A038580 -id:A038580 - cp's OEIS frontend

A076243 Remainder when 3rd-order prime ppp(n) = A038580(n) is divided by n.

0, 1, 1, 3, 2, 5, 4, 3, 8, 9, 5, 7, 10, 5, 7, 3, 2, 11, 17, 1, 20, 21, 11, 19, 12, 17, 14, 17, 18, 19, 18, 23, 28, 27, 11, 19, 15, 7, 2, 21, 40, 25, 31, 1, 19, 15, 9, 31, 46, 47, 10, 15, 43, 23, 14, 9, 17, 19, 18, 41, 24, 27, 50, 3, 14, 29, 13, 3, 4, 39, 21, 1, 47, 19, 31, 13, 6, 17

Offset: 1

Labos Elemer, Oct 08 2002

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A006450, A038580, A049090, A049203, A049202, A057809, A076240, A076241, A076242.

MapIndexed[Mod[#1, First@ #2] &, Nest[Prime, Range@ 79, 3]] (* Michael De Vlieger, Jul 22 2017 *)

a(n) = ppp(n) mod n = A038580(n) mod n.

A076242 Remainder when 3rd order prime A038580(n) is divided by n-th prime=A000040(n).

Original entry on oeis.org

1, 2, 1, 3, 6, 10, 5, 8, 17, 19, 27, 31, 38, 35, 28, 39, 17, 17, 10, 38, 68, 63, 13, 55, 48, 4, 74, 100, 37, 29, 47, 121, 115, 136, 105, 28, 128, 109, 159, 90, 114, 31, 151, 4, 86, 108, 81, 147, 149, 189, 185, 119, 231, 166, 88, 238, 197, 233, 64, 186, 258, 111, 128, 260

Offset: 1

Labos Elemer, Oct 08 2002

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A006450, A038580, A049090, A049203, A049202, A057809, A076240-A076243.

Table[Mod[Prime[Prime[Prime[n]]],Prime[n]],{n,70}] (* Harvey P. Dale, Sep 28 2013 *)

a(n) = Mod[A038580(n), A000040(n)]

A086749 Partial sums of A038580.

Original entry on oeis.org

5, 16, 47, 106, 233, 412, 689, 1020, 1451, 2050, 2759, 3678, 4741, 5894, 7191, 8714, 10501, 12348, 14569, 16950, 19427, 22176, 25177, 28436, 32073, 36016, 40107, 44380, 48777, 53326, 58707, 64330, 70199, 76312, 82973, 89796, 96989, 104596

Offset: 1

Cino Hilliard, Jul 30 2003

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006450, A038580, A083186.

Accumulate[Nest[Prime,Range[45],3]] (* Harvey P. Dale, Oct 25 2011 *)

f(n) = y=0; for(x=1,n,y+=prime(prime(prime(x))); print1(y","))

Edited by N. J. A. Sloane, Apr 17 2007

A245175 Second differences of A038580.

Original entry on oeis.org

14, 8, 40, -16, 46, -44, 46, 68, -58, 100, -66, -54, 54, 82, 38, -204, 314, -214, -64, 176, -20, 6, 120, -72, -158, 34, -58, 28, 680, -590, 4, -2, 304, -386, 208, 44, -180, 146, -74, -114, 408, -458, 136, -158, 652, 340, -902, -66, 60, 42, -30, 490, -466, 228

Offset: 1

N. J. A. Sloane, Jul 17 2014

sign

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Jens Kruse Andersen)
R. G. Batchko, A prime fractal and global quasi-self-similar structure in the distribution of prime-indexed primes, arXiv preprint arXiv:1405.2900, 2014. See Table 2.

Cf. A038580.

a:= ((f-> n-> f(n+1)-f(n))@@2)(ithprime@@3):
seq(a(n), n=1..60);

Differences[Nest[Prime, Range[60], 3], 2] (* Jean-François Alcover, Oct 06 2018 *)

f(n) = prime(prime(prime(n)))
a(n) = f(n+2)-2*f(n+1)+f(n)
vector(50, n, a(n)) \\ Jens Kruse Andersen, Jul 18 2014

A006450 Prime-indexed primes: primes with prime subscripts.

