A078442 -id:A078442 - cp's OEIS frontend

A061775 Number of nodes in rooted tree with Matula-Goebel number n.

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

Offset: 1

N. J. A. Sloane, Jun 22 2001

nonn

Let p(1)=2, ... denote the primes. The label f(T) for a rooted tree T is 1 if T has 1 node, otherwise f(T) = Product p(f(T_i)) where the T_i are the subtrees obtained by deleting the root and the edges adjacent to it. (Cf. A061773 for illustration).

Each n occurs A000081(n) times.

			a(4) = 3 because the rooted tree corresponding to the Matula-Goebel number 4 is "V", which has one root-node and two leaf-nodes, three in total.
See also the illustrations in A061773.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Keith Briggs, Matula numbers and rooted trees
Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to Matula-Goebel numbers

One more than A196050.
Sum of entries in row n of irregular table A214573.
Number of entries in row n of irregular tables A182907, A206491, A206495 and A212620.
One less than the number of entries in row n of irregular tables A184187, A193401 and A193403.
Cf. A005517 (the position of the first occurrence of n).
Cf. A005518 (the position of the last occurrence of n).
Cf. A091233 (their difference plus one).
Cf. A214572 (Numbers k such that a(k) = 8).
Cf. also A000081, A061773, A049084, A020639, A049076, A078442, A091238, A091204, A091205, A109082, A127301, A109129, A193402, A193405, A193406, A196047, A196068, A198333, A206487, A206494, A206496, A214569, A214571, A213670, A214568, A228599, A245817, A245818.
Cf. also A075166, A075167, A216648, A216649.

import Data.List (genericIndex)
a061775 n = genericIndex a061775_list (n - 1)
a061775_list = 1 : g 2 where
   g x = y : g (x + 1) where
      y = if t > 0 then a061775 t + 1 else a061775 u + a061775 v - 1
          where t = a049084 x; u = a020639 x; v = x `div` u
-- Reinhard Zumkeller, Sep 03 2013

with(numtheory): a := proc (n) local u, v: u := n-> op(1, factorset(n)): v := n-> n/u(n): if n = 1 then 1 elif isprime(n) then 1+a(pi(n)) else a(u(n))+a(v(n))-1 end if end proc: seq(a(n), n = 1..108); # Emeric Deutsch, Sep 19 2011

a[n_] := Module[{u, v}, u = FactorInteger[#][[1, 1]]&; v = #/u[#]&; If[n == 1, 1, If[PrimeQ[n], 1+a[PrimePi[n]], a[u[n]]+a[v[n]]-1]]]; Table[a[n], {n, 108}] (* Jean-François Alcover, Jan 16 2014, after Emeric Deutsch *)

A061775(n) = if(1==n, 1, if(isprime(n), 1+A061775(primepi(n)), {my(pfs,t,i); pfs=factor(n); pfs[,1]=apply(t->A061775(t),pfs[,1]); (1-bigomega(n)) + sum(i=1, omega(n), pfs[i,1]*pfs[i,2])}));
for(n=1, 10000, write("b061775.txt", n, " ", A061775(n)));
\\ Antti Karttunen, Aug 16 2014

from functools import lru_cache
from sympy import isprime, factorint, primepi
@lru_cache(maxsize=None)
def A061775(n):
    if n == 1: return 1
    if isprime(n): return 1+A061775(primepi(n))
    return 1+sum(e*(A061775(p)-1) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 19 2022

a(1) = 1; if n = p_t (= the t-th prime), then a(n) = 1+a(t); if n = uv (u,v>=2), then a(n) = a(u)+a(v)-1.
a(n) = A091238(A091204(n)). - Antti Karttunen, Jan 2004
a(n) = A196050(n)+1. - Antti Karttunen, Aug 16 2014

More terms from David W. Wilson, Jun 25 2001

Extended by Emeric Deutsch, Sep 19 2011

A049076 Number of steps in the prime index chain for the n-th prime.

