A245823 -id:A245823 - cp's OEIS frontend

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

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Lubomir Alexandrov, unpublished notes, circa 1960.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

A245821 Permutation of natural numbers: a(n) = A091205(A245703(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 7, 6, 8, 12, 11, 15, 23, 81, 18, 10, 17, 30, 13, 162, 27, 36, 19, 24, 16, 25, 38, 46, 37, 45, 31, 135, 14, 20, 50, 57, 47, 69, 21, 55, 83, 115, 419, 87, 60, 210, 61, 42, 54, 26, 90, 28, 29, 35, 32, 63, 171, 52, 59, 138, 113, 180, 111, 48, 88, 39, 41, 621, 72, 22, 953, 230, 103, 207, 126, 64, 33, 243

Offset: 1

Antti Karttunen, Aug 02 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245822.
Other related permutations:  A091205, A245703, A245815.
Fixed points: A245823.
Cf. A000040, A049084, A078442, A049076, A018252, A062298.

allocatemem(234567890);
v014580 = vector(2^18);
v091226 = vector(2^22);
v091242 = vector(2^22);
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i=0; j=0; n=2; while((n < 2^22), if(isA014580(n), i++; v014580[i] = n; v091226[n] = v091226[n-1]+1, j++; v091242[j] = n; v091226[n] = v091226[n-1]); n++);
A014580(n) = v014580[n];
A091226(n) = v091226[n];
A091242(n) = v091242[n];
A091205(n) = if(n<=1, n, if(isA014580(n), prime(A091205(A091226(n))), {my(irfs, t); irfs=subst(lift(factor(Mod(1, 2)*Pol(binary(n)))), x, 2); irfs[,1]=apply(t->A091205(t), irfs[,1]); factorback(irfs)}));
A245703(n) = if(1==n, 1, if(isprime(n), A014580(A245703(primepi(n))), A091242(A245703(n-primepi(n)-1))));
A245821(n) = A091205(A245703(n));
for(n=1, 10001, write("b245821.txt", n, " ", A245821(n)));

(define (A245821 n) (A091205 (A245703 n)))

a(n) = A091205(A245703(n)).
Other identities. For all n >= 1, the following holds:
A078442(a(n)) = A078442(n), A049076(a(n)) = A049076(n). [Preserves "the order of primeness of n"].
a(p_n) = p_{a(n)} where p_n is the n-th prime, A000040(n).
a(n) = A049084(a(A000040(n))). [Thus the same permutation is induced also when it is restricted to primes].
A245815(n) = A062298(a(A018252(n))). [While restriction to nonprimes induces another permutation].

A245822 Permutation of natural numbers: a(n) = A245704(A091204(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 9, 6, 16, 11, 10, 19, 33, 12, 25, 17, 15, 23, 34, 39, 70, 13, 24, 26, 50, 21, 52, 53, 18, 31, 55, 77, 93, 54, 22, 29, 27, 66, 105, 67, 48, 137, 156, 30, 28, 37, 64, 91, 35, 85, 58, 97, 49, 40, 98, 36, 135, 59, 45, 47, 261, 56, 76, 92, 122, 83, 374, 38, 102, 139, 69, 167, 130, 88, 203, 351, 212, 349, 235, 14

Offset: 1

Antti Karttunen, Aug 02 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245821.
Other related permutations: A091204, A245704, A245816.
Fixed points: A245823.
Cf. A000040, A049084, A078442, A049076, A018252, A062298.

allocatemem(123456789);
default(primelimit, 2^22)
v014580 = vector(2^18); A014580(n) = v014580[n];
v091226 = vector(2^22); A091226(n) = v091226[n];
A002808(n)={ my(k=-1); while( -n + n += -k + k=primepi(n), ); n}; \\ This function from M. F. Hasler
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i=0; j=0; n=2; while((n < 2^22), if(isA014580(n), i++; v014580[i] = n; v091226[n] = v091226[n-1]+1, v091226[n] = v091226[n-1]); n++)
A091204(n) = if(n<=1, n, if(isprime(n), A014580(A091204(primepi(n))), {my(pfs, t, bits, i); pfs=factor(n); pfs[,1]=apply(t->Pol(binary(A091204(t))), pfs[,1]); sum(i=1, #bits=Vec(factorback(pfs))%2, bits[i]<<(#bits-i))}));
A091245(n) = ((n-A091226(n))-1);
A245704(n) = if(1==n, 1, if(isA014580(n), prime(A245704(A091226(n))), A002808(A245704(A091245(n)))));
A245822(n) = A245704(A091204(n));
for(n=1, 10001, write("b245822.txt", n, " ", A245822(n)));

(define (A245822 n) (A245704 (A091204 n)))

a(n) = A245704(A091204(n)).
Other identities. For all n >= 1, the following holds:
A078442(a(n)) = A078442(n), A049076(a(n)) = A049076(n). [Preserves "the order of primeness of n"].
a(p_n) = p_{a(n)} where p_n is the n-th prime, A000040(n).
a(n) = A049084(a(A000040(n))). [Thus the same permutation is induced also when it is restricted to primes].
A245816(n) = A062298(a(A018252(n))). [While restriction to nonprimes induces another permutation].