Original entry on oeis.org

3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, 179, 191, 211, 241, 277, 283, 331, 353, 367, 401, 431, 461, 509, 547, 563, 587, 599, 617, 709, 739, 773, 797, 859, 877, 919, 967, 991, 1031, 1063, 1087, 1153, 1171, 1201, 1217, 1297, 1409, 1433, 1447, 1471

Offset: 1

Jeffrey Shallit, Nov 25 1975

A000040 = A006450 U A007821. - Juri-Stepan Gerasimov, Sep 24 2009
Subsequence of A175247 (primes (A000040) with noncomposite (A008578) subscripts), a(n) = A175247(n+1). - Jaroslav Krizek, Mar 13 2010
Primes p such that p and pi(p) are both primes. - Juri-Stepan Gerasimov, Jul 14 2011
Sum_{n>=1} 1/a(n) converges. In fact, Sum_{n>N} 1/a(n) < 1/log(N), by the integral test. - Jonathan Sondow, Jul 11 2012
The number of such primes not exceeding x > 0 is pi(pi(x)). I conjecture that the sequence a(n)^(1/n) (n = 1,2,3,...) is strictly decreasing. This is an analog of the Firoozbakht conjecture on primes. - Zhi-Wei Sun, Aug 17 2015
Limit_{n->infinity} a(n)/(n*(log(n))^2) = 1. Proof: By Cipolla's asymptotic formula, prime(n) ~ L(n) + R(n), where L(n)/n = log(n) + log(log(n)) - 1 and R(n)/n decreases logarithmically to 0. Hence, for large n, a(n) = prime(prime(n)) ~ L(L(n)+R(n)) + R(L(n)+R(n)) = n*(log(n))^2 + r(n), where r(n) grows as O(n*log(n)*log(log(n))). The rest of the proof is trivial. The convergence is very slow: for k = 1,2,3,4,5,6, sqrt(a(10^k)/10^k)/log(10^k) evaluates to 2.055, 1.844, 1.695, 1.611, 1.545, and 1.493, respectively. - Stanislav Sykora, Dec 09 2015

			a(5) = 31 because a(5) = p(p(5)) = p(11) = 31.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. S. Kimberley, Table of n, a(n) for n = 1..100000
R. 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.
Jonathan Bayless, Dominic Klyve, and Tomás Oliveira e Silva, New Bounds and Computations on Prime-Indexed Primes, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A43, 2013.
K. A. Broughan and A. R. Barnett, On the subsequence of primes having prime subscripts, JIS 12 (2009) 09.2.3.
Paul Cooijmans, Numbers.
Paul Cooijmans, Short Test For Genius.
R. E. Dressler and S. T. Parker, Primes with a prime subscript, J. ACM 22 (1975) 380-381.
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]
N. Fernandez, More terms of this and other sequences related to A049076.
A. B. Frizell, The permutations of the natural numbers can not be well ordered, Bull. Amer. Math. Soc. 22 (1915), no. 2, 71-73.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
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.
Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732 [math.CO], 2012.
J. Shallit, Letter to N. J. A. Sloane, Oct. 1975
Eric Weisstein's World of Mathematics, Prime formulas, see Cipolla formula.

Primes for which A049076 > 1.
Cf. A000040, A007821, A038580, A049090, A049202, A049203, A057847, A057849, A057850, A057851, A058332, A093047.
Cf. A185723 and A214296 for numbers and primes that are sums of distinct a(n); cf. A213356 and A185724 for those that are not.
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.

a006450 = a000040 . a000040
a006450_list = map a000040 a000040_list
-- Reinhard Zumkeller, Jan 12 2013

