A006450 -id:A006450 - cp's OEIS frontend

A076240 Remainder when 2nd order prime pp(n) = A006450(n) is divided by n-th prime = A000040(n).

1, 2, 1, 3, 9, 2, 8, 10, 14, 22, 3, 9, 15, 19, 23, 29, 41, 39, 63, 69, 2, 6, 16, 16, 24, 42, 48, 52, 54, 52, 74, 84, 88, 102, 114, 122, 134, 152, 156, 166, 168, 1, 7, 13, 19, 23, 31, 71, 71, 73, 73, 65, 77, 91, 79, 91, 109, 115, 125, 137, 149, 155, 185, 197, 203, 197, 235

Offset: 1

Labos Elemer, Oct 08 2002

			a(4) = 3 since prime(prime(4)) (mod prime(4)) = prime(7) (mod 7) = 17 (mod 7) = 3. - _Michael De Vlieger_, Mar 25 2017

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A006450, A038580, A049090, A049203, A049202, A057809, A076241, A076242, A076243.

a:= n-> (p-> irem(ithprime(p), p))(ithprime(n)):
seq(a(n), n=1..70);  # Alois P. Heinz, Oct 09 2015

Table[Mod @@ Map[Nest[Prime, n, #] &, {2, 1}], {n, 65}] (* Michael De Vlieger, Mar 25 2017 *)

a(n) = prime(prime(n)) % prime(n); \\ Michel Marcus, Mar 25 2017

a(n) = prime^2(n) mod prime(n) = A006450(n) mod A000040(n).

A096479 "Secondary twin primes": a(n) = A006450(A096477(n)).

Original entry on oeis.org

3, 5, 11, 17, 41, 59, 179, 191, 431, 461, 599, 617, 1031, 1787, 2027, 2081, 2381, 2549, 3299, 4091, 4217, 4421, 4517, 4787, 5021, 5441, 5651, 8999, 9041, 9461, 10457, 13217, 13709, 13757, 14591, 14867, 15641, 16061, 16451, 16901, 17189, 17291

Offset: 1

Labos Elemer, Jun 23 2004

nonn

			a(10) = 461 since prime(10) = 89 and prime(89 + 1) - prime(89) = 463 - 461 = 2.

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

Subsequence of A001359.

Cf. A072677, A006450, A073124, A094068, A096477, A096478.

Prime[Prime[Flatten[Position[Table[Prime[Prime[n]+1] -Prime[Prime[n]], {n, 1, 1000}], 2]]]]

A076241 Remainder when 2nd order prime pp(n)=A006450(n) is divided by n.

Original entry on oeis.org

0, 1, 2, 1, 1, 5, 3, 3, 2, 9, 6, 1, 10, 9, 1, 1, 5, 13, 8, 13, 10, 5, 17, 5, 9, 1, 23, 27, 19, 17, 27, 3, 14, 15, 19, 13, 31, 17, 16, 31, 38, 37, 35, 27, 31, 21, 28, 17, 12, 47, 43, 43, 39, 31, 26, 45, 13, 1, 17, 23, 17, 53, 11, 15, 1, 53, 10, 25, 64, 41, 38, 41, 68, 33, 59, 63, 65

Offset: 1

Labos Elemer, Oct 08 2002

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

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

[NthPrime(NthPrime(n)) mod(n): n in [1..100]]; // Vincenzo Librandi, Jul 10 2017

Table[Mod[Prime[Prime[n]], n], {n, 100}] (* Vincenzo Librandi, Jul 10 2017 *)

a(n) = prime(prime(n)) % n; \\ Michel Marcus, Jul 09 2017

a(n) = A006450(n) mod n.

A104131 a(n) = pip(n)^pip(n) where pip(n) is the n-th prime-indexed prime (see A006450).

Original entry on oeis.org

27, 3125, 285311670611, 827240261886336764177, 17069174130723235958610643029059314756044734431, 1330877630632711998713399240963346255985889330161650994325137953641

Offset: 1

Cino Hilliard, Mar 06 2005

#^#&/@Prime[Prime[Range[7]]] (* Harvey P. Dale, Jun 03 2023 *)

piptopip(n) = { local(x,y); for(x=1,n, y=pip(x)^pip(x); print1(y","); ) } pip() = { return(prime(prime(n))) }

A092769 Squares of A006450: a(n) = prime(prime(n))^2.

Original entry on oeis.org

9, 25, 121, 289, 961, 1681, 3481, 4489, 6889, 11881, 16129, 24649, 32041, 36481, 44521, 58081, 76729, 80089, 109561, 124609, 134689, 160801, 185761, 212521, 259081, 299209, 316969, 344569, 358801, 380689, 502681, 546121, 597529, 635209

Offset: 1

Jorge Coveiro, Apr 14 2004

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

Cf. A006450.

