A006002 -id:A006002 - cp's OEIS frontend

A210647 Least nonnegative m such that k(n) + k(m) is prime, where k(n) = n*(n+1)^2/2.

0, 1, 22, 2, 142, 1, 2, 10, 22, 1, 34, 10, 2, 37, 46, 6, 10, 1, 6, 46, 46, 1, 10, 106, 6, 1, 58, 2, 22, 7, 2, 58, 94, 3, 22, 10, 2, 1, 22, 2, 10, 16, 6, 82, 118, 4, 82, 10, 18, 1, 10, 2, 22, 1, 2, 10, 10, 4, 22, 58, 2, 19, 10, 2, 46, 1, 10, 70, 82, 3, 22, 34

Offset: 1

Gerasimov Sergey, Mar 27 2012

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A006002, A129634, A210646.

f[n_] := n (n + 1)^2/2; Table[k = 0; While[! PrimeQ[f[n] + f[k]], k++]; k, {n, 100}] (* T. D. Noe, Apr 03 2012 *)

a(n)=my(K=n*(n+1)^2/2,m);while(!isprime(K+m*(m+1)^2/2),m++);m \\ Charles R Greathouse IV, Aug 03 2012

Corrected by R. J. Mathar, Mar 31 2012

A281381 a(n) = n(n + 1)(4*n + 5)/2.

Original entry on oeis.org

0, 9, 39, 102, 210, 375, 609, 924, 1332, 1845, 2475, 3234, 4134, 5187, 6405, 7800, 9384, 11169, 13167, 15390, 17850, 20559, 23529, 26772, 30300, 34125, 38259, 42714, 47502, 52635, 58125, 63984, 70224, 76857, 83895, 91350, 99234, 107559, 116337, 125580, 135300, 145509, 156219, 167442, 179190, 191475

Offset: 0

Peter M. Chema, Jan 21 2017

Shares its digital root, zero together with period 9: repeat [3, 3, 3, 6, 6, 6, 9, 9, 9] with A027480.

Final digits cycle a length period 20: repeat [0, 9, 9, 2, 0, 5, 9, 4, 2, 5, 5, 4, 4, 7, 5, 0, 4, 9, 7, 0].

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Partial sums of A195319.

Cf. A006002, A007742, A016061, A027480, A281258.

[n*(n+1)*(4*n+5)/2 : n in [0..50]]; // Wesley Ivan Hurt, Aug 30 2022

Table[n (n + 1) (4 n + 5)/2, {n, 0, 45}] (* or *)
CoefficientList[Series[3 x (3 + x)/(1 - x)^4, {x, 0, 45}], x] (* Michael De Vlieger, Jan 21 2017 *)

concat(0, Vec(3*x*(3 + x) / (1 - x)^4 + O(x^50))) \\ Colin Barker, Jan 21 2017

a(n) = n*(n + 1)*(4*n + 5)/2 \\ Charles R Greathouse IV, Feb 01 2017

a(n) = 2*n^3 + 9*n^2/2 + 5*n/2.
a(n) = 3*A016061(n).
a(n) = A006002(n+1)*(n) - A006002(n)*(n-1).
a(n) = A007742(n)*(n - 1)/2.
From Colin Barker, Jan 21 2017: (Start)
G.f.: 3*x*(3 + x) / (1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. (End)
From Stefano Spezia, Aug 30 2022: (Start)
E.g.f.: exp(x)*x*(18 + 21*x + 4*x^2)/2.
Sum_{n>0} 1/a(n) = 2*(20*log(8) + 10*Pi - 71)/25 = 0.1603805895595720759728288896228498341201... . (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*sqrt(2)*Pi/5 + 4*(3+sqrt(2))*log(2)/5 - 8*sqrt(2)*log(2-sqrt(2))/5 - 178/25. - Amiram Eldar, Sep 22 2022

A302447 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(k+1)^2/2).

Original entry on oeis.org

1, 2, 12, 46, 175, 610, 2107, 6918, 22256, 69498, 212649, 636910, 1874470, 5423332, 15457223, 43433088, 120467606, 330077358, 894193347, 2396636236, 6359325300, 16714566278, 43539016461, 112449776138, 288083439729, 732356943548, 1848098069644, 4630892393996

Offset: 0

Ilya Gutkovskiy, Apr 08 2018

nonn

Euler transform of A006002.

Vaclav Kotesovec, Table of n, a(n) for n = 0..6000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms

Cf. A000294, A006002, A294667.

nmax = 27; CoefficientList[Series[Product[1/(1 - x^k)^(k (k + 1)^2/2), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (d + 1)^2/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 27}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A006002(k).

a(n) ~ exp(5 * (3*Zeta(5))^(1/5) * n^(4/5) / 2^(8/5) + Pi^4 * n^(3/5) / (90 * 2^(1/5) * (3*Zeta(5))^(3/5)) + (Zeta(3) / 2^(9/5) - Pi^8 / (27000 * 2^(4/5) * Zeta(5))) * n^(2/5) / (3*Zeta(5))^(2/5) + (Pi^8 / (12150000 * Zeta(5)) - Zeta(3) / 900) * Pi^4 * n^(1/5) / (2^(2/5) * 3^(1/5) * Zeta(5)^(6/5)) + 1/24 - Zeta(3) / (4*Pi^2) - Pi^16 / (5248800000 * Zeta(5)^3) + Pi^8 * Zeta(3) / (324000 * Zeta(5)^2) - Zeta(3)^2 / (120 * Zeta(5)) + Zeta'(-3)/2) * (3*Zeta(5))^(43/400) / (2^(57/200) * sqrt(5*A*Pi) * n^(243/400)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 08 2018

A360174 Triangle read by rows. T(n, k) = (k + 1) * abs(Stirling1(n, k)).

Original entry on oeis.org

1, 0, 2, 0, 2, 3, 0, 4, 9, 4, 0, 12, 33, 24, 5, 0, 48, 150, 140, 50, 6, 0, 240, 822, 900, 425, 90, 7, 0, 1440, 5292, 6496, 3675, 1050, 147, 8, 0, 10080, 39204, 52528, 33845, 11760, 2254, 224, 9, 0, 80640, 328752, 472496, 336420, 134694, 31752, 4368, 324, 10

Offset: 0

Peter Luschny, Feb 08 2023

			Triangle T(n, k) starts:
[0] 1;
[1] 0,     2;
[2] 0,     2,     3;
[3] 0,     4,     9,     4;
[4] 0,    12,    33,    24,     5;
[5] 0,    48,   150,   140,    50,     6;
[6] 0,   240,   822,   900,   425,    90,    7;
[7] 0,  1440,  5292,  6496,  3675,  1050,  147,   8;
[8] 0, 10080, 39204, 52528, 33845, 11760, 2254, 224, 9;

Cf. A208529 (column 1), A006002 (subdiagonal), A000774 (row sums).

Cf. A069138 (Stirling2 counterpart), A360205 (Lah counterpart).

T := (n, k) -> (k + 1)*abs(Stirling1(n, k)):
for n from 0 to 8 do seq(T(n, k), k = 0..n) od;

A099904 Numerator of sum of all matrix elements of N X N matrix M(i,j) = i^3+j^3, (i,j = 1..n) divided by n!.

Original entry on oeis.org

2, 18, 36, 100, 75, 147, 98, 18, 45, 605, 121, 169, 1183, 7, 1, 289, 289, 361, 361, 1, 11, 5819, 529, 1, 13, 13, 1, 841, 841, 961, 961, 1, 17, 17, 1, 1369, 26011, 19, 1, 1681, 1681, 1849, 1849, 1, 23, 50807, 2209, 1, 1, 1, 1, 2809, 2809, 1, 1, 1, 29, 100949, 3481, 3721

Offset: 1

Alexander Adamchuk, Oct 29 2004

nonn

Sum M(i,j) (i,j = 1..n) is A099903(n). a(n) is an irregular sequence with highest champions belonging to Pentagonal pyramidal numbers n^2*(n+1)/2 (A002411) and n/2*(n+1)^2 (A006002).