[ NthPrime(NthPrime(n)): n in [1..51] ]; // Jason Kimberley, Apr 02 2010

seq(ithprime(ithprime(i)),i=1..50); # Uli Baum (Uli_Baum(AT)gmx.de), Sep 05 2007
# 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[ n ] ], {n, 100} ]
```

i=0;forprime(p=2,1e4,if(isprime(i++),print1(p", "))) \\ Charles R Greathouse IV, Jun 10 2011

a=vector(10^3,n,prime(prime(n))) \\ Stanislav Sykora, Dec 09 2015

from sympy import prime
def a(n): return prime(prime(n))
print([a(n) for n in range(1, 52)]) # Michael S. Branicky, Aug 11 2021

# much faster version for initial segment of sequence
from sympy import nextprime, isprime
def aupton(terms):
    alst, p, pi = [], 2, 1
    while len(alst) < terms:
        if isprime(pi): alst.append(p)
        p, pi = nextprime(p), pi+1
    return alst
print(aupton(10000)) # Michael S. Branicky, Aug 11 2021

a(n) = prime(prime(n)) = A000040(A000040(n)). - Juri-Stepan Gerasimov, Sep 24 2009
a(n) > n*(log(n))^2, as prime(n) > n*log(n) by Rosser's theorem. - Jonathan Sondow, Jul 11 2012
a(n)/log(a(n)) ~ prime(n). - Thomas Ordowski, Mar 30 2015
Sum_{n>=1} 1/a(n) is in the interval (1.04299, 1.04365) (Bayless et al., 2013). - Amiram Eldar, Oct 15 2020

A007821 Primes p such that pi(p) is not prime.

Original entry on oeis.org

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

Offset: 1

Monte J. Zerger (mzerger(AT)cc4.adams.edu), Clark Kimberling

nonn

Primes prime(k) such that A049076(k)=1, sorted along increasing k. - R. J. Mathar, Jan 28 2014

The complement of A006450 (primes with prime index) within the primes A000040.

C. Kimberling, Fractal sequences and interspersions, Ars Combinatoria, vol. 45 p 157 1997.

R. Zumkeller, Table of n, a(n) for n = 1..1000
Lubomir Alexandrov, On the nonasymptotic prime number distribution, arXiv:math/9811096 [math.NT], 1998.
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A049076, A049078, A049079, A049080, A049081, A058322, A058324, A058325, A058326, A058327, A058328, A093046, A006450.

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.

a007821 = a000040 . a018252
a007821_list = map a000040 a018252_list
-- Reinhard Zumkeller, Jan 12 2013

A007821 := proc(n) ithprime(A018252(n)) ; end proc: # R. J. Mathar, Jul 07 2012

Prime[ Select[ Range[75], !PrimeQ[ # ] &]] (* Robert G. Wilson v, Mar 15 2004 *)
With[{nn=100},Pick[Prime[Range[nn]],Table[If[PrimeQ[n],0,1],{n,nn}],1]] (* Harvey P. Dale, Aug 14 2020 *)

forprime(p=2, 1e3, if(!isprime(primepi(p)), print1(p, ", "))) \\ Felix Fröhlich, Aug 16 2014

from sympy import primepi
def A007821(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-(p:=primepi(x))+primepi(p)
    return bisection(f,n,n) # Chai Wah Wu, Oct 19 2024

A137588(a(n)) = n; a(n) = A000040(A018252(n)). - Reinhard Zumkeller, Jan 28 2008
A000040 = A007821 U A006450. - Juri-Stepan Gerasimov, Sep 24 2009
A175247 U { a(n); n > 1 } = A000040. { a(n) } = { 2 } U { primes (A000040) with composite index (A002808) }. - Jaroslav Krizek, Mar 13 2010
G.f. over nonprime powers: Sum_{k >= 1} prime(k)*x^k-prime(prime(k))*x^prime(k). - Benedict W. J. Irwin, Jun 11 2016

Edited by M. F. Hasler, Jul 31 2015

A049078 Primes prime(k) for which A049076(k) = 2.

Original entry on oeis.org

3, 17, 41, 67, 83, 109, 157, 191, 211, 241, 283, 353, 367, 401, 461, 509, 547, 563, 587, 617, 739, 773, 797, 859, 877, 967, 991, 1031, 1087, 1171, 1201, 1217, 1409, 1433, 1447, 1471, 1499, 1597, 1621, 1669, 1723, 1741, 1823, 1913, 2027, 2063, 2081, 2099

Offset: 1

Neil Fernandez

nonn

			For these primes S(p) is a prime but S(S(p)) is not. E.g. S(17)=7, S(7)=4.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A049076, A007821, A049079-A049081, A058322, A058324-A058328, A093046, A006450, A018252, A236542.

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.

A049078 := proc(n) ithprime(A007821(n)) ; end proc: # R. J. Mathar, Aug 14 2008

Nest[ Prime, Select[ Range[70], !PrimeQ[ # ] &], 2] (* Robert G. Wilson v, Mar 15 2004 *)

is(n)=my(p=primepi(n)); isprime(p) && !isprime(primepi(p)) && isprime(n) \\ Charles R Greathouse IV, Apr 29 2015

a(n) = prime(A007821(n)). - Juri-Stepan Gerasimov, Aug 11 2008

a(n) ~ A006450(n) ~ n log^2 n. - Charles R Greathouse IV, Apr 29 2015

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010

A049090 Primes for which A049076 >= 4.

Original entry on oeis.org

11, 31, 127, 277, 709, 1063, 1787, 2221, 3001, 4397, 5381, 7193, 8527, 9319, 10631, 12763, 15299, 15823, 19577, 21179, 22093, 24859, 27457, 30133, 33967, 37217, 38833, 40819, 42043, 43651, 52711, 55351, 57943, 60647, 66851, 68639, 72727

