A060197 -id:A060197 - cp's OEIS frontend

A317964 Prime numbers in the lexicographically earliest sequence of positive integers whose prime indices are not already in the sequence (A304360).

2, 5, 13, 17, 23, 31, 37, 43, 47, 61, 67, 73, 79, 89, 103, 107, 109, 113, 137, 149, 151, 163, 167, 179, 181, 193, 197, 223, 227, 233, 241, 251, 257, 263, 269, 271, 277, 281, 307, 317, 347, 349, 353, 359, 379, 383, 389, 397, 419, 421, 431, 433, 449, 457, 463, 467, 487, 499, 503, 509, 521, 523, 547

Offset: 1

N. J. A. Sloane, Aug 26 2018

nonn

Also primes whose prime index is not in A304360, or is in A324696. A prime index of n is a number m such that prime(m) divides n. - Gus Wiseman, Mar 19 2019

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000720, A001462, A007097, A060197, A079254, A112798, A276625, A277098, A290822, A304360, A306844.

Cf. A324695, A324698, A324704, A324741, A324756, A324758, A324765, A324766, A324768.

count:= 0:
P:= {}: A:= NULL:
for n from 2 while count < 100 do
  pn:= numtheory:-factorset(n);
  if pn intersect P = {} then
    P:= P union {ithprime(n)};
    if isprime(n) then A:= A, n; count:= count+1 fi;
  fi
od:
A; # Robert Israel, Aug 26 2018

aQ[n_]:=n==1||Or@@Cases[FactorInteger[n],{p_,_}:>!aQ[PrimePi[p]]];
Prime[Select[Range[100],aQ]] (* Gus Wiseman, Mar 19 2019 *)

A071578 Number of iterations of Pi(n) needed to reach 1, where Pi(x) denotes the number of primes <= x.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Benoit Cloitre, May 31 2002

Cf. A000720.

Table[Length[NestWhileList[PrimePi[#]&,n,#!=1&]]-1,{n,110}] (* Harvey P. Dale, Dec 24 2018 *)

for(n=2,150,s=n; t=0; while(s!=1,t++; s=sum(i=2,s,isprime(i)); if(s==1,print1(t,","); ); ))

a(n) = a(Pi(n))+1.

a(n) = A060197(n) - 2. - Filip Zaludek, Dec 10 2016

A321132 a(n) is the number of iterations of the mapping of x -> pi(x) until n reaches the main line as defined by A007097.

Original entry on oeis.org

0, 0, 0, 2, 0, 2, 3, 3, 3, 3, 0, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 0, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Robert G. Wilson v, Oct 27 2018

nonn

All primes are either on the main line or will join it before reaching 0, as in A060197 or 1, as in A071578.
First occurrence of k, k=0,1,2,...: 1, 4, 7, 17, 59, 277, 1787, 15299, 167449, 2269733, etc.
A measure of Primeness - see the Fernandez link.

			a(10) is 3 because the tenth prime is 29 -> 10 -> 4 -> 2 and 2 is A007097(1).

Neil Fernandez, An order of primeness, F(p)

Cf. A000720, A007097, A049076, A060197, A071578, A114537.

f[n_] := Length@ NestWhileList[PrimePi, n, ! MemberQ[{1, 2, 3, 5, 11, 31, 127, 709, 5381, 52711, 648391, 9737333, 174440041}, #] &] - 1; Array[f, 105]

a(n) = 0 iff n is a member of A007097.

cp's OEIS Frontend

A317964 Prime numbers in the lexicographically earliest sequence of positive integers whose prime indices are not already in the sequence (A304360).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A071578 Number of iterations of Pi(n) needed to reach 1, where Pi(x) denotes the number of primes <= x.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A321132 a(n) is the number of iterations of the mapping of x -> pi(x) until n reaches the main line as defined by A007097.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula