A135631 -id:A135631 - cp's OEIS frontend

A044102 Multiples of 36.

0, 36, 72, 108, 144, 180, 216, 252, 288, 324, 360, 396, 432, 468, 504, 540, 576, 612, 648, 684, 720, 756, 792, 828, 864, 900, 936, 972, 1008, 1044, 1080, 1116, 1152, 1188, 1224, 1260, 1296, 1332, 1368, 1404, 1440, 1476, 1512, 1548, 1584, 1620, 1656, 1692, 1728

Offset: 0

Clark Kimberling

Also, k such that Fibonacci(k) mod 27 = 0. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 18 2004
A033183(a(n)) = n+1. - Reinhard Zumkeller, Nov 07 2009
A122841(a(n)) > 1 for n > 0. - Reinhard Zumkeller, Nov 10 2013
Sum of the numbers from 4*(n-1) to 4*(n+1). - Bruno Berselli, Oct 25 2018

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

Cf. A005843, A008600, A033183, A122841, A135631, A167632, A174312.

Subsequence of A008588.

a:=[0,36];; for n in [3..50] do a[n]:=2*a[n-1]-a[n-2]; od; a; # Muniru A Asiru, Oct 25 2018

a044102 = (* 36)
a044102_list = [0, 36 ..]  -- Reinhard Zumkeller, Nov 10 2013

[36*n: n in [0..50]]; // Vincenzo Librandi, May 20 2014

seq(coeff(series(36*x/(1-x)^2,x,n+1), x, n), n = 0 .. 50); # Muniru A Asiru, Oct 25 2018

Range[0, 2000, 36] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)
CoefficientList[Series[36 x/(1 - x)^2, {x, 0, 100}], x] (* Vincenzo Librandi, May 20 2014 *)

a(n)=36*n \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 36*x/(1 - x)^2.
a(n) = A167632(n+1). - Reinhard Zumkeller, Nov 07 2009
a(n) = 36*n. - Vincenzo Librandi, Jan 26 2011
From Elmo R. Oliveira, Apr 10 2025: (Start)
E.g.f.: 36*x*exp(x).
a(n) = 18*A005843(n) = 2*A008600(n).
a(n) = 2*a(n-1) - a(n-2). (End)

A195819 Multiples of 29.

Original entry on oeis.org

0, 29, 58, 87, 116, 145, 174, 203, 232, 261, 290, 319, 348, 377, 406, 435, 464, 493, 522, 551, 580, 609, 638, 667, 696, 725, 754, 783, 812, 841, 870, 899, 928, 957, 986, 1015, 1044, 1073, 1102, 1131, 1160, 1189, 1218, 1247, 1276, 1305, 1334

Offset: 0

Omar E. Pol, Oct 12 2011

Length of hypotenuses on the main diagonal of the Pythagorean spiral whose edges have length A195033 and whose vertices are the numbers A195034, if n >= 1.

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. A004922, A004942, A008605, A010868, A135631, A195033, A195034.

29Range[0,50] (* Harvey P. Dale, Oct 25 2011 *)

a(n) = 29*n.
From Elmo R. Oliveira, Mar 21 2024: (Start)
G.f.: 29*x/(x-1)^2.
E.g.f.: 29*x*exp(x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

A174312 a(n) = 32*n.

Original entry on oeis.org

0, 32, 64, 96, 128, 160, 192, 224, 256, 288, 320, 352, 384, 416, 448, 480, 512, 544, 576, 608, 640, 672, 704, 736, 768, 800, 832, 864, 896, 928, 960, 992, 1024, 1056, 1088, 1120, 1152, 1184, 1216, 1248, 1280, 1312, 1344, 1376, 1408, 1440, 1472, 1504, 1536, 1568, 1600

Offset: 0

Paul Curtz, Nov 27 2010

Subsequence of squares is A017066 (see 2nd formula). - Bernard Schott, Mar 03 2023

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

Cf. A001105, A008598, A017066, A135628, A135631, A152691.

[32*n: n in [0..60]]; // Vincenzo Librandi, Aug 04 2011

Range[0, 3000, 32] (* Vladimir Joseph Stephan Orlovsky, Jun 03 2011 *)

a(n)=n<<5 \\ Charles R Greathouse IV, Dec 21 2011

G.f.: 32*x/(1-x)^2.
a(A001105(n)) = A017066(n). - Bernard Schott, Mar 05 2023
From Elmo R. Oliveira, Apr 07 2025: (Start)
E.g.f.: 32*x*exp(x).
a(n) = 2*A008598(n) = A152691(n)/2.
a(n) = 2*a(n-1) - a(n-2). (End)

A135639 a(n) = 839*n.

Original entry on oeis.org

0, 839, 1678, 2517, 3356, 4195, 5034, 5873, 6712, 7551, 8390, 9229, 10068, 10907, 11746, 12585, 13424, 14263, 15102, 15941, 16780, 17619, 18458, 19297, 20136, 20975, 21814, 22653, 23492, 24331, 25170, 26009, 26848, 27687, 28526

Offset: 0

Omar E. Pol, Nov 27 2007

The 146th prime number (839) and some of its multiples are related to the exceptional Lie group E_8 calculation because the result is a matrix with 453060 rows and columns. The size of the matrix is the member a(540)=453060 of this sequence. The number 839 is the largest prime factor of 453060 because we can write 2*2*3*3*3*5*839=453060. The number of entries of the matrix is the member a(244652400)=453060*453060=205263363600.

			a(1)=839. a(540)=540*839=453060. a(244652400)=244652400*839=205263363600.

G. C. Greubel, Table of n, a(n) for n = 0..1000
American Institute of Mathematics, Mathematicians Maps E_8.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A064730, A134888, A134950, A134960, A135631.

839Range[0,40] (* Harvey P. Dale, Sep 13 2011 *)
LinearRecurrence[{2,-1},{0,839},25] (* G. C. Greubel, Oct 25 2016 *)

From G. C. Greubel, Oct 25 2016: (Start)
a(n) = 2*a(n-1) - a(n-2).
G.f.: (839*x)/(1 - x)^2.
E.g.f.: 839*x*exp(x). (End)

A212773 Amounts (in cents) of coins in denominations 1, 5, 10, 25, and 50 (cents) which can consist of equal numbers of coins of all denominations present when two or more denominations are used (or none are used: term 0).

Original entry on oeis.org

0, 6, 11, 12, 15, 16, 18, 22, 24, 26, 30, 31, 32, 33, 35, 36, 40, 41, 42, 44, 45, 48, 51, 52, 54, 55, 56, 60, 61, 62, 64, 65, 66, 70, 72, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 90, 91, 93, 96, 99, 102, 104, 105, 108, 110, 112, 114, 120, 121, 122, 123, 124

Offset: 1

Rick L. Shepherd, May 26 2012

nonn

Nonnegative multiples of each of 6, 11, 15, 16, 26, 31, 35, 40, 41, 51, 56, 61, 65, 76, 81, 85, 86, and 91.

All products of terms are terms.

			4 is not a term because it is not an appropriate multiple. Also 4 = 4*1 cannot be represented with more than one denomination of coin. Similarly 7 is not a term; although 7 = 7*1 = 2*1 + 1*5 does have a representation in terms of two denominations, 1 and 5, there are unequal numbers of each.
a(11) = 30 is a term because it is a multiple of 6. 30 = 5*1 + 5*5 = 2*5 + 2*10 = 1*5 + 1*25, so five coins each of denominations 1 and 5, two each of 5 and 10, or one each of 5 and 25 totals 30.
The term 34924118340711600 (5 times the LCM of the numbers in the first comment, so also divisible by 75) is the smallest which can be expressed in 26 such ways, one for each possible combination of two or more of these five coin denominations. (It also can be expressed as a multiple of each of these five alone of course.)

Rick L. Shepherd, Table of n, a(n) for n = 1..10000
Index entries for sequences related to making change.

Cf. A212774, A008588, A008593, A008597, A008598, A135631.

{c = 0; n = -1; until(c==10000, n++;
if(n%6==0 || n%11==0 || n%15==0 || n%16==0 || n%26==0 ||
  n%31==0 || n%35==0 || n%40==0 || n%41==0 || n%51==0 ||
  n%56==0 || n%61==0 || n%65==0 || n%76==0 || n%81==0 ||
  n%85==0 || n%86==0 || n%91==0,
  c++; write("b212773.txt", c, " ", n)))}

A274824 Triangle read by rows: T(n,k) = (n-k+1)*sigma(k), n>=1, 1<=k<=n.

Original entry on oeis.org

1, 2, 3, 3, 6, 4, 4, 9, 8, 7, 5, 12, 12, 14, 6, 6, 15, 16, 21, 12, 12, 7, 18, 20, 28, 18, 24, 8, 8, 21, 24, 35, 24, 36, 16, 15, 9, 24, 28, 42, 30, 48, 24, 30, 13, 10, 27, 32, 49, 36, 60, 32, 45, 26, 18, 11, 30, 36, 56, 42, 72, 40, 60, 39, 36, 12, 12, 33, 40, 63, 48, 84, 48, 75, 52, 54, 24, 28, 13, 36, 44, 70, 54, 96, 56, 90, 65, 72, 36, 56, 14

Offset: 1

Omar E. Pol, Oct 02 2016

Theorem: for any sequence S the partial sums of the partial sums are also the antidiagonal sums of the square array in which the n-th row gives n times the sequence S. Therefore they are also the row sums of the triangular array in which the n-th diagonal gives n times the sequence S.
In this case the sequence S is A000203.
The n-th diagonal of this triangle gives n times A000203.
The row sums give A175254 which gives the partial sums of A024916 which gives the partial sums of A000203.
T(n,k) is also the total number of unit cubes that are exactly below the terraces of the k-th level (starting from the top) up the base of the stepped pyramid with n levels described in A245092. This fact is because the mentioned terraces have the same shape as the symmetric representation of sigma(k). For more information see A237593 and A237270.
In the definition of this sequence the value n-k+1 is also the height of the terraces associated to sigma(k) in the mentioned pyramid with n levels, or in other words, the distance between the mentioned terraces and the base of the pyramid.
The sum of the n-th row of triangle equals the volume (also the number of cubes) of the mentioned pyramid with n levels.
For an illustration of the pyramid, see the Links section.
The sum of the n-th row is also 1/4 of the volume of the stepped pyramid described in A244050 with n levels.
Column k lists the positive multiples of sigma(k).
The k-th term in the n-th diagonal is equal to n*sigma(k).
Note that this is also a square array read by antidiagonals upwards: T(i,j) = i*sigma(j), i>=1, j>=1. The first row of the array is A000203. So consider that the pyramid is upside down. The value of "i" is the distance between the base of the pyramid and the terraces associated to sigma(j). The antidiagonal sums give the partial sums of the partial sums of A000203.

			Triangle begins:
1;
2,  3;
3,  6,  4;
4,  9,  8,  7;
5,  12, 12, 14, 6;
6,  15, 16, 21, 12, 12;
7,  18, 20, 28, 18, 24,  8;
8,  21, 24, 35, 24, 36,  16, 15;
9,  24, 28, 42, 30, 48,  24, 30,  13;
10, 27, 32, 49, 36, 60,  32, 45,  26,  18;
11, 30, 36, 56, 42, 72,  40, 60,  39,  36,  12;
12, 33, 40, 63, 48, 84,  48, 75,  52,  54,  24, 28;
13, 36, 44, 70, 54, 96,  56, 90,  65,  72,  36, 56,  14;
14, 39, 48, 77, 60, 108, 64, 105, 78,  90,  48, 84,  28, 24;
15, 42, 52, 84, 66, 120, 72, 120, 91,  108, 60, 112, 42, 48, 24;
16, 45, 56, 91, 72, 132, 80, 135, 104, 126, 72, 140, 56, 72, 48, 31;
...
For n = 16 and k = 10 the sum of the divisors of 10 is 1 + 2 + 5 + 10 = 18, and 16 - 10 + 1 = 7, and 7*18 = 126, so T(16,10) = 126.
On the other hand, the symmetric representation of sigma(10) has two parts of 9 cells, giving a total of 18 cells. In the stepped pyramid described in A245092, with 16 levels, there are 16 - 10 + 1 = 7 cubes exactly below every cell of the symmetric representation of sigma(10) up the base of pyramid; hence the total numbers of cubes exactly below the terraces of the 10th level (starting from the top) up the base of the pyramid is equal to 7*18 = 126. So T(16,10) = 126.
The sum of the 16th row of the triangle is 16 + 45 + 56 + 91 + 72 + 132 + 80 + 135 + 104 + 126 + 72 + 140 + 56 + 72 + 48 + 31 = A175254(16) = 1276, equaling the volume (also the number of cubes) of the stepped pyramid with 16 levels described in A245092 (see Links section).