A245815 Permutation of natural numbers induced when A245821 is restricted to nonprime numbers: a(n) = A062298(A245821(A018252(n))).

Original entry on oeis.org

1, 2, 5, 3, 4, 7, 9, 59, 11, 6, 20, 125, 18, 25, 15, 10, 16, 26, 32, 31, 103, 8, 12, 35, 41, 50, 13, 39, 85, 64, 43, 164, 29, 38, 17, 66, 19, 24, 21, 45, 132, 37, 105, 139, 82, 33, 65, 27, 507, 52, 14, 180, 161, 96, 46, 22, 190, 141, 87, 1603, 80, 36, 143, 107, 54, 670, 34, 47, 23, 68, 177, 1337, 40

Offset: 1

Antti Karttunen, Aug 02 2014

nonn

This permutation is induced when A245821 is restricted to nonprimes, A018252, the first column of A114537, but equally, when it is restricted to column 2 (A007821), column 3 (A049078), etc. of that square array, or alternatively, to the successive rows of A236542.

The sequence of fixed points f(n) begins as 1, 2, 15, 142, 548, 1694, 54681. A018252(f(n)) gives the nonprime terms of A245823.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245816.
Related permutations: A245813, A245819, A245821.
Cf. A007821, A018252, A049078, A062298, A114537, A236542, A245823.

allocatemem(234567890);
v014580 = vector(2^18);
v091226 = vector(2^22);
v091242 = vector(2^22);
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i=0; j=0; n=2; while((n < 2^22), if(isA014580(n), i++; v014580[i] = n; v091226[n] = v091226[n-1]+1, j++; v091242[j] = n; v091226[n] = v091226[n-1]); n++);
A002808(n)={my(k); for(k=0, primepi(n), isprime(n++)&&k--); n}; \\ This function from M. F. Hasler
A062298(n) = n-primepi(n);
A018252(n) = if(1==n, 1, A002808(n-1));
A014580(n) = v014580[n];
A091226(n) = v091226[n];
A091242(n) = v091242[n];
A091205(n) = if(n<=1, n, if(isA014580(n), prime(A091205(A091226(n))), {my(irfs, t); irfs=subst(lift(factor(Mod(1, 2)*Pol(binary(n)))), x, 2); irfs[,1]=apply(t->A091205(t), irfs[,1]); factorback(irfs)}));
A245703(n) = if(1==n, 1, if(isprime(n), A014580(A245703(primepi(n))), A091242(A245703(n-primepi(n)-1))));
A245821(n) = A091205(A245703(n));
A245815(n) = A062298(A245821(A018252(n)));
for(n=1, 10001, write("b245815.txt", n, " ", A245815(n)));

(define (A245815 n) (A062298 (A245821 (A018252 n))))

a(n) = A062298(A245821(A018252(n))).
As a composition of related permutations:
a(n) = A245813(A245819(n)).
Also following holds for all n >= 1:
a(n) = A062298(A049084(A245821(A007821(n)))).
a(n) = A062298(A049084(A049084(A245821(A049078(n))))).

A245816 Permutation of natural numbers induced when A245822 is restricted to nonprime numbers: a(n) = A062298(A245822(A018252(n))).

Original entry on oeis.org

1, 2, 4, 5, 3, 10, 6, 22, 7, 16, 9, 23, 27, 51, 15, 17, 35, 13, 37, 11, 39, 56, 69, 38, 14, 18, 48, 78, 33, 120, 20, 19, 46, 67, 24, 62, 42, 34, 28, 73, 25, 103, 31, 206, 40, 55, 68, 92, 300, 26, 76, 50, 99, 65, 157, 281, 165, 184, 8, 121, 134, 277, 423, 30, 47, 36, 223, 70, 514, 75, 101, 116, 236, 139, 74

Offset: 1

Antti Karttunen, Aug 02 2014

nonn

This permutation is induced when A245822 is restricted to nonprimes, A018252, the first column of A114537, but equally, when it is restricted to column 2 (A007821), column 3 (A049078), etc. of that square array, or alternatively, to the successive rows of A236542.

