A246366 -id:A246366 - cp's OEIS frontend

A246368 Permutation of natural numbers: a(n) = A227413(A005941(n)).

1, 2, 4, 3, 6, 7, 8, 5, 9, 13, 10, 17, 20, 19, 12, 11, 46, 23, 166, 41, 15, 29, 858, 59, 14, 71, 16, 67, 6186, 37, 58645, 31, 18, 199, 22, 83, 705348, 983, 32, 179, 10428487, 47, 184718194, 109, 21, 6659, 3840230006, 277, 27, 43, 65, 353

Offset: 1

Antti Karttunen, Aug 26 2014

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse: A246367.
Similar or related permutations: A005941, A156552, A227413, A246364, A246366.
Cf. A000035, A010051.

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
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A005941(n) = A156552(n)+1;
A227413(n) = if(1==n, 1, if(!(n%2), prime(A227413(n/2)), A002808(A227413((n-1)/2))));
A246368(n) = A227413(A005941(n));
for(n=1, 52, write("b246368.txt", n, " ", A246368(n)));

(define (A246368 n) (A227413 (A005941 n)))

a(n) = A227413(A005941(n)) = A227413(1+A156552(n)).
Other identities:
For all n >= 1,  A010051(a(n)) = 1 - A000035(n). [This permutation maps even numbers to primes and odd numbers to nonprimes, in some order, because the permutation A227413 has the same property and A005941 preserves the parity].

A246364 Permutation of natural numbers: a(n) = A064216(A227413(n)).

Original entry on oeis.org

1, 2, 5, 3, 7, 11, 13, 4, 6, 9, 19, 14, 8, 12, 29, 10, 17, 31, 23, 16, 41, 71, 37, 44, 47, 39, 43, 42, 38, 30, 26, 59, 22, 34, 15, 85, 53, 58, 25, 130, 57, 151, 61, 311, 103, 69, 33, 365, 157, 111, 73, 226, 74, 106, 67, 370, 223, 56, 97, 341, 139, 122, 35, 133, 55, 86, 20, 145, 46, 49, 21, 659, 118, 36, 83, 419, 127, 191, 18

Offset: 1

Antti Karttunen, Aug 26 2014

nonn

After a(2) = 2, the rest of the even bisection contains only terms of A246261. However, some of the terms of A246261 are also found in the odd bisection, while terms of A246263, apart from 2, all reside in the odd bisection of this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..4095
Index entries for sequences that are permutations of the natural numbers

Inverse: A246363.
Related or similar permutations: A064216, A227413, A246366, A246368.
Cf. A246261, A246263.

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
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A064216(n) = A064989((2*n)-1);
A227413(n) = if(1==n, 1, if(!(n%2), prime(A227413(n/2)), A002808(A227413((n-1)/2))));
A246364(n) = A064216(A227413(n));
for(n=1, 4095, write("b246364.txt", n, " ", A246364(n)));

(define (A246364 n) (A064216 (A227413 n)))

a(n) = A064216(A227413(n)).

A246380 Permutation of natural numbers: a(1) = 1, a(2n) = nthcomposite(a(n)), a(2n-1) = nthprime(a(A064989(2n-1)-1)), where nthprime = A000040, nthcomposite = A002808, and A064989(n) shifts the prime factorization of n one step towards smaller primes.

Original entry on oeis.org

1, 4, 2, 9, 7, 6, 23, 16, 3, 14, 13, 12, 43, 35, 17, 26, 37, 8, 101, 24, 5, 22, 19, 21, 53, 62, 83, 51, 79, 27, 233, 39, 191, 54, 149, 15, 103, 134, 11, 36, 47, 10, 151, 34, 41, 30, 29, 33, 73, 75, 241, 86, 113, 114, 89, 72, 1153, 108, 443, 40, 593, 296, 547, 56, 167, 245, 173, 76, 563, 194, 1553, 25

Offset: 1

Antti Karttunen, Aug 29 2014

nonn

Has an infinite number of infinite cycles. See comments in A246379.

Antti Karttunen, Table of n, a(n) for n = 1..3098
Index entries for sequences that are permutations of the natural numbers

Inverse: A246379.
Similar or related permutations: A246376, A246378, A246363, A246364, A246366, A246368, A064216, A246682.
Cf. A000035, A000040, A002808, A010051, A064989.

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
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A246380(n) = if(1==n, 1, if(!(n%2), A002808(A246380(n/2)), prime(A246380(A064989(n)-1))));
for(n=1, 3098, write("b246380.txt", n, " ", A246380(n)));
(Scheme, with memoization-macro definec)
(definec (A246380 n) (cond ((< n 2) n) ((even? n) (A002808 (A246380 (/ n 2)))) (else (A000040 (A246380 (- (A064989 n) 1))))))

a(1) = 1, a(2n) = nthcomposite(a(n)), a(2n-1) = nthprime(a(A064989(2n-1)-1)), where nthprime = A000040, nthcomposite = A002808, and A064989(n) shifts the prime factorization of n one step towards smaller primes.
As a composition of related permutations:
a(n) = A246378(A246376(n)).
Other identities. For all n > 1 the following holds:
A010051(a(n)) = A000035(n). [Maps odd numbers larger than one to primes, and even numbers to composites, in some order. Permutations A246378 & A246682 have the same property].

A246365 Permutation of natural numbers: a(n) = A135141(A005940(n)).

Original entry on oeis.org

1, 2, 4, 3, 8, 5, 7, 9, 6, 17, 19, 11, 39, 35, 25, 15, 16, 13, 23, 33, 29, 37, 75, 27, 95, 87, 61, 55, 767, 45, 83, 67, 10, 21, 47, 71, 159, 143, 139, 51, 319, 175, 639, 287, 251, 263, 247, 135, 527, 239, 199, 447, 105, 115, 991, 119, 1015, 443, 4575, 85, 583, 2175, 1343, 151, 12, 31, 63, 69, 131, 77

Offset: 1

Antti Karttunen, Aug 26 2014

nonn

Even terms occur at the positions 2^n + 1 (A000051), in some order, and the odd terms everywhere else.

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

Inverse: A246366.
Related or similar permutations: A005940, A135141, A246363, A246367.
Cf. A000051, A003961.

default(primelimit,(2^31)+(2^30));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
A135141(n) = if(1==n, 1, if(isprime(n), 2*A135141(primepi(n)), 1+(2*A135141(n-primepi(n)-1))));
A246365(n) = A135141(A005940(n));
for(n=1, 10000, write("b246365.txt", n, " ", A246365(n)));

(define (A246365 n) (A135141 (A005940 n)))

a(n) = A135141(A005940(n)).

cp's OEIS Frontend

A246368 Permutation of natural numbers: a(n) = A227413(A005941(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Scheme

Formula

A246364 Permutation of natural numbers: a(n) = A064216(A227413(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Scheme

Formula

A246380 Permutation of natural numbers: a(1) = 1, a(2n) = nthcomposite(a(n)), a(2n-1) = nthprime(a(A064989(2n-1)-1)), where nthprime = A000040, nthcomposite = A002808, and A064989(n) shifts the prime factorization of n one step towards smaller primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A246365 Permutation of natural numbers: a(n) = A135141(A005940(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Scheme

Formula