A333243 -id:A333243 - cp's OEIS frontend

A333244 Prime numbers with prime indices in A333243.

11, 127, 277, 1063, 2221, 3001, 4397, 5381, 7193, 9319, 10631, 12763, 15299, 15823, 21179, 22093, 24859, 30133, 33967, 37217, 38833, 40819, 43651, 55351, 57943, 60647, 66851, 68639, 77431, 80071, 84347, 87803, 90023, 98519, 101701, 103069, 125113, 127643

Offset: 1

Michael P. May, Mar 12 2020

nonn

This sequence can also be generated by the N-sieve.

			a(1) = prime(A333243(1)) = prime(5) = 11.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael P. May, On the Properties of Special Prime Number Subsequences, arXiv:1608.08082 [math.GM], 2016-2020.
Michael P. May, Properties of Higher-Order Prime Number Sequences, Missouri J. Math. Sci. (2020) Vol. 32, No. 2, 158-170; and arXiv version, arXiv:2108.04662 [math.NT], 2021.

Cf. A078442, A262275, A333242, A333243.

b:= proc(n) option remember;
      `if`(isprime(n), 1+b(numtheory[pi](n)), 0)
    end:
a:= proc(n) option remember; local p;
      p:= `if`(n=1, 1, a(n-1));
      do p:= nextprime(p);
        if (h-> h>2 and h::even)(b(p)) then break fi
      od; p
    end:
seq(a(n), n=1..42);  # Alois P. Heinz, Mar 15 2020

b[n_] := b[n] = If[PrimeQ[n], 1+b[PrimePi[n]], 0];
a[n_] := a[n] = Module[{p}, p = If[n==1, 1, a[n-1]]; While[True, p = NextPrime[p]; If[#>2 && EvenQ[#]&[b[p]], Break[]]]; p];
Array[a, 42] (* Jean-François Alcover, Nov 16 2020, after Alois P. Heinz *)

b(n)={my(k=0); while(isprime(n), k++; n=primepi(n)); k};
apply(x->prime(prime(prime(x))), select(n->b(n)%2, [1..500])) \\ Michel Marcus, Nov 18 2022

a(n) = prime(A333243(n)).

A333242 Prime numbers with an odd number of steps in their prime index chain.

Original entry on oeis.org

2, 5, 7, 13, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 71, 73, 79, 89, 97, 101, 103, 107, 113, 131, 137, 139, 149, 151, 163, 167, 173, 179, 181, 193, 197, 199, 223, 227, 229, 233, 239, 251, 257, 263, 269, 271, 281, 293, 307, 311, 313, 317, 331, 337, 347, 349, 359, 373

Offset: 1

Michael P. May, Mar 12 2020

nonn

This sequence can also be generated by the N-sieve.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Michael P. May, Properties of Higher-Order Prime Number Sequences, Missouri J. Math. Sci. (2020) Vol. 32, No. 2, 158-170; and arXiv version, arXiv:2108.04662 [math.NT], 2021.
Michael P. May, Approximating the Prime Counting Function via an Operation on a Unique Prime Number Subsequence, arXiv:2112.08941 [math.GM], 2021.
Michael P. May, Relationship Between the Prime-Counting Function and a Unique Prime Number Sequence, Missouri J. Math. Sci. (2023), Vol. 35, No. 1, 105-116.
Michael P. May, Application of the Inclusion-Exclusion Principle to Prime Number Subsequences, arXiv:2402.13214 [math.GM], 2024.

Cf. A000040, A000720, A078442, A262275 (complement in primes), A333243, A333244.

b:= proc(n) option remember;
       `if`(isprime(n), 1+b(numtheory[pi](n)), 0)
    end:
a:= proc(n) option remember; local p; p:= a(n-1);
      do p:= nextprime(p);
         if b(p)::odd then break fi
      od; p
    end: a(1):=2:
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 15 2020

Select[Prime@ Range@ 75, EvenQ@ Length@ NestWhileList[ PrimePi, #, PrimeQ] &] (* Giovanni Resta, Mar 15 2020 *)

\\ here b(n) is A078442.
b(n)={my(k=0); while(isprime(n), k++; n=primepi(n)); k}
select(n->b(n)%2, [1..500]) \\ Andrew Howroyd, Mar 15 2020

{ p in primes : A078442(p) mod 2 = 1 }.

a(n) = A000720(A262275(n)). - Andrew Howroyd, Mar 15 2020

Terms a(21) and beyond from Andrew Howroyd, Mar 15 2020

A338460 Decimal expansion of the largest real root of e^(x-1) = Gamma(x+1).

