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 1-6 of 6 results.

A082094 A 2nd-order recursion: a(1)=a(2)=1; a(n) = prime(a(n-1)) + primepi(a(n-2)) = A000040(a(n-1)) + A000720(a(n-2)).

Original entry on oeis.org

1, 1, 2, 3, 6, 15, 50, 235, 1498, 12592, 135431, 1806803, 29135476, 555971158, 12336554787, 313733168860, 9034347750986, 291579097035392, 10455240487002922, 413371595329570610
Offset: 1

Views

Author

Labos Elemer, Apr 11 2003

Keywords

Crossrefs

Cf. A069103, A082095.

Programs

  • Magma
    a:= func< n | n lt 4 select Fibonacci(n) else NthPrime(Self(n-1)) + #PrimesUpTo(Self(n-2)) >;
[a(n): n in [1..14]]; // G. C. Greubel, Aug 31 2019
  • Mathematica
    a[n_]:= a[n]= If[n<4, Fibonacci[n], Prime[a[n-1]] + PrimePi[a[n-2]]]; Table[a[n], {n,1,17}] (* modified by G. C. Greubel, Aug 31 2019 *)

Extensions

a(17) from G. C. Greubel, Aug 31 2019
a(18)-a(20) from Chai Wah Wu, Sep 18 2019

A082095 A 2nd order recursion: a(1)=a(2)=1, a(n) = prime(a(n-2)) + pi(a(n-1)) = A000040(a(n-2)) + A000720(a(n-1)).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 15, 25, 56, 113, 293, 679, 2036, 5389, 18447, 54920, 211347, 697252, 2974827, 10741681, 50245401, 196570892, 998427899, 4197026430, 22963115248, 103007695615, 603032992418, 2870053925682, 17876478098333, 89829672327175, 592418610490868, 3129958832408526, 21764504060699104, 120464619408398977, 880014298908322768, 5086633622697900677
Offset: 1

Views

Author

Labos Elemer, Apr 11 2003

Keywords

Crossrefs

Cf. A000040, A000720, A069103, A082094.

Programs

  • Magma
    f:= func< n | n lt 4 select Fibonacci(n) else NthPrime(Self(n-2)) + #PrimesUpTo(Self(n-1)) >;
[f(n): n in [1..25]]; // G. C. Greubel, Aug 30 2019
  • Mathematica
    a[n_]:= a[n]= If[n<4, Fibonacci[n], Prime[a[n-2]] +PrimePi[a[n-1]]]; Table[a[n], {n, 30}] (* modified by G. C. Greubel, Aug 30 2019 *)
nxt[{a_,b_}]:={b,Prime[a]+PrimePi[b]}; NestList[nxt,{1,1},30][[All,1]] (* The program generates the first 31 terms of the sequence. *) (* Harvey P. Dale, May 16 2020 *)
  • PARI
    a(n) = if (n<3, 1, prime(a(n-2)) + primepi(a(n-1))); \\ Michel Marcus, Aug 30 2019
  • PARI
    first(n) = {my(res = vector(max(3, n)), pr = vector(n)); res[1] = res[2] = 1; res[3] = 2; for(i = 1, 3, print1(res[i]", ")); pr[1] = pr[2] = 2; pr[3] = 3; for(i = 4, n, pr[i] = prime(res[i-2]); res[i] = pr[i] + res[i-3] + primedist(pr[i-1], res[i-1]); print1(res[i]", ")); res}
primedist(p1, p2) = {my(res = 0); forprime(p = p1 + 1, p2, res++); res} \\ David A. Corneth, Aug 30 2019

Extensions

a(26) from David A. Corneth, Aug 30 2019
a(27)-a(30) from G. C. Greubel, Aug 30 2019
a(31)-a(36) from Chai Wah Wu, Sep 18 2019

A082096 A 2nd order recursion: a(1)=a(2)=1; a(n) = prime(a(n-2)+a(n-1)) = A000040(a(n-2)+a(n-1)).

Original entry on oeis.org

1, 1, 3, 7, 29, 151, 1069, 9887, 115891, 1666421, 28700933, 580669933, 13578126713, 362075579539, 10886955278951, 365589325548857, 13598064388599629, 556220494250764093
Offset: 1

Views

Author

Labos Elemer, Apr 11 2003

Keywords

Crossrefs

Cf. A069103, A082094, A082095.

Programs

  • Magma
    a:= func< n | n lt 3 select 1 else NthPrime(Self(n-1) + Self(n-2)) >;
[a(n): n in [1..12]]; // G. C. Greubel, Aug 31 2019
  • Mathematica
    a[n_]:= a[n]= If[n<3, 1, Prime[a[n-1]+a[n-2]]]; Table[a[n], {n,13}] (* modified by G. C. Greubel, Aug 31 2019 *)
nxt[{a_,b_}]:={b,Prime[a+b]}; Transpose[NestList[nxt,{1,1},13]][[1]] (* Harvey P. Dale, Oct 02 2013 *)

Extensions

a(15) from G. C. Greubel, Aug 31 2019
a(16)-a(18) from Chai Wah Wu, Sep 18 2019

A082097 a(n) = d(a(n-1)) + n = A000005(a(n-1)) + n, with a(1)=1.