			A099903(n)/n! begins 2, 18, 36, 100/3, 75/4, 147/20, 98/45, 18/35, 45/448, ... So a(6) = 147.

Cf. A099903, A019584, A098077, A002411, A006002.

Table[ Numerator[ Sum[(i^3 + j^3), {i, n}, {j, n}]/n! ], {n, 60}]

a(n) = Numerator[1/n!*Sum[Sum[(i^3+j^3), {i, 1, n}], {j, 1, n}]] a(n) = Numerator[1/2 * (n^3)*(n+1)^2 /n! ].

A109454 Sum of non-Fibonacci numbers between successive Fibonacci numbers: a(n) = Sum_{k=F(n)+1..F(n+1)-1} k.

Original entry on oeis.org

0, 0, 0, 0, 4, 13, 42, 119, 330, 890, 2376, 6291, 16588, 43615, 114492, 300236, 786828, 2061233, 5398470, 14136759, 37015990, 96917974, 253748880, 664346375, 1739318904, 4553656703, 11921726232, 31211643384, 81713400340, 213928875445, 560073740226

Offset: 0

Amarnath Murthy, Aug 27 2005

			F(5) = F(4) + 1 = 4.
F(6) = (F(5) + 1) + (F(5) + 2) = 6+7 = 13.
F(7) = 9+10+11+12 = 42.

Colin Barker, Table of n, a(n) for n = 0..1000
Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (3,1,-5,-1,1).

Cf. A000045, A006002.

CoefficientList[Series[x^4*(x^2 - x - 4)/((x + 1) (x^2 - 3 x + 1) (x^2 + x - 1)), {x, 0, 30}], x] (* Michael De Vlieger, Jul 08 2021 *)
Total[Range[#[[1]]+1,#[[2]]-1]]&/@Partition[Fibonacci[Range[0,40]],2,1] (* or *) LinearRecurrence[{3,1,-5,-1,1},{0,0,0,0,4,13,42},40] (* Harvey P. Dale, Sep 30 2024 *)

concat([0,0,0,0], Vec(x^4*(x^2-x-4) / ((x+1)*(x^2-3*x+1)*(x^2+x-1)) + O(x^100))) \\ Colin Barker, Mar 26 2015

a(n) = Fibonacci(n+2)*(Fibonacci(n-1)-1)/2, n>1. - Vladeta Jovovic, Aug 27 2005
a(n) = 3*a(n-1) + a(n-2) - 5*a(n-3) - a(n-4) + a(n-5) for n>6. - Colin Barker, Mar 26 2015
G.f.: x^4*(x^2-x-4) / ((x+1)*(x^2-3*x+1)*(x^2+x-1)). - Colin Barker, Mar 26 2015

More terms from Franklin T. Adams-Watters, Jun 06 2006

A139309 Array by antidiagonals, sum of non-k-gonal numbers between consecutive k-gonal numbers.

Original entry on oeis.org

0, 0, 2, 0, 5, 9, 0, 9, 26, 24, 0, 14, 51, 75, 50, 0, 20, 84, 153, 164, 90, 0, 27, 125, 258, 342, 305, 147, 0, 35, 174, 390, 584, 645, 510, 224, 0, 44, 231, 549, 890, 1110, 1089, 791, 324, 0, 54, 296, 735, 1260, 1700, 1884, 1701, 1160, 450, 0, 65, 369, 948, 1694

Offset: 0

Jonathan Vos Post, Jun 07 2008

The n=1 column is A000096(k) = n*(n+3)/2. The k=3 row is the sum of nontriangular numbers between successive triangular numbers (A006002) = the sum of n consecutive integers beginning with (n-th triangular number)+1 = (n*(n+1)^2)/2. The k=4 row is the sum of nonsquares between successive squares (A048395) = 2*n^3 + 2*n^2 + n. The k=5 row is the sum of non-pentagonal numbers between successive pentagonal numbers. The k-th row is the sum of non-k-gonal numbers between successive k-gonal numbers. Each column is a quadratic sequence. Each row is a cubic sequence.

