A074271 -id:A074271 - cp's OEIS frontend

A093502 a(1) = 2; for n > 0, a(n+1) is the a(n)-th prime after a(n).

2, 5, 19, 103, 733, 6691, 76831, 1081429, 18242699, 361919671, 8309068723, 217809953467, 6445388418589, 213232943658197, 7821073506524401, 315743571062703689

Offset: 1

Amarnath Murthy, Apr 17 2004

With prepended a(0) = 1 this is the lexicographically earliest infinite sequence such that A056239(a(n)) = a(n-1) + A056239(a(n-1)). - Antti Karttunen, Nov 02 2024

			19 follows 5 as there are 5 primes > 5 and up to 19 inclusive, (7,11,13,17,19).

Cf. A000040, A000720, A056239, A074271.

a[1] := 2; a[n_] := Prime[PrimePi[a[n - 1]] + a[n - 1]]; Table[a[n], {n, 1, 10}] (* Stefan Steinerberger, Apr 10 2006 *)
NestList[Prime[PrimePi[ # ] + # ] &, 2, 13] (* Zak Seidov, Mar 21 2009 *)

from sympy import prime
p, q = 2, 1
A093502_list = [p]
for _ in range(15):
    r = p + q
    p, q = prime(r), r
    A093502_list.append(p) # Chai Wah Wu, Jun 17 2019

a(n) = prime(pi(a(n-1)) + a(n-1)). - Vladeta Jovovic, Jun 19 2004
a(1)=2, a(n) = next a(n-1)th prime after a(n-1). - Zak Seidov, Mar 21 2009
a(n+1) = A000040(a(n) + index of a(n) in A000040). - David James Sycamore, Aug 20 2017
a(n) = A000040(A074271(n)), A056239(a(n)) = A074271(n). - Antti Karttunen, Nov 02 2024

a(10) from Vladeta Jovovic, Jun 19 2004
More terms from Stefan Steinerberger, Apr 10 2006
a(13) from Zak Seidov, Mar 21 2009
a(14)-a(15) from Donovan Johnson, Dec 08 2009
Better definition from Jon E. Schoenfield, Aug 22 2017
a(16) from Chai Wah Wu, Jun 17 2019

A181642 Minimal sequence whose forwards van Eck transform is the sequence of prime numbers.

Original entry on oeis.org

0, 1, 0, 2, 1, 3, 4, 0, 5, 6, 2, 7, 8, 9, 10, 1, 11, 12, 3, 13, 14, 15, 16, 4, 17, 18, 0, 19, 20, 21, 22, 5, 23, 24, 25, 26, 27, 28, 6, 29, 30, 2, 31, 32, 33, 34, 35, 36, 7, 37, 38, 39, 40, 8, 41, 42, 9, 43, 44, 45, 46, 10, 47, 48, 49, 50, 51, 52, 1, 53, 54

Offset: 1

Paolo P. Lava & Giorgio Balzarotti, Nov 03 2010

At each step, the minimum available integer is used.
From Rémy Sigrist, Aug 12 2017: (Start)
a(n)=0 iff n belongs to A074271.
a(n)=1 iff n > 1 and n belongs to A259408.
For any k > 0, A064427(k) = least n such that a(n) = k-1.
(End)

			a(1)=0. Next 0 is at distance 2 (1st prime): a(3)=0.
a(2)=1. Next 1 is at distance 3 (2nd prime): a(5)=1.
a(3)=0. Next 0 is at distance 5 (3rd prime): a(8)=0.
For a(4), we can use neither 0 (distance 1 from previous 0 would lead to an incongruence) nor 1 (distance 1 from subsequent 1 would lead to another incongruence). Therefore we must use 2.
Next 2 must be at distance 7 (4th prime): a(11)=2. And so on.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A000040, A064427, A074271, A181391, A259408.

P:=proc(q,h) local i,k,n,t,x; x:=array(1..h); for k from 1 to h do x[k]:=-1; od; x[1]:=0; i:=0; t:=0;for n from 1 to q do if isprime(n) then  i:=i+1; if x[i]>-1 then x[i+n]:=x[i]; else t:=t+1; x[i]:=t; x[i+n]:=x[i]; fi; fi; od; seq(x[k],k=1..79); end: P(400,500);

a = vector(71, i, -1); u = 0; for (n=1, #a, if (a[n]<0, o = n; while (o <= #a, a[o] = u; o += prime(o)); u++); print1 (a[n] ", ")) \\ Rémy Sigrist, Aug 12 2017

More terms from Rémy Sigrist, Aug 12 2017

A259408 a(1) = 1 thereafter a(n) = Sum_{m=1..n-1} prime(a(m)).

Original entry on oeis.org

1, 2, 5, 16, 69, 416, 3277, 33590, 430131, 6700328, 124069971, 2680915918, 66579269891, 1876496610172, 59387269231505, 2091422223924852, 81321166136299741, 3467614972592015460

Offset: 1

Anders Hellström, Jun 26 2015

Cf. A074271.

a = {1}; Do[AppendTo[a, Sum[Prime[a[[m]]], {m, n - 1}]], {n, 2, 15}];
a (* Michael De Vlieger, Aug 06 2015 *)

a(n) = if (n==1, 1, sum(k=1, n-1, prime(a(k)))); \\ Michel Marcus, Jun 26 2015

first(m)=my(v=vector(m)); v[1]=1; print1(1); for(i=2, m, v[i]=sum(k=1, i-1, prime(v[k])); print1(", ", v[i])); v; \\ Anders Hellström, Aug 01 2015

first(n)=my(v=vector(n,i,i)); for(i=3,n,v[i]=v[i-1]+prime(v[i-1])); v \\ Charles R Greathouse IV, Aug 06 2015

use bignum;
use Math::Prime::Util ':all';
print "1\n2\n";
my $a = 2;
while(1){
  $a += nth_prime($a);
  print "$a\n";
} # Charles R Greathouse IV, Aug 06 2015

from sympy import prime
from functools import lru_cache
@lru_cache()
def a(n): return n if n < 3 else a(n-1) + prime(a(n-1))
print([a(n) for n in range(1, 14)]) # Michael S. Branicky, Oct 07 2022

a(n) = A014688(a(n-1)) for n>2, a(1)=1, a(2)=2.

a(15) from Michael De Vlieger, Jul 01 2015

a(16)-a(18) from Charles R Greathouse IV, Aug 06 2015

cp's OEIS Frontend

A093502 a(1) = 2; for n > 0, a(n+1) is the a(n)-th prime after a(n).

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A181642 Minimal sequence whose forwards van Eck transform is the sequence of prime numbers.

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A259408 a(1) = 1 thereafter a(n) = Sum_{m=1..n-1} prime(a(m)).

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