A206817 -id:A206817 - cp's OEIS frontend

A185382 Sum_{j=1..n-1} P(n)-P(j), where P(j) = A065091(j) is the j-th odd prime.

0, 2, 6, 18, 26, 46, 58, 86, 134, 152, 212, 256, 280, 332, 416, 506, 538, 640, 712, 750, 870, 954, 1086, 1270, 1366, 1416, 1520, 1574, 1686, 2092, 2212, 2398, 2462, 2792, 2860, 3070, 3286, 3434, 3662, 3896, 3976, 4386, 4470, 4642, 4730, 5270

Offset: 1

Clark Kimberling, Feb 13 2012

nonn

It appears 1/3 of a(n) values are divisible by 3 (as measured up to n = 8000). Almost all of these cases occur consecutively (i.e., in "runs"). The sizes of these runs, including runs of 1, in the first 250 primes are given by this sequence: {2, 4, 1, 1, 2, 4, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 5, 6, 3, 2, 2, 3, 3, 9, 1, ..} with two runs up to 12 in length occurring in the first 5000 primes. - Richard R. Forberg, Mar 26 2015

a(n+1) == a(n) (mod 3) iff n == 0 (mod 3) or P(n+1) == P(n) (mod 3); this should have asymptotic probability 2/3, and explains some of the above comment. - Robert Israel, Mar 26 2015

			a(4)=(11-3)+(11-5)+(11-7)=18.

Robert Israel, Table of n, a(n) for n = 1..10000 (corrected by Ray Chandler, Jan 19 2019)

Cf. A001223, A152535, A206802 (a(n)/2), A206803 (= partial sums of this), A206804, A206817.

N:= 1000: # to get terms for all odd primes <= N
P:= select(isprime,[seq(2*i+1, i=1..floor((N-1)/2))]):
Q:= ListTools[PartialSums](P):
seq(n*P[n]-Q[n],n=2..nops(P)); # Robert Israel, Mar 26 2015

s[k_] := Prime[k + 1]; p[n_] := Sum[s[k], {k, 1, n}]; c[n_] := n*s[n] - p[n]; Table[c[n], {n, 2, 100}]

A185382(n)=(n-1)*prime(n+1)-sum(k=2,n-1,prime(k)) \\ M. F. Hasler, May 02 2015

a(n) = (n-1)*A065091(n) - A071148(n-1) = (n-1)*prime(n+1) - sum_{1 < k <= n} prime(k). [Corrected and extended by M. F. Hasler, May 02 2015]
a(n) = A206803(n) - A206803(n-1).
a(n) = Sum_{j=1..n-1} j*A001223(j+1). - Robert Israel, Mar 26 2015

Edited and a(1)=0 prefixed by M. F. Hasler, May 02 2015

A206803 Sum_{0A065091(j) is the j-th odd prime.

Original entry on oeis.org

2, 8, 26, 52, 98, 156, 242, 376, 528, 740, 996, 1276, 1608, 2024, 2530, 3068, 3708, 4420, 5170, 6040, 6994, 8080, 9350, 10716, 12132, 13652, 15226, 16912, 19004, 21216, 23614, 26076, 28868, 31728, 34798, 38084, 41518, 45180, 49076

Offset: 2

Clark Kimberling, Feb 13 2012

nonn

Partial sums of A185382.

Danny Rorabaugh, Table of n, a(n) for n = 2..10000

Cf. A065091, A185382, A206804, A206817.

s[k_] := Prime[k + 1]; t[1] = 0;
p[n_] := Sum[s[k], {k, 1, n}];
c[n_] := n*s[n] - p[n]
t[n_] := t[n - 1] + (n - 1) s[n] - p[n - 1]
Table[c[n], {n, 2, 100}]  (* A185382 *)
%/2  (* A206802 *)
Flatten[Table[t[n], {n, 2, 40}]]  (*  A206803 *)
%/2  (* A206804 *)

[sum([sum([nth_prime(k+1)-nth_prime(j+1) for j in range(1,k)]) for k in range(2,n+1)]) for n in range(2,41)] # Danny Rorabaugh, Apr 18 2015

A206816 a(n) = Sum_{0

Original entry on oeis.org

1, 9, 63, 447, 3447, 29367, 276327, 2856807, 32250087, 395130087, 5225062887, 74201293287, 1126567808487, 18213512883687, 312440245683687, 5668674457011687, 108462341176755687, 2182831421832627687, 46096712669420979687

Offset: 2

Clark Kimberling, Feb 12 2012

nonn

			a(4) = (24-1) + (24-2) + (24-6) = 63.