			The array begins:
========================================================================
...|.n=0.|.n=1.|.n=2.|.n=3.|.n=4.|.n=5.|.n=6.|.n=7.|.n=8.|.n=9.|.in.OEIS
====|=====|=====|=====|=====|=====|=====|=====|=====|=====|=====|========
k=3.|..0..|..2..|..9..|..24.|..50.|..90.|.147.|.224.|.324.|.450.|.A006002
k=4.|..0..|..5..|.26..|..75.|.164.|.305.|.510.|.791.|1160.|1629.|.A048395
k=5.|..0..|..9..|.51..|.153.|.342.|.645.|1089.|...................not.yet
k=6.|..0..|.14..|.84..|.258.|.584.|...............................not.yet
k=7.|..0..|.20..|125..|.390.|.....................................not.yet
k=8.|..0..|.27..|174..|...........................................not.yet
k=9.|..0..|.35..|231..|...........................................not.yet
k=10|..0..|.44..|296..|...........................................not.yet
========================================================================

Cf. A000027, A000096, A006002, A048395.

A139309 := proc(k,n) n*(k-2)*((k-2)*n^2+1+2*n)/2 ; end: for d from 3 to 16 do for n from 0 to d-3 do printf("%d,", A139309(d-n,n)) ; od: od: # R. J. Mathar, Jun 12 2008

T(k,n) = n(k-2)((k-2)n^2+1+2n)/2. - R. J. Mathar, Jun 12 2008

More terms from R. J. Mathar, Jun 12 2008

A181956 Smallest prime greater than n*(n+1)^2/2.

Original entry on oeis.org

2, 3, 11, 29, 53, 97, 149, 227, 331, 457, 607, 797, 1019, 1277, 1579, 1931, 2333, 2767, 3251, 3803, 4421, 5087, 5821, 6637, 7507, 8461, 9479, 10589, 11777, 13063, 14419, 15877, 17431, 19079, 20849, 22691, 24659, 26717, 28901, 31219, 33623, 36187, 38833, 41627

Offset: 0

Gerasimov Sergey, Apr 03 2012

nonn

			a(1)=2 because prime 2 > (0*(0+1)^2/2) = 0, a(2)=3 because prime 3 > (1*(1+1)^2/2) = 2, a(3)=11 because prime 11 > (2*(2+1)^2/2) = 9.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A006002, A097050.

Table[NextPrime[n*(n+1)^2/2], {n, 0, 50}] (* T. D. Noe, Apr 03 2012 *)

a(n)=nextprime(n*(n+1)^2/2+1) \\ Charles R Greathouse IV, Aug 03 2012

a(n) ~ n^3 / 2. - Charles R Greathouse IV, Aug 03 2012

A182091 Numbers n such that (n-1)*n^2/2-1 is prime.

Original entry on oeis.org

4, 6, 8, 10, 13, 17, 18, 21, 22, 25, 30, 34, 36, 38, 41, 42, 45, 46, 56, 57, 58, 62, 64, 69, 70, 72, 74, 86, 88, 101, 105, 113, 114, 116, 120, 122, 126, 132, 133, 141, 157, 158, 174, 176, 181, 182, 186, 192, 198, 200, 205, 216, 217, 221, 224, 240, 253, 254

Offset: 1

Gerasimov Sergey, Apr 11 2012

			a(1)=4 because (4-1)*4^2/2-1=23 is prime, a(2)=6 because (6-1)*6^2/2-1=89 is prime.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A006002.

