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 1-7 of 7 results.

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).

A316322 Sum of piles of first n primes: a(n) = Sum(prime(i)*(2*i-1): 1<=i<=n).

Original entry on oeis.org

2, 11, 36, 85, 184, 327, 548, 833, 1224, 1775, 2426, 3277, 4302, 5463, 6826, 8469, 10416, 12551, 15030, 17799, 20792, 24189, 27924, 32107, 36860, 42011, 47470, 53355, 59568, 66235, 73982, 82235, 91140, 100453, 110734, 121455, 132916, 145141, 158000, 171667, 186166, 201189, 217424, 234215, 251748
Offset: 1

Views

Author

N. J. A. Sloane, Jul 03 2018, based on Reinhard Zumkeller's A083215

Keywords

Examples

			............................................ 7
........................... 5 ............ 7 5 7
............ 3 .......... 5 3 5 ........ 7 5 3 5 7
2 ........ 3 2 3 ...... 5 3 2 3 5 .... 7 5 3 2 3 5 7
a(1)=2 ... a(2)=11 .... a(3)=36 ...... a(4)=85.

Links

Crossrefs

Cf. A083215, A014285, A007504, A167214, A023662, A062020.

Programs

  • Maple
    seq(add((2*i-1)*ithprime(i),i=1..n), n=1..80); # Ridouane Oudra, Feb 19 2025
  • Mathematica
    nxt[{n_, a_}] := {n + 1, a + Prime[n + 1] (2 n + 1)}; NestList[nxt,{1,2},50][[All,2]] (* Harvey P. Dale, Jul 05 2018 *)
  • PARI
    a(n) = sum(i=1, n, prime(i)*(2*i-1)); \\ Michel Marcus, Jan 22 2022

Formula

From Ridouane Oudra, Feb 19 2025: (Start)
a(n) = Sum_{i=1..n} Sum_{j=1..n} max(prime(i), prime(j)).
a(n) = 2*A014285(n) - A007504(n).
a(n) = 2*A167214(n) - A023662(n).
a(n) = A167214(n) + A062020(n). (End)

A062021 a(n) = Sum_{i=1..n} Sum_{j=1..i} (prime(i)^2 - prime(j)^2).

Original entry on oeis.org

0, 5, 42, 151, 548, 1185, 2542, 4403, 7608, 13621, 20834, 32535, 47980, 65609, 88278, 119947, 162368, 208869, 269194, 340007, 416580, 512305, 622286, 756003, 925432, 1114661, 1314498, 1537015, 1771628, 2031993, 2393158, 2786315
Offset: 1

Views

Author

Amarnath Murthy, Jun 02 2001

Keywords

Examples

			a(3) = (5^2 - 2^2) + (5^2 - 3^2) + (3^2 - 2^2) = 42.

Links

Crossrefs

Cf. A000040, A062020, A062022.

Programs

  • Magma
    [(&+[(&+[NthPrime(i)^2 - NthPrime(j)^2: j in [1..i]]): i in [1..n]]): n in [1..40]]; // G. C. Greubel, May 04 2022
  • Maple
    N:= 100: # for a(1)..a(N)
P2:= [seq(ithprime(i)^2,i=1..N)]:
DP2:= P2[2..-1]-P2[1..-2]:
A[1]:= 0: A[2]:= 5:
for n from 3 to N do A[n]:= 2*A[n-1]+(n-1)*DP2[n-1]-A[n-2] od:
seq(A[i],i=1..N); # Robert Israel, Feb 02 2020
  • Mathematica
    RecurrenceTable[{a[1]==0,a[2]==5,a[n]==2a[n-1]-a[n-2]+(n-1)(Prime[n]^2 - Prime[n-1]^2)}, a, {n,40}] (* Harvey P. Dale, May 16 2019 *)
  • SageMath
    @CachedFunction
def a(n):
    if (n<3): return 5*(n-1)
    else: return 2*a(n-1) - a(n-2) + (n-1)*(nth_prime(n)^2 - nth_prime(n-1)^2)
[a(n) for n in (1..40)] # G. C. Greubel, May 04 2022

Formula

a(n) = 2*a(n-1) - a(n-2) + (n-1)*(prime(n)^2 - prime(n-1)^2) with a(1) = 0, a(2) = 5.

Extensions

More terms and formula from Larry Reeves (larryr(AT)acm.org), Jun 06 2001
Name edited by G. C. Greubel, May 04 2022

A062022 a(n) = Sum_{k=1..n} Sum_{j=1..k} (prime(k) - prime(j))^2.

Original entry on oeis.org

0, 1, 14, 59, 256, 581, 1298, 2287, 4004, 7329, 11338, 17915, 26660, 36637, 49406, 67239, 91252, 117585, 151730, 191819, 235112, 289013, 350842, 425919, 521300, 628001, 740666, 865899, 997744, 1143501, 1345454, 1565639, 1815068, 2074761
Offset: 1

Views

Author

Amarnath Murthy, Jun 02 2001

Keywords

Examples

			a(3) = (5-2)^2 + (5-3)^2 + (3-2)^2 = 14, sum of the squared differences of all pairs of the first 3 primes.

Links

Crossrefs

Cf. A000040, A007504, A024450, A062020, A062021.

