A246378 -id:A246378 - cp's OEIS frontend

A237739 a(0) = 1, a(2n) = nthcomposite(a(n)-1), a(2n+1) = nthprime(a(n)), where nthcomposite = A002808, nthprime = A000040.

1, 2, 4, 3, 8, 7, 6, 5, 14, 19, 12, 17, 10, 13, 9, 11, 22, 43, 28, 67, 20, 37, 26, 59, 16, 29, 21, 41, 15, 23, 18, 31, 33, 79, 60, 191, 40, 107, 91, 331, 30, 71, 52, 157, 38, 101, 81, 277, 25, 53, 42, 109, 32, 73, 57, 179, 24, 47, 34, 83, 27, 61, 45, 127, 48

Offset: 0

Reinhard Zumkeller, Apr 30 2014

A071574(a(n)) = n; a(A071574(n)) = n for n > 0.

Reinhard Zumkeller (terms 0-300) & Antti Karttunen, Table of n, a(n) for n = 0..4095
Index entries for sequences that are permutations of the natural numbers

Inverse: A071574.
Cf. A000040, A002808.
Compare also to the permutation A246378.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a237739 = fromIntegral . (+ 1) . fromJust . (`elemIndex` a071574_list)

default(primelimit,(2^31)+(2^30));
A002808(n) = { my(k=-1); while( -n + n += -k + k=primepi(n), ); n }; \\ This function from M. F. Hasler
A237739(n) = if(0==n, 1, if(!(n%2), A002808(A237739(n/2)-1), prime(A237739((n-1)/2))));
for(n=0, 4095, write("b237739.txt", n, " ", A237739(n)));
\\ Antti Karttunen, Apr 04 2015

;; With memoizing definec-macro.
(definec (A237739 n) (cond ((zero? n) 1) ((odd? n) (A000040 (A237739 (/ (- n 1) 2)))) (else (A002808 (+ -1 (A237739 (/ n 2)))))))
;; Antti Karttunen, Apr 04 2015

a(0) = 1, a(2n) = nthcomposite(a(n)-1), a(2n+1) = nthprime(a(n)), where nthcomposite = A002808, nthprime = A000040. - Antti Karttunen, Apr 04 2015

Name replaced by an explicit recurrence. - Antti Karttunen, Apr 04 2015

A257728 Permutation of natural numbers: a(1)=1; a(2n) = not_an_oddprime(1+a(n)), a(2n+1) = oddprime(a(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 11, 9, 13, 10, 17, 12, 19, 14, 23, 18, 37, 15, 29, 21, 43, 16, 31, 26, 61, 20, 41, 28, 71, 22, 47, 34, 89, 27, 67, 52, 163, 24, 53, 42, 113, 32, 79, 60, 193, 25, 59, 45, 131, 38, 103, 84, 293, 30, 73, 57, 181, 40, 109, 95, 359, 33, 83, 65, 223, 49, 149, 119, 463, 39, 107, 91, 337, 72, 241, 209, 971, 35, 97, 74, 251, 58

Offset: 1

Antti Karttunen, May 09 2015

nonn

Here oddprime(n) = n-th odd prime = A065091(n) = A000040(n+1), not_an_oddprime(n) = n-th natural number which is not an odd prime = A065090(n).
This sequence can be represented as a binary tree. Each left hand child is produced as A065090(1+n), and each right hand child as A065091(n), when a parent contains n >= 1:
                                    |
                 ...................1...................
                2                                       3
      4......../ \........5                   6......../ \........7
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  8       11          9       13         10       17         12       19
14 23   18  37      15 29   21  43     16  31   26  61     20  41   28  71
etc.
Because all odd primes are odd, it means that even terms can only occur in even positions (together with odd composites, A071904, for each one of which there is a separate infinite cycle), while terms in odd positions are all odd.

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences that are permutations of the natural numbers

Inverse: A257727.
Cf. A002808, A065090, A065091, A071904.
Related or similar permutations: A246377, A246378, A257726, A257729, A257802.
Differs from A255004 for the first time at n=17, where a(17) = 23, while A255004(17) = 15.

A002808(n) = { my(k=-1); while( -n + n += -k + k=primepi(n), ); n }; \\ This function from M. F. Hasler
A257728(n) = if(n<3, n, if(!(n%2), A002808(A257728(n/2)-1), prime(1+A257728((n-1)/2))));
for(n=1, 4096, write("b257728.txt", n, " ", A257728(n)));

;; With memoizing definec-macro.
(definec (A257728 n) (cond ((< n 2) n) ((even? n) (A065090 (+ 1 (A257728 (/ n 2))))) (else (A065091 (A257728 (/ (- n 1) 2))))))

a(1) = 1; a(2n) = A065090(1+a(n)), a(2n+1) = A065091(a(n)).
As a composition of other permutations:
a(n) = A257729(A246378(n)).
a(n) = A257802(A257726(n)).

A257730 Permutation of natural numbers: a(1)=1; a(oddprime(n)) = prime(a(n)), a(not_an_oddprime(n)) = composite(a(n-1)).

Original entry on oeis.org

1, 4, 2, 9, 7, 6, 3, 16, 14, 12, 23, 8, 17, 26, 24, 21, 13, 35, 5, 15, 27, 39, 53, 36, 33, 22, 51, 10, 43, 25, 37, 40, 56, 75, 52, 49, 83, 34, 72, 18, 19, 62, 59, 38, 54, 57, 101, 78, 102, 74, 69, 114, 89, 50, 98, 28, 30, 86, 73, 82, 41, 55, 76, 80, 134, 106, 149, 135, 100, 94, 11, 150, 47, 120, 70, 130, 42, 45, 103, 117, 99, 112, 167, 58, 77

Offset: 1

Antti Karttunen, May 09 2015

nonn

Here composite(n) = n-th composite = A002808(n), prime(n) = n-th prime = A000040(n), oddprime(n) = n-th odd prime = A065091(n) = A000040(n+1), not_an_oddprime(n) = n-th natural number which is not an odd prime = A065090(n).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A257729.
Cf. A000040, A000720, A002808, A010051, A062298, A065090, A065091.
Related or similar permutations: A246378, A257727, A257732, A257801, A236854.

a(1) = 1; if A000035(n) = 1 and A010051(n) = 1 [i.e., when n is an odd prime], then a(n) = A000040(a(A000720(n)-1)), otherwise a(n) = A002808(a(A062298(n))). [Here A062298(n) gives the index of n among numbers larger than 1 which are not odd primes, 1 for 2, 2 for 4, 3 for 6, etc.]
As a composition of other permutations:
a(n) = A246378(A257727(n)).
a(n) = A257732(A257801(n)).

A260422 a(1) = 1, a(2n) = A205783(1+a(n)), a(2n+1) = A206074(a(n)), where A206074 and A205783 give binary codes for polynomials with coefficients 0 or 1 that are irreducible [resp. reducible] over Q.

Original entry on oeis.org

1, 4, 2, 9, 7, 6, 3, 16, 23, 14, 17, 12, 13, 8, 5, 27, 47, 36, 71, 24, 41, 28, 53, 21, 31, 22, 37, 15, 19, 10, 11, 42, 81, 70, 149, 54, 109, 106, 239, 38, 73, 62, 127, 44, 83, 80, 171, 34, 67, 48, 91, 35, 69, 56, 113, 26, 43, 32, 59, 18, 25, 20, 29, 63, 131, 122, 271, 105, 233, 216, 477, 82, 173, 159, 353, 155, 347, 345, 787, 57

Offset: 1

Antti Karttunen, Jul 25 2015

This sequence can be represented as a binary tree. Each left hand child is produced as A205783(1+n), and each right hand child as A206074(n), when the parent contains n:
                                     |
                  ...................1...................
                 4                                       2
       9......../ \........7                   6......../ \........3
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  16       23         14       17         12       13          8       5
27  47   36  71     24  41   28  53     21  31   22  37      15 19   10 11
etc.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A260421.
Related permutations: A246202, A246378, A260423, A260425.
Cf. A205783, A206074.
Differs from A246378 for the first time at n=16, where a(16)=27, while A246378(16)=26.

uplim = (2^21) + (2^20);
v206074 = vector(uplim);
v205783 = vector(uplim); v205783[1] = 1;
isA206074(n) = polisirreducible(Pol(binary(n)));
i=0; j=1; n=2; while((n < uplim), if(!(n%65536),print1(n,", "));  if(isA206074(n), i++; v206074[i] = n, j++; v205783[j] = n); n++); print(n);
A260422(n) = if(1==n, 1, if(0==(n%2), v205783[1+A260422(n/2)], v206074[A260422((n-1)/2)]));
for(n=1, 8192, write("b260422.txt", n, " ", A260422(n)));

a(1) = 1, a(2n) = A205783(1+a(n)), a(2n+1) = A206074(a(n)).
As a composition of related permutations:
a(n) = A260423(A246378(n)).
a(n) = A260425(A246202(n)).

A257727 Permutation of natural numbers: a(1) = 1, a(oddprime(n)) = 1 + 2a(n), a(not_an_oddprime(n)) = 2a(n-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 9, 14, 11, 16, 20, 24, 13, 18, 15, 28, 22, 32, 17, 40, 48, 26, 36, 30, 21, 56, 25, 44, 64, 34, 80, 96, 19, 52, 72, 60, 29, 42, 23, 112, 50, 88, 33, 128, 68, 160, 192, 38, 41, 104, 144, 120, 58, 84, 49, 46, 27, 224, 100, 176, 66, 256, 37, 136, 320, 384, 31, 76, 57, 82, 208, 288, 240, 116, 45

Offset: 1

Antti Karttunen, May 09 2015

nonn

Here oddprime(n) = n-th odd prime = A065091(n) = A000040(n+1), not_an_oddprime(n) = n-th natural number which is not an odd prime = A065090(n).

