A193231 -id:A193231 - cp's OEIS frontend

A293443 Multiplicative with a(p^e) = A019565(A193231(e)).

1, 2, 2, 6, 2, 4, 2, 3, 6, 4, 2, 12, 2, 4, 4, 10, 2, 12, 2, 12, 4, 4, 2, 6, 6, 4, 3, 12, 2, 8, 2, 5, 4, 4, 4, 36, 2, 4, 4, 6, 2, 8, 2, 12, 12, 4, 2, 20, 6, 12, 4, 12, 2, 6, 4, 6, 4, 4, 2, 24, 2, 4, 12, 15, 4, 8, 2, 12, 4, 8, 2, 18, 2, 4, 12, 12, 4, 8, 2, 20, 10, 4, 2, 24, 4, 4, 4, 6, 2, 24, 4, 12, 4, 4, 4, 10, 2, 12, 12, 36, 2, 8, 2, 6, 8

Offset: 1

Antti Karttunen, Oct 31 2017

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A019565, A028234, A067029, A193231.

Cf. also A270428, A293442, A293231, A293439.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) }; \\ And this from Franklin T. Adams-Watters
vecproduct(v) = { my(m=1); for(i=1,#v,m *= v[i]); m; };
A293443(n) = vecproduct(apply(e -> A019565(A193231(e)), factorint(n)[, 2]));

;; With memoization-macro definec.
(definec (A293443 n) (if (= 1 n) n (* (A019565 (A193231 (A067029 n))) (A293443 (A028234 n)))))

a(1) = 1; for n > 1, a(n) = A019565(A193231(A067029(n))) * a(A028234(n)).
For all n >= 1, A007814(a(n)) = A293439(n).
For all k in A270428, A007814(a(k)) = A001221(k).

A366263 Doudna sequence permuted by Blue code: a(n) = A005940(1+A193231(n)).

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 9, 8, 16, 27, 25, 18, 15, 12, 10, 7, 14, 11, 21, 20, 35, 30, 24, 45, 81, 32, 54, 125, 36, 75, 49, 50, 100, 147, 121, 98, 225, 72, 150, 245, 625, 162, 64, 243, 250, 343, 375, 108, 33, 28, 22, 13, 40, 63, 55, 42, 90, 175, 135, 48, 77, 70, 60, 105, 210, 385, 315, 120, 143, 154, 140, 231, 525, 180

Offset: 0

Antti Karttunen, Oct 06 2023

Cf. A005940, A193231, A234022, A268389, A277818, A286601, A286602, A332450, A366264 (inverse map).

Cf. also A365810, A366260, A366275.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) };
A366263(n) = A005940(1+A193231(n));

a(n) = A332450(A005940(1+n)).
For all n >= 0, A001222(a(n)) = A234022(n) and A046523(a(n)) = A286601(n).
For all n >= 1, A055396(a(n)) = A277818(n) = 1+A268389(n).

A234024 Self-inverse permutation of nonnegative integers, A059893-conjugate of blue code: a(n) = A059893(A193231(A059893(n))).

Original entry on oeis.org

0, 1, 3, 2, 6, 5, 4, 7, 15, 10, 9, 12, 11, 14, 13, 8, 24, 23, 18, 29, 20, 27, 30, 17, 16, 31, 26, 21, 28, 19, 22, 25, 57, 40, 39, 54, 45, 60, 51, 34, 33, 48, 63, 46, 53, 36, 43, 58, 41, 56, 55, 38, 61, 44, 35, 50, 49, 32, 47, 62, 37, 52, 59, 42, 106, 89, 72, 123

Offset: 0

Antti Karttunen, Dec 28 2013

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Cf. A059893, A059894, A193231, A234025-A234027.

(define (A234024 n) (A059893 (A193231 (A059893 n))))

a(n) = A059893(A193231(A059893(n))).

a(n) = A059894(A234027(A059894(n))).