[NthPrime(NthPrime(n))^2: n in [1..40]]; // Vincenzo Librandi, May 26 2016

Table[Prime[Prime[n]]^2, {n, 50}] (* Vincenzo Librandi, May 26 2016 *)

a(n) = prime(prime(n))^2 \\ Michel Marcus, Jun 30 2013

A092770 Cubes of A006450: a(n) = prime(prime(n))^3.

Original entry on oeis.org

27, 125, 1331, 4913, 29791, 68921, 205379, 300763, 571787, 1295029, 2048383, 3869893, 5735339, 6967871, 9393931, 13997521, 21253933, 22665187, 36264691, 43986977, 49430863, 64481201, 80062991, 97972181, 131872229, 163667323

Offset: 1

Jorge Coveiro, Apr 14 2004

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

Cf. A006450.

[NthPrime(NthPrime(n))^3: n in [1..40]]; // Vincenzo Librandi, May 26 2016

Table[Prime[Prime[n]]^3, {n, 50}] (* Vincenzo Librandi, May 26 2016 *)

a(n) = prime(prime(n))^3 \\ Michel Marcus, Jun 30 2013

A245174 Second differences of A006450.

Original entry on oeis.org

4, 0, 8, -4, 8, -10, 8, 10, -8, 12, -8, -10, 8, 10, 6, -30, 42, -26, -8, 20, -4, 0, 18, -10, -22, 8, -12, 6, 74, -62, 4, -10, 38, -44, 24, 6, -24, 16, -8, -8, 42, -48, 12, -14, 64, 32, -88, -10, 10

Offset: 1

N. J. A. Sloane, Jul 17 2014

sign

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000
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. A006450.

Differences[Table[Prime[Prime[n]], {n, 1, 100}], 2] (* Jean-François Alcover, Oct 09 2018 *)

f(n) = 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

A271368 Number of ways to write n as the sum of distinct super-primes (A006450).

Original entry on oeis.org

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

Offset: 1

Felix Fröhlich, Apr 05 2016

nonn

a(n) > 0 for n > 96 (cf. Dressler, Parker, 1975).

			There are two ways to write 31 as the sum of distinct super-primes: 31 (a single summand, as 31 is itself a super-prime) and 17 + 11 + 3 (three summands), so a(31) = 2.

R. E. Dressler and S. T. Parker, Primes with a Prime Subscript, Journal of the ACM, Vol. 22, No. 3 (1975), 380-381.
Wikipedia, Super-prime

Cf. A000586, A006450.

isokp(pt) = {for (k=1, #pt, if (! isprime(pt[k]) || !isprime(primepi(pt[k])), return (0));); #pt == #Set(pt);}
a(n) = {if (n < 3, return (0)); nb = 0; forpart(pt = n, if (isokp(pt), nb++), [3, n]); nb;} \\ Michel Marcus, Apr 06 2016

G.f.: prod(k>=1, 1 + x^A006450(k) ). [Joerg Arndt, Apr 06 2016]

More terms from Michel Marcus, Apr 06 2016

A265124 Integers n such that A002110(n) + A006450(n) is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 12, 45, 64, 121, 144, 238, 261, 415, 2296, 2847

Offset: 1

Altug Alkan, Dec 02 2015

Corresponding primes are 5, 11, 41, 227, 2341 and 30071.

1, 4, 64, 121, 144 are squares for initial terms of sequence. Are there any other squares in this sequence?

			a(1) = 1 because 2 + 3 = 5 is prime.
a(2) = 2 because 2*3 + 5 = 11 is prime.
a(3) = 3 because 2*3*5 + 11 = 41 is prime.

Cf. A002110, A006450.

Select[Range@ 500, PrimeQ[Product[Prime@ k, {k, #}] + Prime@ Prime@ #] &] (* Michael De Vlieger, Dec 02 2015 *)

lista(nn) = {s = 1; for(k=1, nn, s *= prime(k); if(ispseudoprime(s + prime(prime(k))), print1(k, ", ")); ); }

A007097 Primeth recurrence: a(n+1) = a(n)-th prime.

Original entry on oeis.org

1, 2, 3, 5, 11, 31, 127, 709, 5381, 52711, 648391, 9737333, 174440041, 3657500101, 88362852307, 2428095424619, 75063692618249, 2586559730396077, 98552043847093519, 4123221751654370051, 188272405179937051081, 9332039515881088707361, 499720579610303128776791, 28785866289100396890228041

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

A007097(n) = Min {k : A109301(k) = n} = the first k whose rote height is n, the level set leader or minimum inverse function corresponding to A109301. - Jon Awbrey, Jun 26 2005
Lubomir Alexandrov informs me that he studied this sequence in his 1965 notebook. - N. J. A. Sloane, May 23 2008
a(n) is the Matula-Goebel number of the rooted path tree on n+1 vertices. The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T. - Emeric Deutsch, Feb 18 2012
Conjecture: log(a(1))*log(a(2))*...*log(a(n)) ~ a(n). - Thomas Ordowski, Mar 26 2015

Lubomir Alexandrov, unpublished notes, circa 1960.
L. Longeri, Towards understanding nature and the aesthetics of prime numbers, https://www.longeri.org/prime/nature.html [Broken link, but leave the URL here for historical reasons]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Lubomir Alexandrov, On the nonasymptotic prime number distribution, arXiv:math.NT/9811096, 1998.
Lubomir Alexandrov, "The Eratosthenes Progression p(k+1)=π^{-1}(p(k)), k=0,1,2,..., p(0)=1,4,6,... Determines an Inner Prime Number Distribution Law", Second Int. Conf. "Modern Trends in Computational Physics", Jul 24-29, 2000, Dubna, Russia, Book of Abstracts, p. 19. Available at arXiv:math/0105154 [math.NT], 2001.
Lubomir Alexandrov, Prime Number Sequences And Matrices Generated By Counting Arithmetic Functions, Communications of the Joint Institute of Nuclear Research, E5-2002-55, Dubna, 2002.
J. Awbrey, Riffs and Rotes
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.
Peter R. Cappello, A Note on a Bijection between Natural Numbers and Rooted Trees, 4th SIAM Conference on Discrete Mathematics, June 1988. See section 3 set S codes of paths (codes are per Matula-Goebel).
M. Deléglise, Computation of large values of pi(x)
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Robert G. Wilson v, Letter to N. J. A. Sloane, Sep. 1992
Index entries for sequences related to Matula-Goebel numbers

Row 1 of array A114537.
Left edge of tree A227413, right edge of A246378.
Cf. A000040, A000720, A049076-A049081, A049084, A057450, A109301, A131842, A006450, A292667.
Cf. A078442, A109082 (left inverses).
Subsequence of A245823.

P:=Filtered([1..60000],IsPrime);;
a:=[1];; for n in [2..10] do a[n]:=P[a[n-1]]; od; a; # Muniru A Asiru, Dec 22 2018

a007097 n = a007097_list !! n
a007097_list = iterate a000040 1  -- Reinhard Zumkeller, Jul 14 2013

seq((ithprime@@n)(1),n=0..10); # Peter Luschny, Oct 16 2012

NestList[Prime@# &, 1, 16] (* Robert G. Wilson v, May 30 2006 *)

print1(p=1);until(,print1(","p=prime(p)))  \\ M. F. Hasler, Oct 09 2011

A049084(a(n+1)) = a(n). - Reinhard Zumkeller, Jul 14 2013
a(n)/a(n-1) ~ log(a(n)) ~ prime(n). - Thomas Ordowski, Mar 26 2015
a(n) = prime^{[n]}(1), with the prime function prime(k) = A000040(k), with a(0) = 1. See the name and the programs. - Wolfdieter Lang, Apr 03 2018
Sum_{n>=1} 1/a(n) = A292667. - Amiram Eldar, Oct 15 2020

a(15) corrected and a(16)-a(17) added by Paul Zimmermann
a(18)-a(19) found by David Baugh using a program by Xavier Gourdon and Andrey V. Kulsha, Oct 25 2007
a(20)-a(21) found by Andrey V. Kulsha using a program by Xavier Gourdon, Oct 02 2011
a(22) from Henri Lifchitz, Oct 14 2014
a(23) from David Baugh using Kim Walisch's primecount, May 16 2016

cp's OEIS Frontend

A076240 Remainder when 2nd order prime pp(n) = A006450(n) is divided by n-th prime = A000040(n).

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A096479 "Secondary twin primes": a(n) = A006450(A096477(n)).

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A076241 Remainder when 2nd order prime pp(n)=A006450(n) is divided by n.

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Magma

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A104131 a(n) = pip(n)^pip(n) where pip(n) is the n-th prime-indexed prime (see A006450).

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A092769 Squares of A006450: a(n) = prime(prime(n))^2.

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Magma

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A092770 Cubes of A006450: a(n) = prime(prime(n))^3.

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Magma

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A245174 Second differences of A006450.

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A271368 Number of ways to write n as the sum of distinct super-primes (A006450).

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