Danny Rorabaugh, Table of n, a(n) for n = 2..400

Cf. A000142, A206817, A001563, A007489.

seq(add(k^2*k!,k=1..n-1), n=2..30); # Ridouane Oudra, Jun 13 2025

s[k_] := k!; t[1] = 0;
p[n_] := Sum[s[k], {k, 1, n}];
c[n_] := n*s[n] - p[n];
t[n_] := t[n - 1] + (n - 1) s[n] - p[n - 1];
Table[c[n], {n, 2, 32}]           (* A206816 *)
Flatten[Table[t[n], {n, 2, 20}]]  (* A206817 *)

a(n) = sum(j=1, n-1, n!-j!); \\ Michel Marcus, Jun 13 2025

[sum([factorial(n)-factorial(j) for j in range(1,n)]) for n in range(2,21)] # Danny Rorabaugh, Apr 18 2015

a(n) = n*n!-p(n), where p(n) is the n-th partial sum of (j!).
a(n) = t(n)-t(n-1), where t = A206817.
a(n) = Sum_{k=1..n-1} k^2*k!. - Ridouane Oudra, Jun 13 2025
a(n) = A001563(n) - A007489(n). - Ridouane Oudra, Jun 14 2025

A206806 Sum_{0A002620(j) is the j-th quarter-square.

Original entry on oeis.org

1, 4, 13, 30, 62, 112, 190, 300, 455, 660, 931, 1274, 1708, 2240, 2892, 3672, 4605, 5700, 6985, 8470, 10186, 12144, 14378, 16900, 19747, 22932, 26495, 30450, 34840, 39680, 45016, 50864, 57273, 64260, 71877, 80142, 89110, 98800, 109270, 120540, 132671, 145684

Offset: 2

Clark Kimberling, Feb 15 2012

Partial sums of A049774. For a guide to related sequences, see A206817.

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Cf. A206817, A002620.

[(108-36*n-n^2+n^4+(70*n-266)*Ceiling((3-n)/2)-(42*n-234)*Ceiling((3-n)/2)^2+(8*n-88)*Ceiling((3-n)/2)^3+12*Ceiling((3-n)/2)^4-4*n*Floor(n/2)-(12*n-12)*Floor(n/2)^2-(8*n-24)*Floor(n/2)^3+12*Floor(n/2)^4)/12: n in [2..50]]; // Wesley Ivan Hurt, Jul 10 2014

A206806:=n->add(i*(n-i)*(i-ceil((i-1)/2)), i=1..n): seq(A206806(n), n=2..50); # Wesley Ivan Hurt, Jul 10 2014

s[k_] := Floor[k/2]*Ceiling[k/2]; t[1] = 0;
Table[s[k], {k, 1, 20}]    (* A002620 *)
p[n_] := Sum[s[k], {k, 1, n}];
c[n_] := n*s[n] - p[n];
t[n_] := t[n - 1] + (n - 1) s[n] - p[n - 1]
Table[c[n], {n, 2, 50}]    (* A049774 *)
f = Flatten[Table[t[n], {n, 2, 50}]]  (* A206806 *)
Table[Sum[i (n - i) (i - Ceiling[(i - 1)/2]), {i, n}], {n, 2, 50}] (* Wesley Ivan Hurt, Jul 10 2014 *)
CoefficientList[Series[-(2 x^2 + x + 1)/((x - 1)^5 (x + 1)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 10 2014 *)

vector(100, n, ((n+1)*(1+3*(-1)^(n+1)-2*(n+1)+2*(n+1)^2+2*(n+1)^3))/48) \\ Colin Barker, Jul 10 2014

Vec(-x^2*(2*x^2+x+1)/((x-1)^5*(x+1)^2) + O(x^100)) \\ Colin Barker, Jul 10 2014

[sum([sum([floor(k^2/4)-floor(j^2/4) for j in range(1,k)]) for k in range(2,n+1)]) for n in range(2,44)] # Danny Rorabaugh, Apr 18 2015

From Wesley Ivan Hurt, Jul 10 2014: (Start)
a(n) = Sum_{i=1..n} i * (n-i) * (i-ceiling((i-1)/2)).
a(n) = (108 - 36n - n^2 + n^4 + (70n - 266) * ceiling((3 - n)/2) - (42n - 234) * ceiling((3 - n)/2)^2 + (8n - 88) * ceiling((3 - n)/2)^3 + 12 * ceiling((3 - n)/2)^4 - 4n * floor(n/2) - (12n - 12) * floor(n/2)^2 - (8n - 24) * floor(n/2)^3 + 12 * floor(n/2)^4) / 12. (End)
a(n) = (n*(1+3*(-1)^n-2*n+2*n^2+2*n^3))/48. - Colin Barker, Jul 10 2014
G.f.: -x^2*(2*x^2+x+1) / ((x-1)^5*(x+1)^2). - Colin Barker, Jul 10 2014

A206808 Sum_{0

Original entry on oeis.org

7, 45, 156, 400, 855, 1617, 2800, 4536, 6975, 10285, 14652, 20280, 27391, 36225, 47040, 60112, 75735, 94221, 115900, 141120, 170247, 203665, 241776, 285000, 333775, 388557, 449820, 518056, 593775, 677505, 769792, 871200, 982311

Offset: 2

Clark Kimberling, Feb 15 2012

a(n) = n^4-p(n), where p(n) is the n-th partial sum of (j^3).
a(n) = t(n)-t(n-1), where t = A206809.
For a guide to related sequences, see A206817.

			a(2) = 2^3-1^3 = 7.
a(3) = (3^3-1^3) + (3^3-2^3) = 45.

Danny Rorabaugh, Table of n, a(n) for n = 2..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A206817, A206806.

s[k_] := k^3; t[1] = 0;
p[n_] := Sum[s[k], {k, 1, n}];
c[n_] := n*s[n] - p[n];
t[n_] := t[n - 1] + (n - 1) s[n] - p[n - 1]
Table[c[n], {n, 2, 50}]  (* A206808 *)
Flatten[Table[t[n], {n, 2, 35}]]  (* A206809 *)

vector(100, n, (3*n^4+10*n^3+11*n^2+4*n)/4) \\ Colin Barker, Jul 11 2014

Vec(-x^2*(x^2+10*x+7)/(x-1)^5 + O(x^100)) \\ Colin Barker, Jul 11 2014

[sum([n^3-j^3 for j in range(1,n)]) for n in range(2,35)] # Danny Rorabaugh, Apr 18 2015

a(n) = (3*n^4-2*n^3-n^2)/4. G.f.: -x^2*(x^2+10*x+7) / (x-1)^5. - Colin Barker, Jul 11 2014

A206809 a(n) = Sum_{0

Original entry on oeis.org

7, 52, 208, 608, 1463, 3080, 5880, 10416, 17391, 27676, 42328, 62608, 89999, 126224, 173264, 233376, 309111, 403332, 519232, 660352, 830599, 1034264, 1276040, 1561040, 1894815, 2283372, 2733192, 3251248, 3845023, 4522528, 5292320

Offset: 2

Clark Kimberling, Feb 15 2012

Partial sums of A206808. For a guide to related sequences, see A206817.

			a(3) = (8-1) + (27-1) + (27-8) = 52.
a(4) = a(3) + (64-1) + (64-8) + (64-27) = 208.

Danny Rorabaugh, Table of n, a(n) for n = 2..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A206808, A206817.

s[k_] := k^3; t[1] = 0;
p[n_] := Sum[s[k], {k, 1, n}];
c[n_] := n*s[n] - p[n];
t[n_] := t[n - 1] + (n - 1) s[n] - p[n - 1]
Table[c[n], {n, 2, 50}]  (* A206808 *)
Flatten[Table[t[n], {n, 2, 35}]]  (* A206809 *)

vector(100, n, n*(9*n^4+60*n^3+145*n^2+150*n+56)/60) \\ Colin Barker, Jul 11 2014

Vec(x^2*(x^2+10*x+7)/(x-1)^6 + O(x^100)) \\ Colin Barker, Jul 11 2014

[sum([sum([k^3-j^3 for j in range(1,k)]) for k in range(2,n+1)]) for n in range(2,33)] # Danny Rorabaugh, Apr 18 2015

a(n) = (n*(-4-15*n-5*n^2+15*n^3+9*n^4))/60. G.f.: x^2*(x^2+10*x+7) / (x-1)^6. - Colin Barker, Jul 11 2014

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Wesley Ivan Hurt, Jul 23 2025

A206810 Sum_{0

Original entry on oeis.org

15, 145, 670, 2146, 5501, 12131, 23996, 43716, 74667, 121077, 188122, 282022, 410137, 581063, 804728, 1092488, 1457223, 1913433, 2477334, 3166954, 4002229, 5005099, 6199604, 7611980, 9270755, 11206845, 13453650, 16047150

Offset: 2

Clark Kimberling, Feb 15 2012

For a guide to related sequences, see A206817.

			a(2) = 2^4-1^4 = 15.
a(3) = (3^4-1^4) + (3^4-2^4) = 145.

Danny Rorabaugh, Table of n, a(n) for n = 2..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A206817, A206811.

s[k_] := k^4; t[1] = 0;
p[n_] := Sum[s[k], {k, 1, n}];
c[n_] := n*s[n] - p[n];
t[n_] := t[n - 1] + (n - 1) s[n] - p[n - 1]
Table[c[n], {n, 2, 50}]  (* A206810  *)
Flatten[Table[t[n], {n, 2, 35}]] (* A206811 *)

Vec(x^2*(x^3+25*x^2+55*x+15)/(x-1)^6 + O(x^100)) \\ Colin Barker, Jul 11 2014

[sum([n^4-j^4 for j in range(1,n)]) for n in range(2,30)] # Danny Rorabaugh, Apr 18 2015

a(n) = n^5-p(n), where p(n) is the n-th partial sum of (j^4).
a(n) = t(n)-t(n-1), where t = A206811.
a(n) = (n-10*n^3-15*n^4+24*n^5)/30. G.f.: x^2*(x^3+25*x^2+55*x+15) / (x-1)^6. - Colin Barker, Jul 11 2014

A206811 Sum_{0

Original entry on oeis.org

15, 160, 830, 2976, 8477, 20608, 44604, 88320, 162987, 284064, 472186, 754208, 1164345, 1745408, 2550136, 3642624, 5099847, 7013280, 9490614, 12657568, 16659797, 21664896, 27864500, 35476480, 44747235, 55954080, 69407730

Offset: 2

Clark Kimberling, Feb 15 2012

Partial sums of A206810. For a guide to related sequences, see A206817.

			a(4) = 16-1 + 81-1 + 81-16 = 160.

Danny Rorabaugh, Table of n, a(n) for n = 2..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A206810, A206817.

s[k_] := k^4; t[1] = 0;
p[n_] := Sum[s[k], {k, 1, n}];
c[n_] := n*s[n] - p[n];
t[n_] := t[n - 1] + (n - 1) s[n] - p[n - 1]
Table[c[n], {n, 2, 50}]  (* A206810  *)
Flatten[Table[t[n], {n, 2, 35}]] (* A206811 *)

Vec(-x^2*(x^3+25*x^2+55*x+15)/(x-1)^7 + O(x^100)) \\ Colin Barker, Jul 11 2014

[sum([sum([k^4-j^4 for j in range(1,k)]) for k in range(2,n+1)]) for n in range(2,29)] # Danny Rorabaugh, Apr 18 2015

a(n) = (n*(1+n)^2*(1-6*n+n^2+4*n^3))/30. G.f.: -x^2*(x^3+25*x^2+55*x+15) / (x-1)^7. - Colin Barker, Jul 11 2014

A185009 Row sums of A051949 (differences of factorial numbers), seen as a triangle.

Original entry on oeis.org

0, 5, 45, 351, 2847, 25047, 241047, 2534247, 28984167, 358842087, 4785978087, 68453274087, 1045616538087, 16993016806887, 292825130163687, 5333909818803687, 102415654899123687, 2067588695129523687, 43785455761653171687, 970599475776544179687

Offset: 1

Olivier Gérard, Nov 02 2012

G. C. Greubel, Table of n, a(n) for n = 1..445

cf. A051949.

Other summations of differences of factorials : A206816, A206817, A065355.

Table[Plus @@ Prepend[Table[(n + 1)! - i!, {i, n, 2, -1}], (n)! - 1], {n, 0, 20}]

for(n=1,25, print1((n^2-1)*n! - sum(k=1,n-1, k!), ", ")) \\ G. C. Greubel, Jun 09 2017

a(n)= (n-1)*(n+1)*n! - sum( i!, i=1..n-1)

cp's OEIS Frontend

A185382 Sum_{j=1..n-1} P(n)-P(j), where P(j) = A065091(j) is the j-th odd prime.

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A206803 Sum_{0A065091(j) is the j-th odd prime.

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A206816 a(n) = Sum_{0

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A206806 Sum_{0A002620(j) is the j-th quarter-square.

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A206808 Sum_{0

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A206809 a(n) = Sum_{0

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