A161700 -id:A161700 - cp's OEIS frontend

A161702 a(n) = (-n^3 + 9n^2 - 5n + 3)/3.

1, 2, 7, 14, 21, 26, 27, 22, 9, -14, -49, -98, -163, -246, -349, -474, -623, -798, -1001, -1234, -1499, -1798, -2133, -2506, -2919, -3374, -3873, -4418, -5011, -5654, -6349, -7098, -7903, -8766, -9689, -10674, -11723, -12838, -14021, -15274

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 4} = divisors of 14:

a(n) = A027750(A006218(13) + k + 1), 0 <= k < A000005(14).

			Differences of divisors of 14 to compute the coefficients of their interpolating polynomial, see formula:
  1     2     7    14
     1     5     7
        4     2
          -2

G. C. Greubel, Table of n, a(n) for n = 0..1000
R. Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000124, A000125, A000127, A002522, A005408, A006261, A016813, A058331, A080856, A086514, A161701, A161703, A161704, A161706-A161708, A161710, A161711-A161713, A161715.

[(-n^3 + 9*n^2 - 5*n + 3)/3: n in [0..50]]; // Vincenzo Librandi, Dec 27 2010

A161702:=n->(-n^3 + 9*n^2 - 5*n + 3)/3: seq(A161702(n), n=0..60); # Wesley Ivan Hurt, Jul 16 2017

Table[(-n^3+9n^2-5n+3)/3,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,2,7,14},40] (* Harvey P. Dale, Jun 15 2013 *)

a(n)=(-n^3+9*n^2-5*n+3)/3 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = C(n,0) + C(n,1) + 4*C(n,2) - 2*C(n,3).
G.f.: (1-2*x+5*x^2-6*x^3)/(1-x)^4. - Colin Barker, Jan 08 2012
a(0)=1, a(1)=2, a(2)=7, a(3)=14, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Jun 15 2013

A161703 a(n) = (4n^3 - 12n^2 + 14*n + 3)/3.

Original entry on oeis.org

1, 3, 5, 15, 41, 91, 173, 295, 465, 691, 981, 1343, 1785, 2315, 2941, 3671, 4513, 5475, 6565, 7791, 9161, 10683, 12365, 14215, 16241, 18451, 20853, 23455, 26265, 29291, 32541, 36023, 39745, 43715, 47941, 52431, 57193, 62235, 67565, 73191, 79121

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 4} = divisors of 15:

a(n) = A027750(A006218(14) + k + 1), 0 <= k < A000005(15).

			Differences of divisors of 15 to compute the coefficients of their interpolating polynomial, see formula:
  1     3     5    15
     2     2    10
        0     8
           8

G. C. Greubel, Table of n, a(n) for n = 0..1000
R. Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000124, A000125, A000127, A002522, A005408, A006261, A016813, A058331, A080856, A086514, A161701, A161702, A161704, A161706-A161708, A161710, A161711-A161713, A161715.

[(4*n^3 - 12*n^2 + 14*n + 3)/3: n in [0..50]]; // Vincenzo Librandi, Dec 27 2010

A161703:=n->(4*n^3 - 12*n^2 + 14*n + 3)/3: seq(A161703(n), n=0..100); # Wesley Ivan Hurt, Jul 16 2017

CoefficientList[Series[(1 - x - x^2 + 9*x^3)/(1 - x)^4, {x, 0, 50}], x] (* G. C. Greubel, Jul 16 2017 *)
LinearRecurrence[{4,-6,4,-1},{1,3,5,15},50] (* Harvey P. Dale, Oct 04 2024 *)

a(n)=n*(4*n^2-12*n+14)/3+1 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = C(n,0) + 2*C(n,1) + 8*C(n,3).

G.f.: (1-x-x^2+9*x^3)/(1-x)^4. - Colin Barker, Jan 08 2012

A161707 a(n) = (4n^3 - 9n^2 + 11*n + 3)/3.