Original entry on oeis.org

1, 2, 3, 1, 4, 1, 2, 1, 1, 1, 5, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1

Offset: 1

Neil Fernandez

Let p(k) = k-th prime, let S(p) = S(p(k)) = k, the subscript of p; a(n) = order of primeness of p(n) = 1+m where m is largest number such that S(S(..S(p(n))...)) with m S's is a prime.

The record holders correspond to A007097.

			11 is 5th prime, so S(11)=5, 5 is 3rd prime, so S(S(11))=3, 3 is 2nd prime, so S(S(S(11)))=2, 2 is first prime, so S(S(S(S(11))))=1, not a prime. Thus a(5)=4.
Alternatively, a(5) = 4: the 5th prime is 11 and its prime index chain is 11->5->3->2->1->0. a(6) = 1: the 6th prime is 13 and its prime index chain is 13->6->0.

N. Fernandez, Table of n, a(n) for n = 1..500
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]
N. Fernandez, The Exploring Primeness Project

Cf. A000040, A049077 - A049081, A006450, A049084, A236542.

a049076 = (+ 1) . a078442  -- Reinhard Zumkeller, Jul 14 2013

A049076 := proc(n)
    if not isprime(n) then
        1 ;
    else
        1+procname(numtheory[pi](n)) ;
    end if;
end proc:
seq(A049076(n),n=1..30) ; # R. J. Mathar, Jan 28 2014

A049076 f[n_] := Length[ NestWhileList[ PrimePi, n, PrimeQ]]; Table[ f[n], {n, 105}] (* Robert G. Wilson v, Mar 11 2004 *)
Table[Length[NestWhileList[PrimePi[#]&,Prime[n],PrimeQ[#]&]]-1,{n,110}] (* Harvey P. Dale, May 07 2018 *)

apply(p->my(s=1);while(isprime(p=primepi(p)),s++); s, primes(100)) \\ Charles R Greathouse IV, Nov 20 2012

Let b(n) = 0 if n is nonprime, otherwise b(n) = k where n is the k-th prime. Then a(n) is the number of times you can apply b to the n-th prime before you hit a nonprime.
a(n) = 1 + A078442(n). - R. J. Mathar, Jul 07 2012
a(n) = A078442(A000040(n)). - Alois P. Heinz, Mar 16 2020

Additional comments from Gabriel Cunningham (gcasey(AT)mit.edu), Apr 12 2003

A135141 a(1)=1, a(p_n)=2a(n), a(c_n)=2a(n)+1, where p_n = n-th prime, c_n = n-th composite number.

Original entry on oeis.org

1, 2, 4, 3, 8, 5, 6, 9, 7, 17, 16, 11, 10, 13, 19, 15, 12, 35, 18, 33, 23, 21, 14, 27, 39, 31, 25, 71, 34, 37, 32, 67, 47, 43, 29, 55, 22, 79, 63, 51, 20, 143, 26, 69, 75, 65, 38, 135, 95, 87, 59, 111, 30, 45, 159, 127, 103, 41, 24, 287, 70, 53, 139, 151, 131, 77, 36, 271, 191

Offset: 1

Katarzyna Matylla, Feb 13 2008

A permutation of the positive integers, related to A078442.
a(p) is even when p is prime and is divisible by 2^(prime order of p).
From Robert G. Wilson v, Feb 16 2008: (Start)
What is the length of the cycle containing 10? Is it infinite? The cycle begins 10, 17, 12, 11, 16, 15, 19, 18, 35, 29, 34, 43, 26, 31, 32, 67, 36, 55, 159, 1055, 441, 563, 100, 447, 7935, 274726911, 1013992070762272391167, ... Implementation in Mmca: NestList[a(AT)# &, 10, 26] Furthermore, it appears that any non-single-digit number has an infinite cycle.
Records: 1, 2, 4, 8, 9, 17, 19, 35, 39, 71, 79, 143, 159, 287, 319, 575, 639, 1151, 1279, 2303, 2559, 4607, 5119, 9215, 10239, 18431, 20479, 36863, 40959, 73727, 81919, 147455, 163839, 294911, 327679, 589823, 655359, ..., . (End)

			a(20) = 33 = 2*16 + 1 because 20 is 11th composite and a(11)=16. Or, a(20)=33=100001(bin). In other words it is a composite number, its index is a prime number, whose index is a prime....