Programs

  • Maple
    A062022 := proc(n)
    local a,i,j ;
    a := 0 ;
    for j from 1 to n do
        for i from 1 to j-1 do
            a := a+(ithprime(j)-ithprime(i))^2 ;
        end do:
    end do:
    a ;
end proc:
seq(A062022(n), n=1..10); # R. J. Mathar, Oct 03 2014
  • Mathematica
    a[n_]:= a[n]= n*Sum[Prime[k]^2, {k,n}] - (Sum[Prime[j], {j,n}])^2;
Table[a[n], {n, 50}] (* G. C. Greubel, May 04 2022 *)
  • SageMath
    @CachedFunction
def a(n): return n*sum(nth_prime(j)^2 for j in (1..n)) - (sum(nth_prime(j) for j in (1..n)))^2
[a(n) for n in (1..50)] # G. C. Greubel, May 04 2022

Formula

From G. C. Greubel, May 04 2022: (Start)
a(n) = a(n-1) + n*prime(n)^2 + Sum_{k=1..n} prime(k)*(prime(k) - 2*prime(n)), with a(0) = a(1) = 0.
a(n) = n*Sum_{j=1..n} prime(j)^2 - (Sum_{j=1..n} prime(j))^2 = n*A024450(n) - A007504(n)^2. (End)

Extensions

More terms from Matthew Conroy, Jun 11 2001

A023662 Convolution of odd numbers and primes.

Original entry on oeis.org

2, 9, 24, 51, 96, 165, 264, 399, 576, 805, 1094, 1451, 1886, 2405, 3014, 3723, 4544, 5485, 6554, 7761, 9112, 10615, 12280, 14117, 16140, 18361, 20786, 23421, 26272, 29345, 32658, 36229, 40068, 44183, 48586, 53289, 58300, 63631, 69292
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A000040, A005408, A061802 (first differences).
Cf. A167214, A062020, A316322, A014148, A007504.

Programs

  • Maple
    A023662 := proc(n)
    add( ithprime(n-i)*(2*i+1),i=0..n-1) ;
end proc: # R. J. Mathar, Nov 29 2015
  • Mathematica
    Table[Sum[Prime[n - k + 1] (2 k - 1), {k, n}], {n, 39}] (* Michael De Vlieger, Nov 29 2015 *)
  • PARI
    a(n) = sum(i=1, n, prime(n-i+1)*(2*i-1)); \\ Michel Marcus, Sep 30 2013

Formula

a(n) = Sum_{i=0..n-1} A000040(n-i)*A005408(i). - R. J. Mathar, Nov 29 2015
a(n) = Sum_{i=0..n-1} A061802(i). - Marco Zárate, Jun 09 2024
From Ridouane Oudra, Feb 19 2025: (Start)
a(n) = Sum_{i=1..n} Sum_{j=1..n} min(prime(i), prime(j)).
a(n) = A167214(n) - A062020(n).
a(n) = 2*A167214(n) - A316322(n).
a(n) = A014148(n) + A014148(n-1).
a(n) = A007504(n) + 2*A014148(n-1). (End)

A332094 Numerator of the average distance among first n primes.

Original entry on oeis.org

1, 2, 17, 22, 27, 142, 31, 9, 97, 666, 83, 604, 1529, 1906, 791, 367, 3533, 4238, 5019, 584, 617, 7822, 8995, 518, 473, 13342, 1663, 8324, 3689, 20662, 23003, 532, 1655, 31074, 541, 6218, 2145, 44354, 48187, 2613, 18805, 20330, 65651, 356, 15083, 80894, 28979, 23293
Offset: 2

Views

Author

Andres Cicuttin, Nov 20 2020

Keywords

Crossrefs

Cf. A336814 (denominator), A338869, A339022, A062020.

Programs

  • Mathematica
    nmax=64;
Table[Total[Flatten[Table[Table[Prime[k] - Prime[j], {j, 1, k - 1}], {k, 2, n}]]]/(n*(n - 1)/2), {n, 2, nmax}]//Numerator
  • PARI
    lista(nn) = {my(vp = primes(nn)); vector(nn-1, k, k++; numerator((2/(k*(k-1))*sum(j=2, k, sum(i=1, j-1, vp[j] - vp[i])))));} \\ Michel Marcus, Nov 21 2020

Formula

a(n) = numerator((2/(n*(n-1)))*Sum_{j=2..n} Sum_{i=1..j-1} (prime(j) - prime(i))).

A336814 Denominator of the average distance among first n primes.

Original entry on oeis.org

1, 1, 6, 5, 5, 21, 4, 1, 9, 55, 6, 39, 91, 105, 40, 17, 153, 171, 190, 21, 21, 253, 276, 15, 13, 351, 42, 203, 87, 465, 496, 11, 33, 595, 10, 111, 37, 741, 780, 41, 287, 301, 946, 5, 207, 1081, 376, 294, 35, 425, 1326, 689, 477, 33, 1540, 133, 551, 1711, 1770, 915, 1891, 1953, 224
Offset: 2

Views

Author

Andres Cicuttin, Nov 21 2020

Keywords

Crossrefs

Cf. A332094 (numerator), A338869, A339022, A062020.

Programs

  • Mathematica
    nmax=64;
Table[Total[Flatten[Table[Table[Prime[k] - Prime[j], {j, 1, k - 1}], {k, 2, n}]]]/(n*(n - 1)/2), {n, 2, nmax}]//Denominator
(* Also *)
denavepdist[n_]:=Module[{pset,p2s,diffp2s},
pset=Prime[Range[n]];
p2s=Subsets[pset,{2}];
diffp2s=Map[Differences,p2s]//Flatten//Tally;
Sum[diffp2s[[j]][[1]]*diffp2s[[j]][[2]],{j,1,Length[diffp2s]}]/Length[p2s]//Denominator//Return];
Table[denavepdist[n],{n,2,2^6}]

Formula

a(n) = denominator((2/(n*(n-1)))*Sum_{j=2..n} Sum_{i=1..j-1} (prime(j) - prime(i))).
Showing 1-7 of 7 results.

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

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