Original entry on oeis.org

1, 3, 7, 21, 53, 111, 203, 337, 521, 763, 1071, 1453, 1917, 2471, 3123, 3881, 4753, 5747, 6871, 8133, 9541, 11103, 12827, 14721, 16793, 19051, 21503, 24157, 27021, 30103, 33411, 36953, 40737, 44771, 49063, 53621, 58453, 63567, 68971, 74673

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 4} = divisors of 21:

a(n) = A027750(A006218(20) + k + 1), 0 <= k < A000005(21).

			Differences of divisors of 21 to compute the coefficients of their interpolating polynomial, see formula:
  1     3     7    21
     2     4    14
        2    10
           8

G. C. Greubel, Table of n, a(n) for n = 0..1000
Reinhard Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000005, A006218, A027750.

[(4*n^3 - 9*n^2 + 11*n + 3)/3: n in [0..50]]; // Vincenzo Librandi, Dec 27 2010

A161707:=n->(4*n^3 - 9*n^2 + 11*n + 3)/3: seq(A161707(n), n=0..100); # Wesley Ivan Hurt, Jan 19 2017

Table[(4n^3-9n^2+11n+3)/3,{n,0,40}] (* or *)
CoefficientList[Series[(7x^3+x^2-x+1)/(x-1)^4, {x,0,60}], x] (* Harvey P. Dale, Mar 28 2011 *)

a(n)=(4*n^3-9*n^2+11*n)/3+1 \\ Charles R Greathouse IV, Jul 16 2011

a(n) = C(n,0) + 2*C(n,1) + 2*C(n,2) + 8*C(n,3).
G.f.: (7*x^3 + x^2 - x + 1)/(x-1)^4. - Harvey P. Dale, Mar 28 2011
E.g.f.: (1/3)*(4*x^3 + 3*x^2 + 6*x + 3)*exp(x). - G. C. Greubel, Jul 16 2017

A161711 a(n) = (-4n^3 + 27n^2 - 20*n + 3)/3.

Original entry on oeis.org

1, 2, 13, 26, 33, 26, -3, -62, -159, -302, -499, -758, -1087, -1494, -1987, -2574, -3263, -4062, -4979, -6022, -7199, -8518, -9987, -11614, -13407, -15374, -17523, -19862, -22399, -25142, -28099, -31278, -34687, -38334, -42227, -46374, -50783

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 4} = divisors of 26:

a(n) = A027750(A006218(25) + k + 1), 0 <= k < A000005(26).

			Differences of divisors of 26 to compute the coefficients of their interpolating polynomial, see formula:
  1     2    13    26
     1    11    13
       10     2
          -8

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
R. Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005408, A000124, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A080856, A161712, A161713, A161715, A006261.

[(-4*n^3 + 27*n^2 - 20*n + 3)/3: n in [0..40]]; // Vincenzo Librandi, Jul 17 2011

LinearRecurrence[{4,-6,4,-1},{1,2,13,26},40] (* Harvey P. Dale, Jul 02 2017 *)

x='x+O('x^50); Vec((1-2*x+11*x^2-18*x^3)/(1-x)^4) \\ G. C. Greubel, Jul 16 2017

a(n) = C(n,0) + C(n,1) + 10*C(n,2) - 8*C(n,3).

G.f.: (1-2*x+11*x^2-18*x^3)/(1-x)^4. - Bruno Berselli, Jul 17 2011

A161712 a(n) = (4n^3 - 6n^2 + 8*n + 3)/3.

Original entry on oeis.org

1, 3, 9, 27, 65, 131, 233, 379, 577, 835, 1161, 1563, 2049, 2627, 3305, 4091, 4993, 6019, 7177, 8475, 9921, 11523, 13289, 15227, 17345, 19651, 22153, 24859, 27777, 30915, 34281, 37883, 41729, 45827, 50185, 54811, 59713, 64899, 70377, 76155

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 4} = divisors of 27:
a(n) = A027750(A006218(26) + k + 1), 0 <= k < A000005(27).
a(n), n > 0 is the number of points of the half-integer lattice in R^n that lie in the open unit ball. - Tom Harris, Jun 15 2021