Original entry on oeis.org

3, 6, 1, 4, 7, 9, 3, 7, 0, 3, 1, 9, 2, 5, 2, 5, 4, 4, 7, 3, 8, 6, 5, 3, 6, 6, 2, 5, 6, 0, 3, 4, 5, 4, 6, 3, 3, 5, 3, 1, 5, 1, 6, 5, 9, 6, 9, 4, 7, 5, 0, 2, 2, 6, 6, 1, 1, 1, 5, 9, 9, 9, 7, 7, 4, 6, 2, 5, 1, 8, 2, 9, 8, 6, 1, 3, 6, 1, 8, 5, 7, 5, 4, 3, 2, 8, 1, 8, 6, 2, 8, 2, 1, 5, 7, 1, 1, 5, 9, 6, 3, 3, 0, 8, 1

Offset: 1

Michael P. May, Jan 31 2021

Decimal expansion of the constant value for which the average and the minimum prime gaps are equal for the prime number sequences of higher order P', P'', P''', and P'''' as represented by A333242, A262275, A333243 and A333244.

x-1 is the smallest average gap size for the set of all prime numbers P. - Michael P. May, Jan 26 2025

			3.61479370319252544738653662560345463353151659694750...

Cf. A333242, A262275, A333243, A333244.

Cf. A001113, A078335, A336763.

Digits:= 155:
fsolve(exp(x-1)=GAMMA(x+1), x=3..4);  # Alois P. Heinz, Feb 01 2021

RealDigits[x /. FindRoot[LogGamma[x + 1] - x + 1, {x, 3}, WorkingPrecision -> 110], 10, 105][[1]] (* Amiram Eldar, Feb 01 2021 *)

solve(x=3,4,lngamma(x+1)-x+1) \\ Hugo Pfoertner, Feb 01 2021

x| (log(x!))^n * (log(x!) + 1) = x * (x-1)^n, for n >= 0

A358179 Prime numbers with prime indices in A333244.

Original entry on oeis.org

31, 709, 1787, 8527, 19577, 27457, 42043, 52711, 72727, 96797, 112129, 137077, 167449, 173867, 239489, 250751, 285191, 352007, 401519, 443419, 464939, 490643, 527623, 683873, 718807, 755387, 839483, 864013, 985151, 1021271, 1080923, 1128889, 1159901, 1278779, 1323503, 1342907, 1656649, 1693031

Offset: 1

Michael P. May, Nov 11 2022

nonn

This sequence can also be generated by the N-sieve.

			a(1) = prime(A333244(1)) = prime(11) = 31.

Michael P. May, Properties of Higher-Order Prime Number Sequences, Missouri J. Math. Sci. (2020) Vol. 32, No. 2, 158-170; and arXiv version, arXiv:2108.04662 [math.NT], 2021.

Cf. A078442, A262275, A333242, A333243, A333244.

b[n_] := b[n] = If[PrimeQ[n], 1+b[PrimePi[n]], 0];
a[n_] := a[n] = Module[{p}, p = If[n==1, 1, a[n-1]]; While[True, p = NextPrime[p]; If[#>3 && OddQ[#]&[b[p]], Break[]]]; p];
Array[a, 50]

b(n)={my(k=0); while(isprime(n), k++; n=primepi(n)); k};
apply(x->prime(prime(prime(prime(x)))), select(n->b(n)%2, [1..500])) \\ Michel Marcus, Nov 18 2022

a(n) = prime(A333244(n)).
a(n) = A049090(A333242(n)).
a(n) = A038580(A262275(n)).
a(n) = A006450(A333243(n)).

cp's OEIS Frontend

A333244 Prime numbers with prime indices in A333243.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A333242 Prime numbers with an odd number of steps in their prime index chain.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A338460 Decimal expansion of the largest real root of e^(x-1) = Gamma(x+1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A358179 Prime numbers with prime indices in A333244.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula