A257729 Permutation of natural numbers: a(1)=1; a(prime(n)) = oddprime(a(n)), a(composite(n)) = not_an_oddprime(1+a(n)).

1, 3, 7, 2, 19, 6, 5, 12, 4, 28, 71, 10, 17, 9, 20, 8, 13, 40, 41, 95, 16, 26, 11, 15, 30, 14, 21, 56, 109, 57, 359, 125, 25, 38, 18, 24, 31, 44, 22, 32, 61, 77, 29, 143, 78, 445, 73, 162, 36, 54, 27, 35, 23, 45, 62, 33, 46, 84, 43, 104, 179, 42, 185, 105, 545, 98, 181, 208, 51, 75, 503, 39, 59, 50, 34, 63, 85, 48, 103, 64, 114, 60, 37

Offset

1,2

Comments

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).

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences that are permutations of the natural numbers

Formula

a(1) = 1; if A010051(n) = 1 [i.e., if n is a prime], then a(n) = A065091(a(A000720(n))), otherwise a(n) = A065090(1+a(A065855(n))).

As a composition of other permutations:

a(n) = A257728(A246377(n)).

a(n) = A257802(A257731(n)).

Example

As an initial value we have a(1) = 1.
2 is the first prime (= A000040(1)), so we take the a(1)-th odd prime, A065091(1) = 3, thus a(2) = 3.
3 is the second prime, thus we take a(2)-th odd prime, A065091(3) = 7, thus a(3) = 7.
4 is the first composite, thus we take a(1)-th number larger than one which is not an odd prime, and that is A065090(1+1) = 2, thus a(4) = 2.
5 is the third prime, thus we take a(3)-th odd prime, which is A065091(7) = 19, thus a(5) = 19.

Prog

(PARI)
A002808(n) = { my(k=-1); while( -n + n += -k + k=primepi(n), ); n};
A257729(n) = { if(1==n, n, if(isprime(n), prime(1+A257729(primepi(n))), if(4==n, 2, A002808(A257729(n-primepi(n)-1)-1))))};
for(n=1, 10000, write("b257729.txt", n, " ", A257729(n)));
\\ After M. F. Hasler's PARI-code for A236854.
(Scheme, with memoizing definec-macro)
(definec (A257729 n) (cond ((<= n 1) n) ((= 1 (A010051 n)) (A065091 (A257729 (A000720 n)))) (else (A065090 (+ 1 (A257729 (A065855 n)))))))
;; Alternatively, by composing other permutations:
(define (A257729 n) (A257728 (A246377 n)))

Crossrefs

Inverse: A257730.

Cf. A000040, A002808, A000720, A010051, A065090, A065091, A065855.

Related or similar permutations: A257728, A246377, A257731, A257802, A236854.

Sequence in context: A324713 A245611 A063041 * A328502 A308480 A347772

Adjacent sequences: A257726 A257727 A257728 * A257730 A257731 A257732

Keyword

nonn

Author

Antti Karttunen, May 09 2015

Status

approved