			Differences of divisors of 27 to compute the coefficients of their interpolating polynomial, see formula:
  1     3     9    27
     2     6    18
        4    12
           8

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Reinhard Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

[(4*n^3 - 6*n^2 + 8*n + 3)/3: n in [0..40]]; // Vincenzo Librandi, Jul 17 2011

Table[(4n^3-6n^2+8n+3)/3,{n,0,80}] (* Harvey P. Dale, Apr 13 2011 *)

a(n)=(4*n^3-6*n^2+8*n)/3+1 \\ Charles R Greathouse IV, Jul 16 2011

a(n) = C(n,0) + 2*C(n,1) + 4*C(n,2) + 8*C(n,3).
G.f.: (x+1)*(1+x*(5*x-2))/(x-1)^4. - Harvey P. Dale, Apr 13 2011
E.g.f.: (1/3)*(4*x^3 + 6*x^2 + 6*x + 3)*exp(x). - G. C. Greubel, Jul 16 2017
a(n) -a(n-1) = A005899(n-1), n>=2. - R. J. Mathar, Aug 03 2025

A128470 a(n) = 30*n + 1.

Original entry on oeis.org

1, 31, 61, 91, 121, 151, 181, 211, 241, 271, 301, 331, 361, 391, 421, 451, 481, 511, 541, 571, 601, 631, 661, 691, 721, 751, 781, 811, 841, 871, 901, 931, 961, 991, 1021, 1051, 1081, 1111, 1141, 1171, 1201, 1231, 1261, 1291, 1321, 1351, 1381, 1411, 1441, 1471

Offset: 0

Cino Hilliard, May 06 2007

Possible upper bounds of twin primes pairs ending in 1: For a 30k + r "wheel", k > 0, r = 1, 13, 19 are the only possible values that can form an upper bound of a twin prime pair. The 30k+r wheel gives the sequence 1, 7, 11, 13, 17, 19, 23, 29 31, 37, 41, 43, 47, 49, 53, 59, ... which is frequently used in prime number sieves to skip multiples of 2, 3, 5. The fact that subtracting 2 from 30k+7, 11, 17, 23 will give us a multiple of 3 or 5 precludes these numbers from being an upper bound of a twin prime pair. This leaves us with r = 1, 13, 19 for k > 0 as the only possible cases to form an upper bound of a twin prime pair. 1, 13, 19 concludes the 6 numbers of the 8 number wheel that can form part of a twin prime pair.

			61 = 30 * 2 + 1, the upper part of the twin prime pair 59, 61.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Albert van der Horst, Counting Twin Primes
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A161700, A005408, A016813, A016921, A017281, A017533, A158057, A161705, A161709, A161714.

[30*n+1: n in [0..50]]; // Vincenzo Librandi, Jun 16 2011

Range[1, 3001, 30] (* Vladimir Joseph Stephan Orlovsky, Jun 15 2011 *)
CoefficientList[Series[(1 + 29 x) / (1 - x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 30 2014 *)
LinearRecurrence[{2, -1}, {1, 31}, 100] (* G. C. Greubel, Apr 04 2016 *)

a(n)=30*n+1 \\ Charles R Greathouse IV, Oct 07 2015

(0 to 49).map(30 *  + 1) // _Alonso del Arte, Jun 02 2019

a(n) = 2*a(n-1) - a(n-2) for n > 1. - Vincenzo Librandi, Dec 30 2014
G.f.: (1 + 29*x)/(1 - x)^2. - Vincenzo Librandi, Dec 30 2014
E.g.f.: (1 + 30*x)*exp(x). - G. C. Greubel, Apr 04 2016

A161709 a(n) = 22*n + 1.

