A245704 -id:A245704 - cp's OEIS frontend

A255421 Permutation of natural numbers: a(1) = 1, a(p_n) = ludic(1+a(n)), a(c_n) = nonludic(a(n)), where p_n = n-th prime, c_n = n-th composite number and ludic = A003309, nonludic = A192607.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 23, 19, 20, 21, 25, 22, 24, 26, 27, 28, 29, 34, 37, 30, 31, 32, 36, 33, 41, 35, 38, 39, 43, 40, 47, 42, 49, 52, 53, 44, 45, 46, 51, 48, 61, 57, 50, 54, 55, 59, 67, 56, 71, 64, 58, 66, 70, 72, 97, 60, 62, 63, 77, 69, 83, 65, 81

Offset: 1

Antti Karttunen, Feb 23 2015

nonn

This can be viewed as yet another "entanglement permutation", where two pairs of complementary subsets of natural numbers are interwoven with each other. In this case a complementary pair ludic/nonludic numbers (A003309/A192607) is intertwined with a complementary pair prime/composite numbers (A000040/A002808).

			When n = 19 = A000040(8) [the eighth prime], we look for the value of a(8), which is 8 [all terms less than 19 are fixed because the beginnings of A003309 and A008578 coincide up to A003309(8) = A008578(8) = 17], and then take the eighth ludic number larger than 1, which is A003309(1+8) = 23, thus a(19) = 23.
When n = 20 = A002808(11) [the eleventh composite], we look for the value of a(11), which is 11 [all terms less than 19 are fixed, see above], and then take the eleventh nonludic number, which is A192607(11) = 19, thus a(20) = 19.
When n = 30 = A002808(19) [the 19th composite], we look for the value of a(19), which is 23 [see above], and then take the 23rd nonludic number, which is A192607(23) = 34, thus a(30) = 34.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A255422.
Cf. A000040, A000720, A002808, A003309, A007097, A008578, A065855, A010051, A192607, A255420, A255324.
Related or similar permutations: A237126, A246377, A245703, A245704, A255407, A255408.

a(1) = 1, and for n > 1, if A010051(n) = 1 [i.e. when n is a prime], a(n) = A003309(1+a(A000720(n))), otherwise a(n) = A192607(a(A065855(n))).
As a composition of other permutations:
a(n) = A237126(A246377(n)).
Other identities.
a(A007097(n)) = A255420(n). [Maps iterates of primes to the iterates of Ludic numbers.]

A255422 Permutation of natural numbers: a(1) = 1 and for n > 1, if n is k-th ludic number larger than 1 [i.e., n = A003309(k+1)], a(n) = nthprime(a(k)), otherwise, when n is k-th nonludic number [i.e., n = A192607(k)], a(n) = nthcomposite(a(k)), where nthcomposite = A002808, nthprime = A000040.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 19, 25, 23, 26, 27, 28, 29, 32, 33, 34, 36, 30, 38, 35, 31, 39, 40, 42, 37, 44, 41, 48, 49, 50, 43, 52, 45, 55, 51, 46, 47, 56, 57, 60, 54, 63, 58, 68, 53, 69, 70, 62, 74, 64, 59, 77, 72, 65, 61, 66, 78, 80, 84, 76, 71, 87, 81

Offset: 1

Antti Karttunen, Feb 23 2015

The graph has a comet appearance. - Daniel Forgues, Dec 15 2015

			When n = 19 = A192607(11) [the eleventh nonludic number], we look for the value of a(11), which is 11 [all terms less than 19 are fixed because the beginnings of A003309 and A008578 coincide up to A003309(8) = A008578(8) = 17], and then take the eleventh composite number, which is A002808(11) = 20, thus a(19) = 20.
When n = 25 = A003309(10) = A003309(1+9) [the tenth ludic number, and ninth after one], we look for the value of a(9), which is 9 [all terms less than 19 are fixed, see above], and then take the ninth prime number, which is A000040(9) = 23, thus a(25) = 23.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Antti Karttunen, Entanglement Permutations, 2016-2017
Index entries for sequences that are permutations of the natural numbers

Inverse: A255421.
Cf. A000040, A002808, A003309, A192490, A192512, A192607, A236863.
Related or similar permutations: A237427, A246378, A245703, A245704 (compare the scatterplots), A255407, A255408.

a(1)=1; and for n > 1, if A192490(n) = 1 [i.e., n is ludic], a(n) = A000040(a(A192512(n)-1)), otherwise a(n) = A002808(a(A236863(n))) [where A192512 and A236863 give the number of ludic and nonludic numbers <= n, respectively].

As a composition of other permutations: a(n) = A246378(A237427(n)).

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.

A260424 a(1) = 1, a(A206074(n)) = prime(a(n)), a(A205783(1+n)) = composite(a(n)), where A206074 and A205783 give binary codes for polynomials with coefficients 0 or 1 that are irreducible [resp. reducible] over Q.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 29, 25, 26, 27, 31, 28, 37, 30, 32, 33, 34, 35, 41, 36, 44, 38, 43, 39, 47, 40, 46, 42, 53, 54, 45, 48, 49, 50, 59, 51, 61, 58, 52, 63, 67, 55, 71, 62, 56, 66, 57, 65, 73, 60, 79, 75, 83, 76, 89, 64, 68, 69, 109, 70, 97, 82, 101, 72, 103, 85, 81, 74, 127

Offset: 1

Antti Karttunen, Jul 25 2015

nonn

After 1, each term of A206075 resides in a separate infinite cycle. This follows because primes (A000040) is a subsequence of A206074 [see Thomas Ordowski's Feb 19 2014 comment in A206074] and thus each composite in A206074 is trapped into a trajectory containing only primes.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A260423.
Related permutations: A245704, A246378, A260421, A260426.
Cf. A000040, A002808, A205783, A206074, A206075, A257000, A255572, A255574.

allocatemem(123456789);
default(primelimit,4294965247);
uplim = 2^20;
v255574 = vector(uplim); A255574 = n -> v255574[n];
A255572 = n -> (n - A255574(n) - 1);
A257000(n) = polisirreducible(Pol(binary(n)));
v255574[1] = 0; i=0; j=0; n=2; while((n < uplim), v255574[n] = v255574[n-1]+A257000(n); n++);
A002808(n)={ my(k=-1); while( -n + n += -k + k=primepi(n), ); n}; \\ This function from M. F. Hasler
A260424(n) = if(1==n, 1, if(A257000(n), prime(A260424(A255574(n))), A002808(A260424(A255572(n)))));
for(n=1, 8192, write("b260424.txt", n, " ", A260424(n)));

a(1) = 1; for n > 1, if A257000(n) = 1 [when n is in A206074], then a(n) = A000040(a(A255574(n))), otherwise [when n is in A205783], a(n) = A002808(a(A255572(n))).
As a composition of related permutations:
a(n) = A246378(A260421(n)).
a(n) = A245704(A260426(n)).

A260425 a(1) = 1, a(A014580(n)) = A206074(a(n)), a(A091242(n)) = A205783(1+a(n)), where A014580(n) [resp. A091242(n)] give binary codes for n-th irreducible [resp. reducible] polynomial over GF(2), while A206074 and A205783 give similar codes for polynomials with coefficients 0 or 1 that are irreducible [resp. reducible] over Q.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 5, 9, 12, 15, 7, 10, 13, 16, 21, 26, 14, 18, 19, 22, 27, 34, 40, 24, 11, 30, 32, 35, 42, 51, 23, 60, 38, 20, 46, 49, 31, 52, 63, 76, 43, 36, 92, 57, 33, 68, 17, 74, 48, 78, 95, 114, 64, 54, 25, 135, 86, 50, 37, 102, 47, 28, 111, 72, 118, 140, 67, 165, 96, 82, 39, 195, 79, 128, 75, 56, 150, 70, 44

Offset: 1

Antti Karttunen, Jul 26 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4244
Index entries for sequences that are permutations of the natural numbers

Inverse: A260426.
Related permutations: A246201, A245704, A260422, A260423.
Cf. A014580, A091225, A091226, A091242, A091245, A205783, A206074.
Differs from A245704 for the first time at n=16, where a(16) = 26, while A245704(16) = 25.

allocatemem(234567890);
vecsize = (2^24)-4;
uplim = 2^25;
v091226 = vector(vecsize); A091226 = n -> v091226[n];
A091245(n) = ((n-A091226(n))-1);
v206074 = vector(vecsize); A206074 = n -> v206074[n];
v205783 = vector(vecsize); A205783 = n -> v205783[n];
isA014580(n) = polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
isA206074(n) = polisirreducible(Pol(binary(n)));
v091226[1] = 0; v205783[1] = 1; i=0; j=1; n=2; while((n < uplim), if(!(n%65536),print1(n,", ")); if(isA206074(n), i++; v206074[i] = n, j++; v205783[j] = n); if(isA014580(n), v091226[n] = v091226[n-1]+1, v091226[n] = v091226[n-1]); n++); print(n);
A260425(n) = if(1==n, 1, if(isA014580(n), A206074(A260425(A091226(n))), A205783(1+A260425(A091245(n)))));
for(n=1, 4244, write("b260425.txt", n, " ", A260425(n)));

(definec (A260425 n) (cond ((<= n 1) n) ((= 1 (A091225 n)) (A206074 (A260425 (A091226 n)))) (else (A205783 (+ 1 (A260425 (A091245 n)))))))

a(1) = 1; for n > 1, if A091225(n) = 1 [when n is in A014580], then a(n) = A206074(a(A091226(n))), otherwise [when n is in A091242], a(n) = A205783(1+a(A091245(n))).
As a composition of related permutations:
a(n) = A260422(A246201(n)).
a(n) = A260423(A245704(n)).

cp's OEIS Frontend

A255421 Permutation of natural numbers: a(1) = 1, a(p_n) = ludic(1+a(n)), a(c_n) = nonludic(a(n)), where p_n = n-th prime, c_n = n-th composite number and ludic = A003309, nonludic = A192607.

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A255422 Permutation of natural numbers: a(1) = 1 and for n > 1, if n is k-th ludic number larger than 1 [i.e., n = A003309(k+1)], a(n) = nthprime(a(k)), otherwise, when n is k-th nonludic number [i.e., n = A192607(k)], a(n) = nthcomposite(a(k)), where nthcomposite = A002808, nthprime = A000040.

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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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A260424 a(1) = 1, a(A206074(n)) = prime(a(n)), a(A205783(1+n)) = composite(a(n)), where A206074 and A205783 give binary codes for polynomials with coefficients 0 or 1 that are irreducible [resp. reducible] over Q.

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