Robert G. Wilson v, Table of n, a(n) for n = 1..10000
Antti Karttunen, Entanglement Permutations, 2016-2017
Index entries for sequences computed from indices in prime factorization
Index entries for sequences that are permutations of the natural numbers

Cf. A000720, A007814, A010051, A026238, A078442.
Cf. A246346, A246347 (record positions and values).
Cf. A227413 (inverse).
Cf. A071574, A245701, A245702, A245703, A245704, A246377, A236854, A237427 for related and similar permutations.

import Data.List (genericIndex)
a135141 n = genericIndex a135141_list (n-1)
a135141_list = 1 : map f [2..] where
   f x | iprime == 0 = 2 * (a135141 $ a066246 x) + 1
       | otherwise   = 2 * (a135141 iprime)
       where iprime = a049084 x
-- Reinhard Zumkeller, Jan 29 2014

a[1] = 1; a[n_] := If[PrimeQ@n, 2*a[PrimePi[n]], 2*a[n - 1 - PrimePi@n] + 1]; Array[a, 69] (* Robert G. Wilson v, Feb 16 2008 *)

/* Let pc = prime count (which prime it is), cc = composite count: */
pc[1]:0;
cc[1]:0;
pc[2]:1;
cc[4]:1;
pc[n]:=if primep(n) then 1+pc[prev_prime(n)] else 0;
cc[n]:=if primep(n) then 0 else if primep(n-1) then 1+cc[n-2] else 1+cc[n-1];
a[1]:1;
a[n]:=if primep(n) then 2*a[pc[n]] else 1+2*a[cc[n]];

A135141(n) = if(1==n, 1, if(isprime(n), 2*A135141(primepi(n)), 1+(2*A135141(n-primepi(n)-1)))); \\ Antti Karttunen, Dec 09 2019

from sympy import isprime, primepi
def a(n): return 1 if n==1 else 2*a(primepi(n)) if isprime(n) else 2*a(n - 1 - primepi(n)) + 1 # Indranil Ghosh, Jun 11 2017, after Mathematica code

a(n) = 2*A135141((A049084(n))*chip + A066246(n)*(1-chip)) + 1 - chip, where chip = A010051(n). - Reinhard Zumkeller, Jan 29 2014
From Antti Karttunen, Dec 09 2019: (Start)
A007814(a(n)) = A078442(n).
A070939(a(n)) = A246348(n).
A080791(a(n)) = A246370(n).
A054429(a(n)) = A246377(n).
A245702(a(n)) = A245703(n).
a(A245704(n)) = A245701(n). (End)

A262275 Prime numbers with an even number of steps in their prime index chain.

Original entry on oeis.org

3, 11, 17, 41, 67, 83, 109, 127, 157, 191, 211, 241, 277, 283, 353, 367, 401, 461, 509, 547, 563, 587, 617, 739, 773, 797, 859, 877, 967, 991, 1031, 1063, 1087, 1171, 1201, 1217, 1409, 1433, 1447, 1471, 1499, 1597, 1621, 1669, 1723, 1741, 1823, 1913, 2027, 2063, 2081, 2099, 2221, 2269, 2341, 2351

Offset: 1

Zak Seidov and Robert G. Wilson v, Sep 17 2015

nonn

Old (incorrect) name was: Primes not appearing in A121543.

Number of terms less than 10^n: 1, 6, 30, 165, 1024, ... .

			11 is a term: 11 -> 5 -> 3 -> 2 -> 1, four (an even number of) steps "->" = pi = A000720.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1025 terms from Zak Seidov and Robert G. Wilson v)
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, Application of the Inclusion-Exclusion Principle to Prime Number Subsequences, arXiv:2402.13214 [math.GM], 2024.

Cf. A000040, A000720, A078442, A121543, A333242 (complement in primes).