Original entry on oeis.org

1, 23, 45, 67, 89, 111, 133, 155, 177, 199, 221, 243, 265, 287, 309, 331, 353, 375, 397, 419, 441, 463, 485, 507, 529, 551, 573, 595, 617, 639, 661, 683, 705, 727, 749, 771, 793, 815, 837, 859, 881, 903, 925, 947, 969, 991, 1013, 1035, 1057, 1079, 1101, 1123

Offset: 0

Reinhard Zumkeller, Jun 17 2009

Italo Ghersi, Matematica dilettevole e curiosa, p. 139, Hoepli, Milano, 1967. [From Vincenzo Librandi, Dec 02 2009]

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

Cf. A005408, A008604, A016813, A016921, A017281, A017533, A128470, A158057, A161700, A161705, A161714.

List([0..60], n-> 1+22*n ); # G. C. Greubel, Sep 18 2019

[1+22*n: n in [0..60]]; // G. C. Greubel, Sep 18 2019

seq(1+22*n, n=0..60); # G. C. Greubel, Sep 18 2019

22*Range[0,60]+1 (* Harvey P. Dale, Jan 09 2011 *)

vector(60, n, 22*n-21) \\ G. C. Greubel, Sep 18 2019

[1+22*n for n in (0..60)] # G. C. Greubel, Sep 18 2019

From G. C. Greubel, Sep 18 2019: (Start)
a(n) = 2*a(n-1) - a(n-2).
G.f.: (1 + 21*x)/(1-x)^2.
E.g.f.: (1 + 22*x)*exp(x). (End)

A161714 a(n) = 28*n + 1.

Original entry on oeis.org

1, 29, 57, 85, 113, 141, 169, 197, 225, 253, 281, 309, 337, 365, 393, 421, 449, 477, 505, 533, 561, 589, 617, 645, 673, 701, 729, 757, 785, 813, 841, 869, 897, 925, 953, 981, 1009, 1037, 1065, 1093, 1121, 1149, 1177, 1205, 1233, 1261, 1289, 1317, 1345, 1373

Offset: 0

Reinhard Zumkeller, Jun 17 2009

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

Cf. A005408, A016813, A016921, A017281, A017533, A128470, A158057, A161700, A161705, A161709.

List([0..60], n-> 1+28*n); # G. C. Greubel, Sep 18 2019

[28*n + 1: n in [0..60]]; // Vincenzo Librandi, May 04 2011

seq(1+28*n, n=0..60); # G. C. Greubel, Sep 18 2019

Range[1, 1700, 28] (* Vladimir Joseph Stephan Orlovsky, Jun 15 2011 *)
CoefficientList[Series[(1+27x)/(1-x)^2, {x,0,60}], x] (* Michael De Vlieger, Apr 05 2017 *)

vector(60, n, 28*n-27) \\ G. C. Greubel, Sep 18 2019

[1+28*n for n in (0..60)] # G. C. Greubel, Sep 18 2019

G.f.: (1 + 27*x)/(1-x)^2. - Indranil Ghosh, Apr 05 2017

E.g.f.: (1 + 28*x)*exp(x). - G. C. Greubel, Sep 18 2019

A161856 Triangle read by rows in which row n lists the coefficients of the interpolating polynomial for its divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 0, 2, 1, 6, 1, 1, 1, 1, 1, 2, 4, 1, 1, 2, 0, 1, 10, 1, 1, 0, 0, 1, 1, 1, 12, 1, 1, 4, -2, 1, 2, 0, 8, 1, 1, 1, 1, 1, 1, 16, 1, 1, 0, 2, -4, 12, 1, 18, 1, 1, 1, -2, 7, -11, 1, 2, 2, 8, 1, 1, 8, -6, 1, 22, 1, 1, 0, 0, 1, -3, 8, -12, 1, 4, 16, 1, 1, 10, -8, 1, 2, 4, 8, 1, 1

Offset: 1

Reinhard Zumkeller, Jun 20 2009

EDP(n,x) = SUM(a(A006218(n)-1+i)*A007318(x,i-1): 1<=i<=A000005(n)) is the interpolating polynomial for the divisors of n, see also A161700;
A000005(n) = length of n-th row, i.e. same length as n-th row in A027750;
sum of n-th row, n>1: A161857(n) = SUM(a(A006218(n-1)+i): 1<=i<=A000005(n));
a(A006218(n)+1) = 1.

			1; 1,1; 1,2; 1,1,1; 1,4; 1,1,0,2; 1,6; 1,1,1,1; 1,2,4; ... .

R. Zumkeller, Table of rows 1..1000, flattened
R. Zumkeller, Enumerations of Divisors

A005408, A000124, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A006261, A161715.

A273135 Difference table of the divisors of the positive integers (with every table read by antidiagonals downwards).

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 2, 1, 2, 1, 4, 2, 1, 1, 5, 4, 1, 2, 1, 3, 1, 0, 6, 3, 2, 2, 1, 7, 6, 1, 2, 1, 4, 2, 1, 8, 4, 2, 1, 1, 3, 2, 9, 6, 4, 1, 2, 1, 5, 3, 2, 10, 5, 2, 0, 1, 11, 10, 1, 2, 1, 3, 1, 0, 4, 1, 0, 0, 6, 2, 1, 1, 1, 12, 6, 4, 3, 2, 1, 1, 13, 12, 1, 2, 1, 7, 5, 4, 14, 7, 2, -2, 1, 3, 2, 5, 2, 0, 15, 10, 8, 8

Offset: 1

Omar E. Pol, May 18 2016

This is an irregular tetrahedron T(n,j,k) in which the slice n lists the elements of the j-th antidiagonal of the difference triangle of the divisors of n.
The first row of the slice n is also the n-th row of the triangle A027750.
The bottom entry of the slice n is A187202(n).
The number of elements in the n-th slice is A000217(A000005(n)) = A184389(n).
The sum of the elements of the n-th slice is A273103(n).
The antidiagonal sums give A273262.
If n is a power of 2 the antidiagonals are also the divisors of the powers of 2 from 1 to n in decreasing order, for example if n = 8 the finite sequence of antidiagonals is [1], [2, 1], [4, 2, 1], [8, 4, 2, 1].
First differs from A272121 at a(92).

			The tables of the first nine positive integers are
  1; 1, 2; 1, 3; 1, 2, 4; 1, 5; 1, 2, 3, 6; 1, 7; 1, 2, 4, 8; 1, 3, 9;
     1;    2;    1, 2;    4;    1, 1, 3;    6;    1, 2, 4;    2, 6;
                 1;             0, 2;             1, 2;       4;
                                2;                1;
For n = 18 the difference table of the divisors of 18 is
  1,  2, 3, 6, 9, 18;
  1,  1, 3, 3, 9;
  0,  2, 0, 6;
  2, -2, 6;
 -4,  8;
 12;
This table read by antidiagonals downwards gives the finite subsequence [1], [2, 1], [3, 1, 0], [6, 3, 2, 2], [9, 3, 0, -2, -4], [18, 9, 6, 6, 8, 12].

Cf. A000005, A000217, A027750, A161700, A184389, A187202, A272210, A272121, A273102, A273103, A273262.

Table[Table[#[[m - k + 1, k]], {m, Length@ #}, {k, m, 1, -1}] &@ NestWhileList[Differences, Divisors@ n, Length@ # > 1 &], {n, 15}] // Flatten (* Michael De Vlieger, Jun 26 2016 *)

cp's OEIS Frontend

A161702 a(n) = (-n^3 + 9n^2 - 5n + 3)/3.

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A161703 a(n) = (4*n^3 - 12*n^2 + 14*n + 3)/3.

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A161707 a(n) = (4*n^3 - 9*n^2 + 11*n + 3)/3.

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A161711 a(n) = (-4*n^3 + 27*n^2 - 20*n + 3)/3.

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A161712 a(n) = (4*n^3 - 6*n^2 + 8*n + 3)/3.

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A128470 a(n) = 30*n + 1.

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A161703 a(n) = (4n^3 - 12n^2 + 14*n + 3)/3.

A161707 a(n) = (4n^3 - 9n^2 + 11*n + 3)/3.

A161711 a(n) = (-4n^3 + 27n^2 - 20*n + 3)/3.

A161712 a(n) = (4n^3 - 6n^2 + 8*n + 3)/3.