A038580 Primes with indices that are primes with prime indices.

5, 11, 31, 59, 127, 179, 277, 331, 431, 599, 709, 919, 1063, 1153, 1297, 1523, 1787, 1847, 2221, 2381, 2477, 2749, 3001, 3259, 3637, 3943, 4091, 4273, 4397, 4549, 5381, 5623, 5869, 6113, 6661, 6823, 7193, 7607, 7841, 8221, 8527, 8719, 9319, 9461, 9739

Offset: 1

Author

Brian Galebach

Keywords

• easy
• nice
• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• Robert G. Batchko, A prime fractal and global quasi-self-similar structure in the distribution of prime-indexed primes, arXiv preprint arXiv:1405.2900 [math.GM], 2014.
• Robert E. Dressler and S. Thomas Parker, Primes with a prime subscript, J. ACM, Volume 22 Issue 3, July 1975, 380-381.
• Neil Fernandez, An order of primeness, F(p).
• Neil Fernandez, An order of primeness. [cached copy, included with permission of the author]
• Neil Fernandez, More terms of this and other sequences related to A049076.
• 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.

Crossrefs

Primes p for which A049076(p) > 3.

Cf. A049090, A049202, A049203, A057847, A057849, A057850, A057851, A058332, A086749, A093047.

Second differences give A245175.

Let A = primes A000040, B = nonprimes A018252. The 2-level compounds are AA = A006450, AB = A007821, BA = A078782, BB = A102615. The 3-level compounds AAA, AAB, ..., BBB are A038580, A049078, A270792, A102617, A270794, A270795, A270796, A102616.

Programs

Magma

[NthPrime(NthPrime(NthPrime(n))): n in [1..50]]; // Vincenzo Librandi, Jul 17 2016

Maple

a:= ithprime@@3;
seq(a(n), n=1..50);  # Alois P. Heinz, Jun 14 2015
# For Maple code for the prime/nonprime compound sequences (listed in cross-references) see A003622. - N. J. A. Sloane, Mar 30 2016

Mathematica

Table[ Prime[ Prime[ Prime[ n ] ] ], {n, 1, 60} ]
Nest[Prime, Range[45], 3] (* Robert G. Wilson v, Mar 15 2004 *)

PARI

a(n) = prime(prime(prime(n))) \\ Charles R Greathouse IV, Apr 28 2015

PARI

list(lim)=my(v=List(),q,r); forprime(p=2,lim, if(isprime(q++) && isprime(r++), listput(v,p))); Set(v) \\ Charles R Greathouse IV, Feb 14 2017

Formula

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

a(n) ~ n*log(n)^3. - Ilya Gutkovskiy, Jul 17 2016