cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A206802 a(n) = (1/2)*A185382(n).

Original entry on oeis.org

1, 3, 9, 13, 23, 29, 43, 67, 76, 106, 128, 140, 166, 208, 253, 269, 320, 356, 375, 435, 477, 543, 635, 683, 708, 760, 787, 843, 1046, 1106, 1199, 1231, 1396, 1430, 1535, 1643, 1717, 1831, 1948, 1988, 2193, 2235, 2321, 2365, 2635, 2911, 3005
Offset: 2

Views

Author

Clark Kimberling, Feb 13 2012

Keywords

Comments

See A185382.

Crossrefs

Cf. A185382.

Programs

  • Mathematica
    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 *)
Table[Sum[Prime[n + 1] - Prime[j + 1], {j, n - 1}]/2, {n, 2, 48}] (* Michael De Vlieger, Sep 13 2016 *)

A206817 Sum_{0

Original entry on oeis.org

1, 10, 73, 520, 3967, 33334, 309661, 3166468, 35416555, 430546642, 5655609529, 79856902816, 1206424711303, 19419937594990, 331860183278677, 6000534640290364, 114462875817046051, 2297294297649673738, 48394006967070653425
Offset: 2

Views

Author

Clark Kimberling, Feb 12 2012

Keywords

Comments

In the following guide to related sequences,
c(n) = Sum_{0
t(n) = Sum_{0
s(k).................c(n)........t(n)
k....................A000217.....A000292
k^2..................A016061.....A004320
k^3..................A206808.....A206809
k^4..................A206810.....A206811
k!...................A206816.....A206817
prime(k).............A152535.....A062020
prime(k+1)...........A185382.....A206803
2^(k-1)..............A000337.....A045618
k(k+1)/2.............A007290.....A034827
k-th quarter-square..A049774.....A206806

Examples

			a(3) = (2-1) + (6-1) + (6-2) = 10.

Links

Crossrefs

Cf. A000142, A206816.

Programs

  • Mathematica
    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 *)
  • PARI
    a(n)=sum(j=1,n,j!*(2*j-n-1)) \\ Charles R Greathouse IV, Oct 11 2015
  • PARI
    a(n)=my(t=1); sum(j=1,n,t*=j; t*(2*j-n-1)) \\ Charles R Greathouse IV, Oct 11 2015
  • Sage
    [sum([sum([factorial(k)-factorial(j) for j in range(1,k)]) for k in range(2,n+1)]) for n in range(2,21)] # Danny Rorabaugh, Apr 18 2015

Formula

a(n) = a(n-1)+(n-1)s(n)-p(n-1), where s(n) = n! and p(k) = 1!+2!+...+k!.
a(n) = Sum_{k=2..n} A206816(k).

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

Views

Author

Clark Kimberling, Feb 13 2012

Keywords

Comments

Partial sums of A185382.

Links

Crossrefs

Cf. A065091, A185382, A206804, A206817.

Programs

  • Mathematica
    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 *)
  • Sage
    [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

A206804 (1/2)*A206803.

Original entry on oeis.org

1, 4, 13, 26, 49, 78, 121, 188, 264, 370, 498, 638, 804, 1012, 1265, 1534, 1854, 2210, 2585, 3020, 3497, 4040, 4675, 5358, 6066, 6826, 7613, 8456, 9502, 10608, 11807, 13038, 14434, 15864, 17399, 19042, 20759, 22590, 24538, 26526, 28719
Offset: 2

Views

Author

Clark Kimberling, Feb 13 2012

Keywords

Crossrefs

Cf. A206803.

Programs

  • Mathematica
    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 *)
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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