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)::even then break fi
      od; p
    end: a(1):=3:
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 15 2020

fQ[n_] := If[ !PrimeQ[n] || (PrimeQ[n] && FreeQ[lst, PrimePi[n]]), AppendTo[lst, n]]; k = 2; lst = {1}; While[k < 2401, fQ@ k; k++]; Select[lst, PrimeQ]

b(n)={my(k=0); while(isprime(n), k++; n=primepi(n)); k};
apply(prime, select(n->b(n)%2, [1..500])) \\ Michel Marcus, Jan 03 2022; after A333242

From Alois P. Heinz, Mar 15 2020: (Start)
{ p in primes : A078442(p) mod 2 = 0 }.
a(n) = prime(A333242(n)). (End)

New name from Alois P. Heinz, Mar 15 2020

A250249 Permutation of natural numbers: a(1) = 1, a(n) = A083221(a(A055396(n)), a(A246277(n))).

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, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 39, 34, 35, 36, 37, 38, 63, 40, 41, 54, 43, 44, 33, 46, 47, 48, 49, 50, 75, 52, 53, 42, 65, 56, 99, 58, 59, 60, 61, 62, 57, 64, 95, 78, 67, 68, 111, 70, 71, 72, 103, 74, 51, 76, 77, 126, 79, 80, 45, 82

Offset: 1

Antti Karttunen, Nov 17 2014

nonn

This is a "doubly-recursed" version of A249817.
For primes p_n, a(p_n) = p_{a(n)}.
The first 7-cycle occurs at: (33 39 63 57 99 81 45), which is mirrored by the cycle (66 78 126 114 198 162 90) with terms double the size and also by the cycle (137 167 307 269 523 419 197), consisting of primes (p_33, p_39, p_63, ...).

			For n = 42 = 2*3*7, we see that it occurs as the 21st term on the top row of A246278 (A055396(42) = 1 and A246277(42) = 21), recursing on both yields a(1) = 1, a(21) = 27, thus we find A083221(1,27), the 27th term on A083221's topmost row (also A005843) which is 54, thus a(42) = 54.
Examples for cases where n is a prime:
a(3709) = a(p_518) = p_{a(518)} = A000040(1162) = 9397.
a(3719) = a(p_519) = p_{a(519)} = A000040(1839) = 15767.

Antti Karttunen, Table of n, a(n) for n = 1..3718
Index entries for sequences that are permutations of the natural numbers

Inverse: A250250.
Fixed points: A250251, their complement: A249729.
Cf. A000035, A000040, A055396, A078442, A083221, A049076.
Differs from A250250 for the first time at n=33, where a(33) = 39, while A250250(33) = 45.
Differs from the "vanilla version" A249817 for the first time at n=42, where a(42) = 54, while A249817(42) = 42.

a(1) = 1, a(n) = A083221(a(A055396(n)), a(A246277(n))).
Other identities. For all n >= 1:
a(2n) = 2*a(n), or equally, a(n) = a(2n)/2. [The even bisection halved gives the sequence back].
a(p_n) = p_{a(n)}, or equally, a(n) = A049084(a(A000040(n))). [Restriction to primes induces the same sequence].
A078442(a(n)) = A078442(n), A049076(a(n)) = A049076(n). [Preserves the "order of primeness of n"].
A000035(n) = A000035(a(n)). [Preserves the parity].

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].

A250250 Permutation of natural numbers: a(1) = 1, a(n) = A246278(a(A055396(n)),a(A078898(n))).

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, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 45, 34, 35, 36, 37, 38, 33, 40, 41, 54, 43, 44, 81, 46, 47, 48, 49, 50, 75, 52, 53, 42, 125, 56, 63, 58, 59, 60, 61, 62, 39, 64, 55, 90, 67, 68, 135, 70, 71, 72, 103, 74, 51, 76, 77, 66, 79, 80, 99, 82, 83

Offset: 1

Antti Karttunen, Nov 17 2014

nonn

This is a "doubly-recursed" version of A249818.