Original entry on oeis.org

1, 1, 1, 5, 7, 8, 11, 10, 13, 12, 17, 14, 17, 16, 20, 22, 21, 22, 23, 22, 25, 25, 26, 28, 31, 28, 33, 32, 35, 34, 35, 36, 42, 42, 43, 38, 41, 40, 47, 42, 49, 45, 49, 47, 47, 48, 57, 52, 55, 54, 59, 54, 61, 56, 63, 62, 61, 60, 71, 62, 65, 66, 71, 66, 73, 68, 73, 70, 77, 74
Offset: 1

Views

Author

Labos Elemer, Apr 11 2003

Keywords

Links

Crossrefs

Cf. A069103, A082094, A082095, A082096.

Programs

  • GAP
    a:= function(n)
    if n<4 then return 1;
    else return Tau(a(n-1)) + n;
    fi;
  end;
List([1..70], n-> a(n) ); # G. C. Greubel, Aug 31 2019
  • Magma
    a:= func< n | n lt 4 select 1 else n + NumberOfDivisors(Self(n-1)) >;
[a(n): n in [1..70]]; // G. C. Greubel, Aug 31 2019
  • Maple
    with(numtheory);
a:= proc(n) option remember;
   if n < 4 then 1
   else tau(a(n-1)) + n
   fi
 end:
seq(a(n), n=1..70); # G. C. Greubel, Aug 31 2019
  • Mathematica
    a[n_]:= If[n<4, 1, DivisorSigma[0, a[n-1]] + n]; Table[a[n], {n, 70}] (* modified by G. C. Greubel, Aug 31 2019 *)
nxt[{n_,a_}]:={n+1,DivisorSigma[0,a]+n+1}; Join[{1,1},NestList[nxt,{3,1},70][[;;,2]]] (* Harvey P. Dale, Mar 28 2024 *)
  • PARI
    a(n) = if(n<4,1, numdiv(a(n-1)) + n); \\ G. C. Greubel, Aug 31 2019
  • Sage
    def a(n):
    if n<4: return 1
    else: return sigma(a(n-1), 0) + n
[a(n) for n in (1..70)] # G. C. Greubel, Aug 31 2019

A122594 a(0) = 2; a(1) = 5; a(n) = prime(a(n-1)) + prime(a(n-2)) if n > 1.

Original entry on oeis.org

2, 5, 14, 54, 294, 2182, 21168, 258584, 3866528, 69094248, 1446704598, 34928585060, 959616504858, 29669964598708, 1022713744909606, 38985339660148638, 1631985092027500964
Offset: 0

Views

Author

Carlos Alves, Oct 29 2006

Keywords

Examples

			a(2) = prime(2) + prime(5) = 3 + 11 = 14.

Crossrefs

Cf. A069103, A107327.

Programs

  • Maple
    A122594 := proc(n) option remember: if(n=0)then return 2: elif(n=1)then return 5: else return ithprime(procname(n-1)) + ithprime(procname(n-2)): fi: end:
seq(A122594(n),n=0..8); # Nathaniel Johnston, Apr 26 2011
  • Mathematica
    Transpose[NestList[{#[[2]],Prime[ #[[1]]]+Prime[ #[[2]]]}&,{1, 2},13]][[2]]

Extensions

a(14)-a(16) from Jinyuan Wang, Nov 21 2020

A285742 a(0) = 0, a(1) = 1; a(2*n) = prime(a(n)), a(2*n+1) = prime(a(n)) + prime(a(n+1)).

Original entry on oeis.org

0, 1, 2, 5, 3, 14, 11, 16, 5, 48, 43, 74, 31, 84, 53, 64, 11, 234, 223, 414, 191, 564, 373, 500, 127, 560, 433, 674, 241, 552, 311, 342, 31, 1512, 1481, 2890, 1409, 4260, 2851, 4004, 1153, 5246, 4093, 6642, 2549, 6120, 3571, 4280, 709, 4766, 4057, 7076, 3019, 8042, 5023, 6546, 1523, 5526, 4003, 6066, 2063
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 25 2017

Keywords

Comments

A variation on Stern's diatomic sequence (A002487) and primeth recurrence (A007097).

Examples

			a(0) = 0;
a(1) = 1;
a(2) = a(2*1) = prime(a(1)) = prime(1) = 2;
a(3) = a(2*1+1) = prime(a(1)) + prime(a(2)) = prime(1) + prime(2) = 2 + 3 = 5;
a(4) = a(2*2) = prime(a(2)) = prime(2) = 3;
a(5) = a(2*2+1) = prime(a(2)) + prime(a(3)) = prime(2) + prime(5) = 3 + 11 = 14, etc.

Links

Crossrefs

Cf. A002487, A007097, A069103, A285743.

Programs

  • Mathematica
    a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], Prime[a[n/2]], Prime[a[(n - 1)/2]] + Prime[a[(n + 1)/2]]]; Table[a[n], {n, 0, 60}]
Showing 1-6 of 6 results.

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