[n: n in [1..300] |IsPrime((n-1)*n^2 div 2-1)]; // Vincenzo Librandi, Aug 21 2017

Select[Range[300], PrimeQ[(# - 1)*#^2/2 - 1] &] (* T. D. Noe, Apr 11 2012 *)

is(n)=isprime((n-1)*n^2/2-1) \\ Charles R Greathouse IV, Jun 13 2017

A255687 a(n) = n(n + 1)(7*n + 11)/6.

Original entry on oeis.org

0, 6, 25, 64, 130, 230, 371, 560, 804, 1110, 1485, 1936, 2470, 3094, 3815, 4640, 5576, 6630, 7809, 9120, 10570, 12166, 13915, 15824, 17900, 20150, 22581, 25200, 28014, 31030, 34255, 37696, 41360, 45254, 49385, 53760, 58386, 63270, 68419, 73840, 79540, 85526

Offset: 0

Luce ETIENNE, Mar 02 2015

This sequence gives the number of triangles of all sizes in (3*n^2+2*n)-polyiamonds in a pentagonal or heptagonal configuration.

Also sum of 2*n*(n+1)*(n+2)/3 triangles oriented in one direction and n*(n+1)^2/2 oriented in the opposite direction.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

First bisection of A212977.
Partial sums of A179986.
Cf. A000292, A006002, A007584, A255211.

[n*(n+1)*(7*n+11)/6: n in [0..50]]; // Bruno Berselli, Mar 02 2015

A255687:=n->n*(n+1)*(7*n+11)/6: seq(A255687(n), n=0..50); # Wesley Ivan Hurt, Mar 03 2015

Table[n (n + 1) (7 n + 11)/6, {n, 0, 50}] (* Bruno Berselli, Mar 02 2015 *)
LinearRecurrence[{4,-6,4,-1},{0,6,25,64},50] (* Harvey P. Dale, Jul 17 2015 *)

PARI
```
vector(50, n, n--; n*(n+1)*(7*n+11)/6)
```

concat(0, Vec(x*(x+6)/(x-1)^4 + O(x^100))) \\ Colin Barker, Mar 02 2015

[n*(n+1)*(7*n+11)/6 for n in (0..50)] # Bruno Berselli, Mar 02 2015

a(n) = (1/2)*(Sum_{j=0..n} (n+1-j)*(3*n-j) + Sum_{j=0..n-1} (n-j)*(3*n+1-3*j)).
From Colin Barker, Mar 02 2015: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: x*(x + 6)/(x - 1)^4. (End)
a(n) = -A007584(-n-1). - Bruno Berselli, Mar 02 2015
From Elmo R. Oliveira, Aug 18 2025: (Start)
E.g.f.: exp(x)*x*(36 + 39*x + 7*x^2)/6.
a(n) = A212977(2*n). (End)

cp's OEIS Frontend

A210647 Least nonnegative m such that k(n) + k(m) is prime, where k(n) = n*(n+1)^2/2.

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A281381 a(n) = n*(n + 1)*(4*n + 5)/2.

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Magma

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A302447 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(k+1)^2/2).

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A360174 Triangle read by rows. T(n, k) = (k + 1) * abs(Stirling1(n, k)).

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Maple

A099904 Numerator of sum of all matrix elements of N X N matrix M(i,j) = i^3+j^3, (i,j = 1..n) divided by n!.

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A109454 Sum of non-Fibonacci numbers between successive Fibonacci numbers: a(n) = Sum_{k=F(n)+1..F(n+1)-1} k.

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A139309 Array by antidiagonals, sum of non-k-gonal numbers between consecutive k-gonal numbers.

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Maple

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A181956 Smallest prime greater than n*(n+1)^2/2.

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A281381 a(n) = n(n + 1)(4*n + 5)/2.

A255687 a(n) = n(n + 1)(7*n + 11)/6.