Offset: 1

Neil Fernandez

nonn

Union of A049080, A049081, A058322, A058324, etc. - R. J. Mathar, Jul 07 2012

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 1000 terms from Robert G. Wilson v)
R. 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.
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]
N. Fernandez, More terms of this and other sequences related to A049076.

Cf. A000040, A006450, A038580, A049076, A049202, A049203, A057847, A057849, A057850, A057851, A058332, A093047.

map(ithprime@@3, select(isprime, [$1..157])); # Peter Luschny, Feb 17 2014

Nest[ Prime, Range[40], 4] (* Robert G. Wilson v, Mar 15 2004 *)

list(lim)=my(v=List(),q,r,s); forprime(p=2,lim,if(isprime(q++)&&isprime(r++)&&isprime(s++),listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Feb 16 2017

a(n) = A006450(A006450(n)). - James G. Merickel, Feb 14 2010
a(n) = A000040(A038580(n)). - R. J. Mathar, Jul 07 2012
a(n) ~ n (log n)^4. - Charles R Greathouse IV, Feb 16 2017

Name corrected by Sean A. Irvine, Jul 18 2021

A049203 Primes for which A049076(p) >= 5.

Original entry on oeis.org

31, 127, 709, 1787, 5381, 8527, 15299, 19577, 27457, 42043, 52711, 72727, 87803, 96797, 112129, 137077, 167449, 173867, 219613, 239489, 250751, 285191, 318211, 352007, 401519, 443419, 464939, 490643, 506683, 527623, 648391, 683873, 718807

Offset: 1

Neil Fernandez

nonn

Union of A049081, A058322, A058324-A058328, A093046, etc. - R. J. Mathar, Jul 07 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert G. Wilson v)
R. 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.
N. Fernandez, An order of primeness.
N. Fernandez, An order of primeness [cached copy, included with permission of the author]
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. A000040, A006450, A038580, A049076, A049090, A049202, A057847, A057849, A057850, A057851, A058332, A093047.

map(ithprime@@4, select(isprime, [$1..137])); # Peter Luschny, Feb 17 2014

Nest[ Prime, Range[35], 5] (* Robert G. Wilson v, Mar 15 2004 *)

list(lim)=my(v=List(),q,r,s,t); forprime(p=2,lim, if(isprime(q++) && isprime(r++) && isprime(s++) && isprime(t++), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Feb 16 2017

a(n) = A000040(A049090(n)). - R. J. Mathar, Jul 07 2012

a(n) ~ n (log n)^5. - Charles R Greathouse IV, Feb 16 2017

More terms from Robert G. Wilson v, Nov 10 2000

Name corrected by Sean A. Irvine, Jul 21 2021

A049202 Primes p whose order of primeness A049076(p) is >= 6.

Original entry on oeis.org

127, 709, 5381, 15299, 52711, 87803, 167449, 219613, 318211, 506683, 648391, 919913, 1128889, 1254739, 1471343, 1828669, 2269733, 2364361, 3042161, 3338989, 3509299, 4030889, 4535189, 5054303, 5823667, 6478961, 6816631

Offset: 1

Neil Fernandez

nonn

Union of A058322, A058324-A058328, A093046 etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 1000 terms from Robert G. Wilson v)
R. 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.
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A049076, A000040, A006450, A038580, A049090, A049203, A057849, A057850, A057851, A057847, A058332, A093047.

map(ithprime@@4,select(isprime, [$1..137])); # Peter Luschny, Feb 17 2014

Nest[ Prime, Range[35], 6] (* Robert G. Wilson v, Mar 15 2004 *)

list(lim)=my(v=List(), q, r, s, t, u); forprime(p=2, lim, if(isprime(q++) && isprime(r++) && isprime(s++) && isprime(t++) && isprime(u++), listput(v, p))); Vec(v) \\ Charles R Greathouse IV, Feb 16 2017

More terms from Robert G. Wilson v, Nov 10 2000

Name corrected by Sean A. Irvine, Jul 21 2021

cp's OEIS Frontend

A076243 Remainder when 3rd-order prime ppp(n) = A038580(n) is divided by n.

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A076242 Remainder when 3rd order prime A038580(n) is divided by n-th prime=A000040(n).

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A086749 Partial sums of A038580.

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A245175 Second differences of A038580.

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A006450 Prime-indexed primes: primes with prime subscripts.

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A007821 Primes p such that pi(p) is not prime.

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A049078 Primes prime(k) for which A049076(k) = 2.

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