A091204 -id:A091204 - cp's OEIS frontend

A235199 Self-inverse and multiplicative permutation of integers: For n < 4, a(n)=n, a(5)=7 and a(7)=5, a(p_i) = p_{a(i)} for primes with index i > 4, and a(u * v) = a(u) * a(v) for u, v > 0.

0, 1, 2, 3, 4, 7, 6, 5, 8, 9, 14, 17, 12, 13, 10, 21, 16, 11, 18, 19, 28, 15, 34, 23, 24, 49, 26, 27, 20, 43, 42, 59, 32, 51, 22, 35, 36, 37, 38, 39, 56, 41, 30, 29, 68, 63, 46, 73, 48, 25, 98, 33, 52, 53, 54, 119, 40, 57, 86, 31, 84, 61, 118, 45, 64, 91, 102

Offset: 0

Antti Karttunen, Jan 04 2014

The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n except n=3 or 4, and is self-inverse. It swaps 5 & 7, maps all larger primes p_i (with index i > 4) to p_{a(i)}, and lets the multiplicativity take care of the rest.
It can be viewed also as a "signature-permutation" for a bijection of non-oriented rooted trees, mapped through Matula-Goebel numbers (cf. A061773). The bijection will swap the subtrees encoded by primes 5 and 7, wherever they occur as the terminal branches of the tree:
....................
.o..................
.|..................
.o.............o...o
.|..............\./.
.o.....<--->.....o..
.|...............|..
.x...............x..
.5...............7..
That is, any branch which ends at least in three edges long unbranched stem, will be changed so that its last two edges will become V-branch. Vice versa, any branch of the tree that ends with three edges in Y-formation, will be transformed so that those three edges will be straightened to an unbranching stem of three edges.
This permutation commutes with A235201, i.e. a(A235201(n)) = A235201(a(n)) for all n.
Permutation fixes n! for n=0, 1, 2, 3, 4, 7, 8 and 9.
Note also that a(5!) = a(120) = 168 = 120+(2*4!) and a(10!) = 5080320 = 3628800+(4*9!).

Composition with A234840 gives A234743 & A234744.
List below gives similarly constructed permutations, which all force a swap of two small numbers, with (the rest of) primes permuted with the sequence itself and the new positions of composite numbers defined by the multiplicative property:
A234840 (swaps 2 & 3, conjugates A008578 back to itself).
A235200 (swaps 3 & 5, conjugates A065091 back to itself).
A235201 (swaps 3 & 4, conjugates A000040 back to itself).
A235487 (swaps 7 & 8, conjugates A000040 back to itself).
A235489 (swaps 8 & 9, conjugates A000040 back to itself).
Cf. also A000040, A010051, A000720, A235485/A235486, A235493/A235494, A091204/A091205, A072026, A061773.

For n < 4, a(n)=n, a(5)=7 and a(7)=5, a(p_i) = p_{a(i)} for primes with index i > 4, and a(u * v) = a(u) * a(v) for u, v > 0.

A000035(a(n)) = A000035(n) = (n mod 2) for all n. [Even terms occur only on even indices and odd terms only on odd indices, respectively]

A235489 Self-inverse and multiplicative permutation of integers: For n < 8, a(n) = n, a(8)=9 and a(9)=8, a(p_i) = p_{a(i)} for primes with index i, and for composites > 9, a(uv) = a(u) a(v).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 8, 10, 11, 12, 13, 14, 15, 18, 17, 16, 23, 20, 21, 22, 19, 27, 25, 26, 24, 28, 29, 30, 31, 36, 33, 34, 35, 32, 37, 46, 39, 45, 41, 42, 43, 44, 40, 38, 47, 54, 49, 50, 51, 52, 61, 48, 55, 63, 69, 58, 59, 60, 53, 62, 56, 81, 65, 66, 83, 68, 57, 70, 71, 72, 73, 74, 75, 92, 77, 78, 79, 90, 64

Offset: 0

Antti Karttunen, Jan 11 2014

The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n, and is self-inverse. It swaps 8 & 9, maps any prime p_i with index i to p_{a(i)}, and lets the multiplicativity take care of the rest.
This can be viewed also as a "signature-permutation" for a bijection of non-oriented rooted trees, mapped through Matula-Goebel numbers (cf. A061773). This bijection will swap the subtrees encoded by numbers 8 and 9, wherever they occur as the terminal branches of the tree:
.......................
.................o...o.
.................|...|.
.o.o.o...........o...o.
..\|/.............\./..
...x.....<--->.....x...
...8...............9...
Thus, any terminal configuration in the tree that consists of three or more single edges next to each other (in "parallel") will be transformed so that maximal 3k number of those single edges will be replaced by k subtrees Matula-Goebel-encoded by 9 (see above, or equally: replaced by 2k two-edges-long branches encoded by 3), and one or two left-over single edges, if present, will stay as they are. Vice versa, any terminal configuration in the tree that consists of more than one two-edges-long branches next to each other (in "parallel") will be transformed so that maximal even number (2k) of those double-edges will be replaced by 3k single edges, and an extra odd double-edge, if present, will stay as it is.
Note how in contrast to A235487, A235201 and A235199, this bijection is not size-preserving (the number of edges will change), which has implications when composing this with other such permutations (cf. e.g. A235493/A235494).

Composition with A235201 gives A235493/A235494.
List below gives similarly constructed permutations, which all force a swap of two small numbers, with (the rest of) primes permuted with the sequence itself and the new positions of composite numbers defined by the multiplicative property:
A234840 (swaps 2 & 3, conjugates A008578 back to itself).
A235200 (swaps 3 & 5, conjugates A065091 back to itself).
A235201 (swaps 3 & 4, conjugates A000040 back to itself).
A235199 (swaps 5 & 7, conjugates A000040 back to itself).
A235487 (swaps 7 & 8, conjugates A000040 back to itself).
Cf. also A000040, A010051, A000720, A091204/A091205, A061773.

Multiplicative with a(3^(2k)) = 2^3k = 8^k, a(3^(2k+1)) = 3*2^3k, a(2^(3k)) = 3^2k = 9^k, a(2^(3k+1)) = 2*9^k, a(2^(3k+2)) = 4*9^k, a(p_i) = p_{a(i)} for primes with index i, and a(u*v) = a(u) * a(v) for composites other than 8 or 9.

A235487 Self-inverse and multiplicative permutation of integers: For n < 7, a(n)=n, a(7)=8 and a(8)=7, a(p_i) = p_{a(i)} for primes with index i <> 4, and for composites > 8, a(uv) = a(u) a(v).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 11, 12, 13, 16, 15, 14, 19, 18, 17, 20, 24, 22, 23, 21, 25, 26, 27, 32, 29, 30, 31, 28, 33, 38, 40, 36, 37, 34, 39, 35, 41, 48, 53, 44, 45, 46, 47, 42, 64, 50, 57, 52, 43, 54, 55, 56, 51, 58, 67, 60, 61, 62, 72, 49, 65, 66, 59, 76, 69, 80, 71, 63, 89, 74, 75, 68, 88, 78, 79, 70, 81

Offset: 0

Antti Karttunen, Jan 11 2014

The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n except n=4, and is self-inverse. It swaps 7 & 8, maps any prime p_i with index i > 4 to p_{a(i)}, and lets the multiplicativity take care of the rest.
This can be viewed also as a "signature-permutation" for a bijection on non-oriented rooted trees, mapped through the Matula-Goebel numbers (cf. A061773). This bijection will swap the subtrees encoded by numbers 7 and 8, wherever they occur as the terminal configurations anywhere in the tree:
.......................
.o...o.................
..\./..................
...o.............o.o.o.
...|..............\|/..
...x.....<--->.....x...
...7...............8...
Thus any branch of the tree that ends with three edges in Y-formation, will be transformed so that those three edges will emanate "in parallel" from the same vertex. Vice versa, any terminal configuration in the tree that consists of more than two single edges next to each other (in "parallel") will be transformed so that maximal 3k number of those single edges will be transformed to k Y-formations, and one or two left-over edges, if present, will stay as they are.

Composition with A235201 gives A235485/A235486.
List below gives similarly constructed permutations, which all force a swap of two small numbers, with (the rest of) primes permuted with the sequence itself and the new positions of composite numbers defined by the multiplicative property:
A234840 (swaps 2 & 3, conjugates A008578 back to itself).
A235200 (swaps 3 & 5, conjugates A065091 back to itself).
A235201 (swaps 3 & 4, conjugates A000040 back to itself).
A235199 (swaps 5 & 7, conjugates A000040 back to itself).
A235489 (swaps 8 & 9, conjugates A000040 back to itself).
Cf. also A000040, A010051, A000720, A091204/A091205, A061773.

Multiplicative with a(p_i) = p_{a(i)} for primes with index i <> 4, a(7) = 8, a(2^(3k)) = 7^k, a(2^(3k+1)) = 2*7^k, a(2^(3k+2)) = 4*7^k, and for other composites, a(u * v) = a(u) * a(v).

A091230 Iterates of A014580, starting with a(0) = 1, a(n) = A014580^(n)(1). [Here A014580^(n) means the n-th fold application of A014580].

Original entry on oeis.org

1, 2, 3, 7, 25, 137, 1123, 13103, 204045, 4050293, 99440273

Offset: 0

Antti Karttunen, Jan 03 2004

nonn

Cf. A000079, A007097, A091238, A091204-A091205, A245701-A245702, A245703-A245704.

isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
prev=1; i=0; print1(1, ", "); for(n=1, 123456789, if(isA014580(n), i++; if((i == prev), print1(n, ", "); prev=n))) \\ Antti Karttunen, Aug 02 2014

a(0)=1, a(n) = A014580(a(n-1)). [The defining recurrence].
From Antti Karttunen, Aug 03 2014: (Start)
Other identities. For all n >= 0, the following holds:
A091238(a(n)) = n+1.
a(n) = A091204(A007097(n)) and A091205(a(n)) = A007097(n).
a(n) = A245703(A007097(n)) and A245704(a(n)) = A007097(n).
a(n) = A245702(A000079(n)) and A245701(a(n)) = A000079(n).
(End)

Terms a(8)-a(10) computed by Antti Karttunen, Aug 02 2014

A245816 Permutation of natural numbers induced when A245822 is restricted to nonprime numbers: a(n) = A062298(A245822(A018252(n))).

Original entry on oeis.org

1, 2, 4, 5, 3, 10, 6, 22, 7, 16, 9, 23, 27, 51, 15, 17, 35, 13, 37, 11, 39, 56, 69, 38, 14, 18, 48, 78, 33, 120, 20, 19, 46, 67, 24, 62, 42, 34, 28, 73, 25, 103, 31, 206, 40, 55, 68, 92, 300, 26, 76, 50, 99, 65, 157, 281, 165, 184, 8, 121, 134, 277, 423, 30, 47, 36, 223, 70, 514, 75, 101, 116, 236, 139, 74

Offset: 1

Antti Karttunen, Aug 02 2014

nonn

This permutation is induced when A245822 is restricted to nonprimes, A018252, the first column of A114537, but equally, when it is restricted to column 2 (A007821), column 3 (A049078), etc. of that square array, or alternatively, to the successive rows of A236542.

The sequence of fixed points f(n) begins as 1, 2, 15, 142, 548, 1694, 54681. A018252(f(n)) gives the nonprime terms of A245823.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245815.
Related permutations: A245814, A245820, A245822.
Cf. A007821, A018252, A049078, A062298, A114537, A236542, A245823.

allocatemem(123456789);
default(primelimit, 2^22)
A014580 = vector(2^18);
A091226 = vector(2^22);
A002808(n)={my(k); for(k=0, primepi(n), isprime(n++)&&k--); n}; \\ This function from M. F. Hasler, Oct 31 2008
A062298(n) = n-primepi(n);
A018252(n) = if(1==n,1,A002808(n-1));
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++; A014580[i] = n; A091226[n] = A091226[n-1]+1, A091226[n] = A091226[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));
A245816(n) = A062298(A245822(A018252(n)));
for(n=1, 10001, write("b245816.txt", n, " ", A245816(n)));

(define (A245816 n) (A062298 (A245822 (A018252 n))))

a(n) = A062298(A245822(A018252(n))).
As a composition of related permutations:
a(n) = A245820(A245814(n)).
Also following holds for all n >= 1:
a(n) = A062298(A049084(A245822(A007821(n)))).
a(n) = A062298(A049084(A049084(A245822(A049078(n))))).
etc.

cp's OEIS Frontend

A235199 Self-inverse and multiplicative permutation of integers: For n < 4, a(n)=n, a(5)=7 and a(7)=5, a(p_i) = p_{a(i)} for primes with index i > 4, and a(u * v) = a(u) * a(v) for u, v > 0.

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A235489 Self-inverse and multiplicative permutation of integers: For n < 8, a(n) = n, a(8)=9 and a(9)=8, a(p_i) = p_{a(i)} for primes with index i, and for composites > 9, a(u*v) = a(u) * a(v).

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A235487 Self-inverse and multiplicative permutation of integers: For n < 7, a(n)=n, a(7)=8 and a(8)=7, a(p_i) = p_{a(i)} for primes with index i <> 4, and for composites > 8, a(u*v) = a(u) * a(v).

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A091230 Iterates of A014580, starting with a(0) = 1, a(n) = A014580^(n)(1). [Here A014580^(n) means the n-th fold application of A014580].

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A245816 Permutation of natural numbers induced when A245822 is restricted to nonprime numbers: a(n) = A062298(A245822(A018252(n))).

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A235489 Self-inverse and multiplicative permutation of integers: For n < 8, a(n) = n, a(8)=9 and a(9)=8, a(p_i) = p_{a(i)} for primes with index i, and for composites > 9, a(uv) = a(u) a(v).

A235487 Self-inverse and multiplicative permutation of integers: For n < 7, a(n)=n, a(7)=8 and a(8)=7, a(p_i) = p_{a(i)} for primes with index i <> 4, and for composites > 8, a(uv) = a(u) a(v).