The sequence of fixed points f(n) begins as 1, 2, 15, 142, 548, 1694, 54681. A018252(f(n)) gives the nonprime terms of A245823.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245815.
Related permutations: A245814, A245820, A245822.
Cf. A007821, A018252, A049078, A062298, A114537, A236542, A245823.

allocatemem(123456789);
default(primelimit, 2^22)
A014580 = vector(2^18);
A091226 = vector(2^22);
A002808(n)={my(k); for(k=0, primepi(n), isprime(n++)&&k--); n}; \\ This function from M. F. Hasler, Oct 31 2008
A062298(n) = n-primepi(n);
A018252(n) = if(1==n,1,A002808(n-1));
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i=0; j=0; n=2; while((n < 2^22), if(isA014580(n), i++; A014580[i] = n; A091226[n] = A091226[n-1]+1, A091226[n] = A091226[n-1]); n++)
A091204(n) = if(n<=1, n, if(isprime(n), A014580[A091204(primepi(n))], {my(pfs,t,bits,i); pfs=factor(n); pfs[, 1]=apply(t->Pol(binary(A091204(t))), pfs[, 1]); sum(i=1, #bits=Vec(factorback(pfs))%2, bits[i]<<(#bits-i))}));
A091245(n) = ((n-A091226[n])-1);
A245704(n) = if(1==n, 1, if(isA014580(n), prime(A245704(A091226[n])), A002808(A245704(A091245(n)))));
A245822(n) = A245704(A091204(n));
A245816(n) = A062298(A245822(A018252(n)));
for(n=1, 10001, write("b245816.txt", n, " ", A245816(n)));

(define (A245816 n) (A062298 (A245822 (A018252 n))))

a(n) = A062298(A245822(A018252(n))).
As a composition of related permutations:
a(n) = A245820(A245814(n)).
Also following holds for all n >= 1:
a(n) = A062298(A049084(A245822(A007821(n)))).
a(n) = A062298(A049084(A049084(A245822(A049078(n))))).
etc.

A250251 Fixed points of A250249 and A250250.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 40, 41, 43, 44, 46, 47, 48, 49, 50, 52, 53, 56, 58, 59, 60, 61, 62, 64, 67, 68, 70, 71, 72, 74, 76, 77, 79, 80, 82, 83, 86, 88, 89, 92, 94, 96, 97, 98, 100, 101, 104, 106, 107, 109, 112, 113, 116, 118, 120, 121

Offset: 1

Antti Karttunen, Nov 18 2014

nonn

Numbers for which A250249(n) = n (equally: A250250(n) = n).

If n is a member, then 2n is also a member. If any 2n is a member, then n is also a member. If n is a member, then the n-th prime, p_n (= A000040(n)) is also a member. If p_n is a member, then its index n is also a member. Thus the sequence is completely determined by its odd nonprime terms: 1, 9, 15, 25, ..., (A249730) and is obtained as a union of their multiples with powers of 2, and all prime recurrences that start with those values: A007097 U A057450 U A057451 U A057452 U A057453 U ..., etc.

Antti Karttunen, Table of n, a(n) for n = 1..801

Complement: A249729.
Subsequences: A249730, and also A007097, A057450, A057451, A057452, A057453, etc.
Cf. also A245823, A250249, A250250.

A245817 Difference in size between rooted trees which are encoded as Matula-Goebel numbers A245821(n) and n: a(n) = A061775(A245821(n)) - A061775(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 16 2014

sign

Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A000040, A061775, A245818, A245821, A245823.

\\ Execute first the code given in A061775 and A245821.
A245817(n) = A061775(A245821(n)) - A061775(n);
for(n=1, 10001, write("b245817.txt", n, " ", A245817(n)));

(define (A245817 n) (- (A061775 (A245821 n)) (A061775 n)))

a(n) = A061775(A245821(n)) - A061775(n).
Other identities. For all n >= 1, the following holds:
a(A000040(n)) = a(n). [The result for the n-th prime is same as for n itself].
a(A245823(n)) = 0. [A245823 gives a (proper) subsequence of the positions of the zeros].

A245818 Difference in size between rooted trees which are encoded with Matula-Goebel numbers A245822(n) and n: a(n) = A061775(A245822(n)) - A061775(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 16 2014

sign

Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A000040, A061775, A245817, A245822, A245823.

\\ Execute first the code given in A061775 and A245822.
A245818(n) = A061775(A245822(n)) - A061775(n);
for(n=1, 10001, write("b245818.txt", n, " ", A245818(n)));

(define (A245818 n) (- (A061775 (A245822 n)) (A061775 n)))

a(n) = A061775(A245822(n)) - A061775(n).
Other identities. For all n >= 1, the following holds:
a(A000040(n)) = a(n). [The result for the n-th prime is same as for n itself].
a(A245823(n)) = 0. [A245823 gives a (proper) subsequence of the positions of the zeros].

cp's OEIS Frontend

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

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A245821 Permutation of natural numbers: a(n) = A091205(A245703(n)).

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A245822 Permutation of natural numbers: a(n) = A245704(A091204(n)).

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A245815 Permutation of natural numbers induced when A245821 is restricted to nonprime numbers: a(n) = A062298(A245821(A018252(n))).

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A245816 Permutation of natural numbers induced when A245822 is restricted to nonprime numbers: a(n) = A062298(A245822(A018252(n))).

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A250251 Fixed points of A250249 and A250250.

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A245817 Difference in size between rooted trees which are encoded as Matula-Goebel numbers A245821(n) and n: a(n) = A061775(A245821(n)) - A061775(n).

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A245818 Difference in size between rooted trees which are encoded with Matula-Goebel numbers A245822(n) and n: a(n) = A061775(A245822(n)) - A061775(n).

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