Antti Karttunen, Table of n, a(n) for n = 1..3432
Index entries for sequences that are permutations of the natural numbers

Inverse: A250249.
Fixed points: A250251, their complement: A249729.
Cf. A000035, A000040, A055396, A078442, A078898, A049076, A246278.
See also other (somewhat) similar permutations: A245821, A057505.
Differs from the "vanilla version" A249818 for the first time at n=42, where a(42) = 54, while A249818(42) = 42.

a(1) = 1, a(n) = A246278(a(A055396(n)), a(A078898(n))).
Other identities. For all n >= 1:
a(2n) = 2*a(n), or equally, a(n) = a(2n)/2. [The even bisection halved gives the sequence back].
a(p_n) = p_{a(n)}, or equally, a(n) = A049084(a(A000040(n))). [Restriction to primes induces the same sequence].
A078442(a(n)) = A078442(n), A049076(a(n)) = A049076(n). [Preserves the "order of primeness of n"].
A000035(n) = A000035(a(n)). [Preserves the parity].

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].

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

A246681 Permutation of natural numbers: a(0) = 1, a(1) = 2, a(p_n) = A003961(a(n)), a(c_n) = 2*a(n), where p_n = n-th prime = A000040(n), c_n = n-th composite number = A002808(n), and A003961(n) shifts the prime factorization of n one step towards larger primes.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 9, 10, 8, 14, 11, 12, 15, 18, 20, 16, 25, 28, 21, 22, 24, 30, 27, 36, 40, 32, 50, 56, 33, 42, 13, 44, 48, 60, 54, 72, 45, 80, 64, 100, 35, 112, 75, 66, 84, 26, 63, 88, 96, 120, 108, 144, 81, 90, 160, 128, 200, 70, 49, 224, 99, 150, 132, 168, 52, 126, 55, 176, 192, 240, 39

Offset: 0

Antti Karttunen, Sep 01 2014

Note the indexing: the domain starts from 0, while the range excludes zero.
Iterating a(n) from n=0 gives the sequence: 1, 2, 3, 5, 7, 9, 8, 10, 14, 18, 28, 56, 128, 156, 1344, 16524, 2706412500, ..., which is the only one-way cycle of this permutation.
Because 2 is the only even prime, it implies that, apart from a(0)=1 and a(2)=3, odd numbers occur in odd positions only (along with many even numbers that also occur in odd positions). This in turn implies that there exists an infinite number of infinite cycles like (... 648391 31 13 15 20 22 30 42 112 196 1350 ...) which contain just one odd composite (A071904). Apart from 9 which is in that one-way cycle, each odd composite occurs in a separate infinite two-way cycle, like 15 in the example above.

Antti Karttunen, Table of n, a(n) for n = 0..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A246682.
Similar or related permutations: A163511, A246377, A246379, A246367, A245821.
Cf. A000040, A000720, A003961, A007097, A010051, A065855, A071904, A078442, A049076, A055396.

a(0) = 1, a(1) = 2, and for n > 1, if A010051(n) = 1 [i.e. when n is a prime], a(n) = A003961(a(A000720(n))), otherwise a(n) = 2*a(A065855(n)).
Other identities.
For all n >= 0, the following holds:
a(A007097(n)) = A000040(n+1). [Maps the iterates of primes to primes].
A078442(a(n)) > 0 if and only if n is in A007097. [Follows from above].
For all n >= 1, the following holds:
a(n) = A163511(A246377(n)).
A000035(a(n)) = A010051(n). [Maps primes to odd numbers > 1, and composites to even numbers, in some order. Permutations A246377 & A246379 have the same property].
A055396(a(n)) = A049076(n). [An "order of primeness" is mapped to the index of the smallest prime dividing n].

cp's OEIS Frontend

A061775 Number of nodes in rooted tree with Matula-Goebel number n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

Extensions

A049076 Number of steps in the prime index chain for the n-th prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A135141 a(1)=1, a(p_n)=2*a(n), a(c_n)=2*a(n)+1, where p_n = n-th prime, c_n = n-th composite number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Maxima

PARI

Python

Formula

A262275 Prime numbers with an even number of steps in their prime index chain.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A250249 Permutation of natural numbers: a(1) = 1, a(n) = A083221(a(A055396(n)), a(A246277(n))).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Scheme

Formula

A135141 a(1)=1, a(p_n)=2a(n), a(c_n)=2a(n)+1, where p_n = n-th prime, c_n = n-th composite number.