A246364 -id:A246364 - cp's OEIS frontend

A246363 Permutation of natural numbers: a(n) = A135141(A048673(n)).

1, 2, 4, 8, 3, 9, 5, 13, 10, 16, 6, 14, 7, 12, 35, 20, 17, 79, 11, 67, 71, 33, 19, 271, 39, 31, 139, 87, 15, 30, 18, 311, 47, 34, 63, 74, 23, 29, 26, 351, 21, 28, 27, 24, 303, 69, 25, 2431, 70, 223, 135, 319, 37, 1663, 65, 58, 41, 38, 32, 219, 43, 127, 367, 327, 287, 239, 55, 107, 46, 283, 22, 413, 51, 53, 147

Offset: 1

Antti Karttunen, Aug 26 2014

nonn

Apart from 2, even numbers occur only in positions given by A246261 (together with some odd numbers).

Also, apart from A246263(1) = 2, the positions given by the rest of A246263: 5, 6, 7, 8, 15, 17, 18, 19, 20, 21, ... contain odd numbers only.

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

Inverse: A246364.
Related or similar permutations: A048673, A135141, A246365, A246367.
Cf. A000035, A003961, A246260, A246261, A246263.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A048673(n) = (A003961(n)+1)/2;
A135141(n) = if(1==n, 1, if(isprime(n), 2*A135141(primepi(n)), 1+(2*A135141(n-primepi(n)-1))));
A246363(n) = A135141(A048673(n));
for(n=1, 10000, write("b246363.txt", n, " ", A246363(n)));

(define (A246363 n) (A135141 (A048673 n)))

a(n) = A135141(A048673(n)).

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

Original entry on oeis.org

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].

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].

A246366 Permutation of natural numbers: a(n) = A005941(A227413(n)).

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 7, 5, 8, 33, 12, 65, 18, 257, 16, 17, 10, 129, 11, 4097, 34, 2049, 19, 65537, 15, 8193, 24, 4194305, 21, 32769, 66, 1025, 20, 513, 14, 262145, 22, 16385, 13, 1099511627777, 1026, 2097153, 130, 68719476737, 30, 1048577, 35, 288230376151711745, 8194, 67108865, 40, 4398046511105, 2050, 8388609, 28

Offset: 1

Antti Karttunen, Aug 26 2014

nonn

Maps even numbers to terms of A000051 (2^n + 1) in some order.

Antti Karttunen, Table of n, a(n) for n = 1..95
Index entries for sequences that are permutations of the natural numbers

Inverse: A246365.
Related or similar permutations: A005941, A156552, A227413, A246364, A246368.
Cf. A000051.

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))));
A246366(n) = A005941(A227413(n));
for(n=1, 95, write("b246366.txt", n, " ", A246366(n)));

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

a(n) = A005941(A227413(n)) = 1 + A156552(A227413(n)).

A246379 Permutation of natural numbers: a(1) = 1, a(p_n) = A003961(1+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, 3, 9, 2, 21, 6, 5, 18, 4, 42, 39, 12, 11, 10, 36, 8, 15, 84, 23, 78, 24, 22, 7, 20, 72, 16, 30, 168, 47, 46, 189, 156, 48, 44, 14, 40, 17, 144, 32, 60, 45, 336, 13, 94, 92, 378, 41, 312, 96, 88, 28, 80, 25, 34, 288, 64, 120, 90, 81, 672, 133, 26, 188, 184, 756, 82, 135, 624, 192, 176, 83, 56, 49

Offset: 1

Antti Karttunen, Aug 29 2014

nonn

Because 2 is the only even prime, it implies that, apart from 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 each odd composite (A071904) resides in a separate infinite cycle in this permutation, except 9, which is in a finite cycle (2 3 9 4).

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

Inverse: A246380.
Similar or related permutations: A246375, A246377, A246363, A246364, A246365, A246367, A246681.
Cf. A000720, A003961, A010051, A065855, A071904.

default(primelimit, (2^31)+(2^30));
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A246379(n) = if(1==n, 1, if(isprime(n), A003961(1+A246379(primepi(n))), 2*A246379(n-primepi(n)-1)));
for(n=1, 10000, write("b246379.txt", n, " ", A246379(n)));
(Scheme, with memoization-macro definec)
(definec (A246379 n) (cond ((< n 2) n) ((= 1 (A010051 n)) (A003961 (+ 1 (A246379 (A000720 n))))) (else (* 2 (A246379 (A065855 n))))))

a(1) = 1, and for n > 1, if A010051(n) = 1 [i.e. when n is a prime], a(n) = A003961(1+a(A000720(n))), otherwise a(n) = 2*a(A065855(n)).
As a composition of related permutations:
a(n) = A246375(A246377(n)).
Other identities. For all n > 1 the following holds:
A000035(a(n)) = A010051(n). [Maps primes to odd numbers > 1, and composites to even numbers, in some order. Permutations A246377 & A246681 have the same property].

cp's OEIS Frontend

A246363 Permutation of natural numbers: a(n) = A135141(A048673(n)).

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

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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.

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A246366 Permutation of natural numbers: a(n) = A005941(A227413(n)).

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A246379 Permutation of natural numbers: a(1) = 1, a(p_n) = A003961(1+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.

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