Indranil Ghosh, Rows 1..100 of triangle, flattened
Omar E. Pol, Illustration of the stepped pyramid with 16 levels

Row sums of triangle give A175254.
Diagonals 1-6: A000203, A074400, A272027, A239050, A274535, A274536.
Column 1 is A000027.
Initial zeros should be omitted in the following sequences related to the columns of triangle:
Columns 2-5: A008585, A008586, A008589, A008588.
Columns 6 and 11: A008594.
Columns 7-9: A008590, A008597, A008595.
Columns 10 and 17: A008600.
Columns 12-13: A135628, A008596.
Columns 14, 15 and 23: A008606.
Columns 16 and 25: A135631.
(There are many other OEIS sequences that are also columns of this triangle.)
Cf. A004736, A024916, A054973, A196020, A236104, A235791, A237591, A237593, A237270, A237271, A244050, A245092, A245093, A262612.

T(n,k) = (n-k+1) * A000203(k).

T(n,k) = A004736(n,k) * A245093(n,k).

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 *)

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)

A138127 Multiples of 127.

Original entry on oeis.org

0, 127, 254, 381, 508, 635, 762, 889, 1016, 1143, 1270, 1397, 1524, 1651, 1778, 1905, 2032, 2159, 2286, 2413, 2540, 2667, 2794, 2921, 3048, 3175, 3302, 3429, 3556, 3683, 3810, 3937, 4064, 4191, 4318, 4445, 4572, 4699, 4826, 4953, 5080

Offset: 0

Omar E. Pol, Mar 09 2008

127 is the 4th Mersenne prime A000668.

			a(64)=8128 (The 4th perfect number A000396) because 127*64=8128.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000396, A000668, A135631.

127*Range[0,40] (* Harvey P. Dale, Jan 17 2013 *)

a(n) = 127*n.
From Chai Wah Wu, Aug 10 2023: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 1.
G.f.: 127*x/(x - 1)^2. (End)

A160547 Numbers coprime to 31.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Zerinvary Lajos, May 18 2009

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).

Row 31 of A126572.

Complement of A135631.

With[{nn=100},Complement[Range[nn],31*Range[Floor[nn/31]]]] (* Harvey P. Dale, Nov 29 2017 *)

[i for i in range(0,100) if gcd(31, i) == 1]

a(n) = n+floor((n-1)/(p-1)) where p=31 in this case. - Roger M Ellingson, Nov 14 2023

Name edited by Roger M Ellingson, Nov 14 2023

cp's OEIS Frontend

A044102 Multiples of 36.

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A195819 Multiples of 29.

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A174312 a(n) = 32*n.

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A135639 a(n) = 839*n.

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A212773 Amounts (in cents) of coins in denominations 1, 5, 10, 25, and 50 (cents) which can consist of equal numbers of coins of all denominations present when two or more denominations are used (or none are used: term 0).

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A274824 Triangle read by rows: T(n,k) = (n-k+1)*sigma(k), n>=1, 1<=k<=n.

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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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Maple

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A305548 a(n) = 27*n.