			For n=2, which is the second natural number >= 1 that is not an odd prime [2 = A065090(2)], we compute 2*a(1) = 2 = a(2).
For n=4, which is A065090(3), we compute 2*a(3-1) = 2*2 = 4.
For n=5, and 5 is the second odd prime [5 = A065091(2)], thus a(5) = 1 + 2*a(2) = 5.
For n=9, which is the sixth natural number >= 1 not an odd prime (9 = A065090(6)), we compute 2*a(6-1) = 2*5 = 10.
For n=11, which is the fourth odd prime [11 = A065091(4)], we compute 1 + 2*a(4) = 1 + 2*4 = 9, thus a(11) = 9.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A257728.
Cf. A000720, A010051, A062298, A065090, A065091.
Related or similar permutations: A246377, A246378, A257725, A257730, A257801.

a(1) = 1; a(2) = 2; and for n > 2, if A010051(n) = 1 [i.e., when n is a prime], then a(n) = 1 + 2*a(A000720(n)-1), otherwise a(n) = 2*a(A062298(n)).
As a composition of other permutations:
a(n) = A246377(A257730(n)).
a(n) = A257725(A257801(n)).

A260424 a(1) = 1, a(A206074(n)) = prime(a(n)), a(A205783(1+n)) = composite(a(n)), where A206074 and A205783 give binary codes for polynomials with coefficients 0 or 1 that are irreducible [resp. reducible] over Q.

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, 21, 22, 23, 24, 29, 25, 26, 27, 31, 28, 37, 30, 32, 33, 34, 35, 41, 36, 44, 38, 43, 39, 47, 40, 46, 42, 53, 54, 45, 48, 49, 50, 59, 51, 61, 58, 52, 63, 67, 55, 71, 62, 56, 66, 57, 65, 73, 60, 79, 75, 83, 76, 89, 64, 68, 69, 109, 70, 97, 82, 101, 72, 103, 85, 81, 74, 127

Offset: 1

Antti Karttunen, Jul 25 2015

nonn

After 1, each term of A206075 resides in a separate infinite cycle. This follows because primes (A000040) is a subsequence of A206074 [see Thomas Ordowski's Feb 19 2014 comment in A206074] and thus each composite in A206074 is trapped into a trajectory containing only primes.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A260423.
Related permutations: A245704, A246378, A260421, A260426.
Cf. A000040, A002808, A205783, A206074, A206075, A257000, A255572, A255574.

allocatemem(123456789);
default(primelimit,4294965247);
uplim = 2^20;
v255574 = vector(uplim); A255574 = n -> v255574[n];
A255572 = n -> (n - A255574(n) - 1);
A257000(n) = polisirreducible(Pol(binary(n)));
v255574[1] = 0; i=0; j=0; n=2; while((n < uplim), v255574[n] = v255574[n-1]+A257000(n); n++);
A002808(n)={ my(k=-1); while( -n + n += -k + k=primepi(n), ); n}; \\ This function from M. F. Hasler
A260424(n) = if(1==n, 1, if(A257000(n), prime(A260424(A255574(n))), A002808(A260424(A255572(n)))));
for(n=1, 8192, write("b260424.txt", n, " ", A260424(n)));

a(1) = 1; for n > 1, if A257000(n) = 1 [when n is in A206074], then a(n) = A000040(a(A255574(n))), otherwise [when n is in A205783], a(n) = A002808(a(A255572(n))).
As a composition of related permutations:
a(n) = A246378(A260421(n)).
a(n) = A245704(A260426(n)).

cp's OEIS Frontend

A237739 a(0) = 1, a(2n) = nthcomposite(a(n)-1), a(2n+1) = nthprime(a(n)), where nthcomposite = A002808, nthprime = A000040.

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A257728 Permutation of natural numbers: a(1)=1; a(2n) = not_an_oddprime(1+a(n)), a(2n+1) = oddprime(a(n)).

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A257730 Permutation of natural numbers: a(1)=1; a(oddprime(n)) = prime(a(n)), a(not_an_oddprime(n)) = composite(a(n-1)).

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A260422 a(1) = 1, a(2n) = A205783(1+a(n)), a(2n+1) = A206074(a(n)), where A206074 and A205783 give binary codes for polynomials with coefficients 0 or 1 that are irreducible [resp. reducible] over Q.

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A257727 Permutation of natural numbers: a(1) = 1, a(oddprime(n)) = 1 + 2*a(n), a(not_an_oddprime(n)) = 2*a(n-1).

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A260424 a(1) = 1, a(A206074(n)) = prime(a(n)), a(A205783(1+n)) = composite(a(n)), where A206074 and A205783 give binary codes for polynomials with coefficients 0 or 1 that are irreducible [resp. reducible] over Q.

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Programs

PARI

Formula

A257727 Permutation of natural numbers: a(1) = 1, a(oddprime(n)) = 1 + 2a(n), a(not_an_oddprime(n)) = 2a(n-1).