481 and A008595.

Cf. A094447.

filter:= proc(n) local t,F;
  F:= select(t -> t[2]::odd, ifactors(n)[2]);
  andmap(t -> t[1] = 2 or t[1] mod 4 = 1, F);
end proc:
select(filter, [seq(i,i=13..10000, 13)]); # Robert Israel, Dec 02 2022

With[{nn=50},Take[Union[Select[Total/@Subsets[Range[nn]^2,{2}], Divisible[ #,13]&]],nn]] (* Harvey P. Dale, Aug 15 2014 *)

a(n) = 13 * A001481(n+1) for n >= 1. - Joerg Arndt, Dec 03 2022

A101492 Triangle read by rows: T(n,k) = (n-k+1)(4k+1).

Original entry on oeis.org

1, 2, 5, 3, 10, 9, 4, 15, 18, 13, 5, 20, 27, 26, 17, 6, 25, 36, 39, 34, 21, 7, 30, 45, 52, 51, 42, 25, 8, 35, 54, 65, 68, 63, 50, 29, 9, 40, 63, 78, 85, 84, 75, 58, 33, 10, 45, 72, 91, 102, 105, 100, 87, 66, 37, 11, 50, 81, 104, 119, 126, 125, 116, 99, 74, 41, 12, 55, 90, 117

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 5 0 0...
1 5 9 0...
1 5 9 13...
...
T(n+0,0) = 1*n = A000027(n+1),
T(n+0,1) = 5*n = A008587(n),
T(n+1,2) = 9*n = A008591(n),
T(n+2,3) = 13*n = A008595(n),
so, for example,
T(n,n) = 4*n+1 = A016813(n),
T(n+1,n) = 8*n+2 = A017089(n),
T(n,0)*T(n,1)/10 = A000217(n) (triangular numbers),
T(n,n)*T(n,0) = A001107(n+1) (10-gonal numbers: 4*n^2 - 3*n),
T(n,n)*T(n,1)/5 = A007742(n).

Muniru A Asiru, Rows n=0..150 of triangle, flattened

Row sums give hexagonal pyramidal numbers A002412.

Cf. A101493 for product B*A, A002412.

Flat(List([0..11],n->List([0..n],k->(n+1-k)*(4*k+1)))); # Muniru A Asiru, Mar 07 2019

[[(n+1-k)*(4*k+1): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Mar 07 2019

Flatten[Table[(n+1-k)(4k+1),{n,0,15},{k,0,n}]] (* Harvey P. Dale, Jun 09 2011 *)

T(n, k) = if(k>n,0,(n-k+1)*(4*k+1));
for(i=0,10, for(j=0,i,print1(T(i,j),", "));print())

[[(n-k+1)*(4*k+1) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 07 2019

A199799 Totatives of 111111.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 16, 17, 19, 20, 23, 25, 29, 31, 32, 34, 38, 40, 41, 43, 46, 47, 50, 53, 58, 59, 61, 62, 64, 67, 68, 71, 73, 76, 79, 80, 82, 83, 85, 86, 89, 92, 94, 95, 97, 100, 101, 103, 106, 107, 109, 113, 115, 116, 118, 122, 124, 125, 127, 128, 131, 134

Offset: 1

Reinhard Zumkeller, Nov 11 2011

a(n) and 111111 are coprime, 111111 = 3*7*11*13*37; empty intersections with A008585, A008593, A008595, or A085959; sequence is finite with 51840 terms, A000010(111111) = 51840, last term: a(51840) = 111110.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Repunit
Eric Weisstein's World of Mathematics, Totative

Cf. A109492 (divisors of 111111).

a199799 n = a199799_list !! (n-1)
a199799_list = [x | x <- [1..111111], gcd x 111111 == 1]

Select[Range[200],CoprimeQ[#,111111]&] (* Paolo Xausa, Sep 27 2023 *)

A166397 Multiples of 13 whose reversal - 1 is also a multiple of 13.

Original entry on oeis.org

156, 247, 338, 429, 1560, 1651, 1742, 1833, 1924, 2470, 2561, 2652, 2743, 2834, 2925, 3380, 3471, 3562, 3653, 3744, 3835, 3926, 4290, 4381, 4472, 4563, 4654, 4745, 4836, 4927, 5291, 5382, 5473, 5564, 5655, 5746, 5837, 5928, 6292, 6383, 6474, 6565, 6656

Offset: 1

Claudio Meller, Oct 13 2009

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

Subsequence of A008595.

Select[13Range[700],Divisible[FromDigits[Reverse[IntegerDigits[#]]] -1, 13]&]  (* Harvey P. Dale, Apr 23 2011 *)

cp's OEIS Frontend

A317313 Multiples of 13 and odd numbers interleaved.

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A133371 Triangle read by rows: T(i,j) is the number of i-permutations of 14 objects a,b,c,d,e,f,g,h,i,j,k,l,m,n, with repetition allowed, containing j a's.

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A319073 Square array read by antidiagonals upwards: T(n,k) = k*sigma(n), n >= 1, k >= 1.

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A062903 Numbers n such that n and its reversal are both multiples of 13.

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A283737 Numbers with digit sum 13 that are multiples of 13.

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

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A098961 Sums of two squares and divisible by 13.

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A101492 Triangle read by rows: T(n,k) = (n-k+1)(4k+1).