A245815 -id:A245815 - cp's OEIS frontend

A245821 Permutation of natural numbers: a(n) = A091205(A245703(n)).

1, 2, 3, 4, 5, 9, 7, 6, 8, 12, 11, 15, 23, 81, 18, 10, 17, 30, 13, 162, 27, 36, 19, 24, 16, 25, 38, 46, 37, 45, 31, 135, 14, 20, 50, 57, 47, 69, 21, 55, 83, 115, 419, 87, 60, 210, 61, 42, 54, 26, 90, 28, 29, 35, 32, 63, 171, 52, 59, 138, 113, 180, 111, 48, 88, 39, 41, 621, 72, 22, 953, 230, 103, 207, 126, 64, 33, 243

Offset: 1

Antti Karttunen, Aug 02 2014

nonn

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

Inverse: A245822.
Other related permutations:  A091205, A245703, A245815.
Fixed points: A245823.
Cf. A000040, A049084, A078442, A049076, A018252, A062298.

allocatemem(234567890);
v014580 = vector(2^18);
v091226 = vector(2^22);
v091242 = vector(2^22);
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++; v014580[i] = n; v091226[n] = v091226[n-1]+1, j++; v091242[j] = n; v091226[n] = v091226[n-1]); n++);
A014580(n) = v014580[n];
A091226(n) = v091226[n];
A091242(n) = v091242[n];
A091205(n) = if(n<=1, n, if(isA014580(n), prime(A091205(A091226(n))), {my(irfs, t); irfs=subst(lift(factor(Mod(1, 2)*Pol(binary(n)))), x, 2); irfs[,1]=apply(t->A091205(t), irfs[,1]); factorback(irfs)}));
A245703(n) = if(1==n, 1, if(isprime(n), A014580(A245703(primepi(n))), A091242(A245703(n-primepi(n)-1))));
A245821(n) = A091205(A245703(n));
for(n=1, 10001, write("b245821.txt", n, " ", A245821(n)));

(define (A245821 n) (A091205 (A245703 n)))

a(n) = A091205(A245703(n)).
Other identities. For all n >= 1, the following holds:
A078442(a(n)) = A078442(n), A049076(a(n)) = A049076(n). [Preserves "the order of primeness of n"].
a(p_n) = p_{a(n)} where p_n is the n-th prime, A000040(n).
a(n) = A049084(a(A000040(n))). [Thus the same permutation is induced also when it is restricted to primes].
A245815(n) = A062298(a(A018252(n))). [While restriction to nonprimes induces another permutation].

A245813 Permutation of natural numbers induced when A091205 is restricted to {1} and binary codes for polynomials reducible over GF(2): a(1) = 1, a(n) = A062298(A091205(A091242(n-1))).

Original entry on oeis.org

1, 2, 5, 3, 4, 9, 11, 7, 6, 18, 10, 59, 20, 25, 16, 8, 50, 15, 32, 31, 12, 13, 38, 21, 41, 125, 85, 43, 17, 45, 52, 35, 22, 19, 103, 105, 33, 24, 14, 190, 68, 27, 66, 28, 161, 29, 80, 26, 54, 46, 177, 84, 258, 34, 180, 64, 90, 70, 507, 37, 196, 96, 39, 110, 430, 92, 78, 75, 600, 48, 40, 82, 213, 218, 71, 23, 87, 72, 51, 132, 30

Offset: 1

Antti Karttunen, Aug 16 2014

nonn

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

Inverse: A245814.
Cf. A062298, A091242.
Related permutations: A091205, A245815, A245820.

allocatemem(234567890);
v091226 = vector(2^22);
v091242 = vector(2^22);
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++; v091226[n] = v091226[n-1]+1, j++; v091242[j] = n; v091226[n] = v091226[n-1]); n++);
A062298(n) = n-primepi(n);
A091226(n) = v091226[n];
A091242(n) = v091242[n];
A091205(n) = if(n<=1, n, if(isA014580(n), prime(A091205(A091226(n))), {my(irfs, t); irfs=subst(lift(factor(Mod(1, 2)*Pol(binary(n)))), x, 2); irfs[,1]=apply(t->A091205(t), irfs[,1]); factorback(irfs)}));
A245813(n) = if(n<=1, n, A062298(A091205(A091242(n-1))));
for(n=1, 10001, write("b245813.txt", n, " ", A245813(n)));

(define (A245813 n) (if (<= n 1) n (A062298 (A091205 (A091242 (- n 1))))))

a(1) = 1, and for n > 1, a(n) = A062298(A091205(A091242(n-1))).
As a composition of related permutations:
a(n) = A245815(A245820(n)).

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.

A245819 Permutation of natural numbers induced when A245703 is restricted to nonprime numbers: a(n) = 1+A091245(A245703(A018252(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 6, 12, 7, 9, 13, 26, 10, 14, 18, 11, 15, 48, 19, 20, 35, 16, 21, 32, 25, 17, 22, 63, 27, 56, 28, 138, 46, 23, 29, 43, 34, 38, 24, 30, 80, 60, 36, 88, 72, 37, 167, 42, 59, 31, 39, 55, 45, 62, 50, 33, 40, 100, 77, 320, 47, 92, 109, 90, 49, 201, 54, 98, 76, 41, 51

Offset: 1

Antti Karttunen, Aug 16 2014

nonn

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

Inverse: A245820.
Related permutations: A245703, A245814, A245815.
Cf. A008578, A018252, A091226, A091245.

\\ The rest of code can be found in A245703.
A245819(n) = if(1==n, 1, 1 + A245703(n-1));

(define (A245819 n) (if (<= n 1) n (+ 1 (A245703 (- n 1)))))

(define (A245819 n) (+ 1 (A091245 (A245703 (A018252 n)))))

(define (A245819 n) (+ 1 (A091226 (A245703 (A008578 n)))))

a(1) = 1, and for n > 1, a(n) = 1 + A245703(n-1).
a(n) = 1+A091245(A245703(A018252(n))). [Induced when A245703 is restricted to nonprime numbers].
a(n) = 1+A091226(A245703(A008578(n))). [Induced also when A245703 is restricted to noncomposite numbers].
As a composition of related permutations:
a(n) = A245814(A245815(n)).

cp's OEIS Frontend

A245821 Permutation of natural numbers: a(n) = A091205(A245703(n)).

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A245813 Permutation of natural numbers induced when A091205 is restricted to {1} and binary codes for polynomials reducible over GF(2): a(1) = 1, a(n) = A062298(A091205(A091242(n-1))).

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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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A245819 Permutation of natural numbers induced when A245703 is restricted to nonprime numbers: a(n) = 1+A091245(A245703(A018252(n))).

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