A234026 Permutation of nonnegative integers: a(n) = A193231(A054429(n)).

Original entry on oeis.org

0, 1, 2, 3, 7, 6, 4, 5, 8, 9, 11, 10, 13, 12, 14, 15, 25, 24, 26, 27, 28, 29, 31, 30, 22, 23, 21, 20, 19, 18, 16, 17, 42, 43, 41, 40, 47, 46, 44, 45, 37, 36, 38, 39, 32, 33, 35, 34, 59, 58, 56, 57, 62, 63, 61, 60, 52, 53, 55, 54, 49, 48, 50, 51, 127, 126, 124

Offset: 0

Antti Karttunen, Dec 28 2013

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A234025. Cf. A234024.

(define (A234026 n) (A193231 (A054429 n)))

a(n) = A193231(A054429(n)).

a(n) = A054429(A234027(n)).

A245454 Self-inverse permutation of nonnegative integers, A075158-conjugate of blue code: a(n) = 1 + A075157(A193231(A075158(n-1))).

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 18, 8, 9, 25, 11, 16, 64, 14, 27, 12, 96, 7, 288, 21, 20, 243, 891, 45, 10, 405, 15, 162, 33750, 30, 78650, 75, 625, 2025, 35, 81, 390390, 224, 875, 250, 41, 375, 16384, 270, 24, 300125, 24576, 150, 125, 54, 6125, 1350, 73728, 50, 108, 350, 594, 140777, 5845851, 98, 221433750, 1446445, 343, 13

Offset: 1

Antti Karttunen, Jul 22 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..512
Index entries for sequences that are permutations of the natural numbers

Cf. A193231, A075157, A075158, A234747, A234748.

Similar permutations: A245451, A245452, A122111, A241909, A241916.

(define (A245454 n) (+ 1 (A075157 (A193231 (A075158 (- n 1))))))

a(n) = 1 + A075157(A193231(A075158(n-1))).

A280501 "Blue kernel" of n: a(n) = A091255(n, A193231(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 7, 1, 7, 6, 1, 6, 1, 7, 1, 1, 1, 18, 19, 20, 21, 1, 1, 6, 1, 1, 7, 7, 1, 6, 1, 1, 1, 6, 7, 18, 1, 19, 1, 20, 1, 21, 1, 1, 7, 6, 1, 6, 7, 1, 1, 1, 19, 18, 1, 7, 1, 6, 1, 20, 1, 1, 21, 1, 21, 6, 1, 20, 1, 7, 1, 18, 1, 1, 1, 19, 1, 6, 7, 20, 1, 1, 7, 21, 1, 6, 1, 1, 1, 18, 1, 6, 7, 1, 19, 6, 1, 7, 1, 1, 7, 6, 1, 1, 1, 106, 107, 108, 109, 1, 1, 7

Offset: 1

Antti Karttunen, Jan 09 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences operating on polynomials in ring GF(2)[X]

Cf. A048720, A091255, A193231, A280502, A280505, A331167.

A091255sq(a,b) = fromdigits(Vec(lift(gcd(Pol(binary(a))*Mod(1, 2),Pol(binary(b))*Mod(1, 2)))),2);
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) }; \\ From A193231
A280501(n) = A091255sq(n, A193231(n)); \\ Antti Karttunen, Jan 12 2020

(define (A280501 n) (A091255bi n (A193231 n))) ;; A091255bi implements the 2-argument GF(2)[X] GCD-function (A091255).

a(n) = A091255(n, A193231(n)).
Other identities. For all n >= 1:
a(A193231(n)) = a(n).
A048720(a(n), A280502(n)) = n.

A332450 Self-inverse permutation of natural numbers, "Blue code" conjugated by A156552 and its inverse: a(n) = A005940(1+A193231(A156552(n))).

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 16, 8, 9, 27, 14, 18, 100, 11, 25, 7, 210, 12, 256, 20, 21, 147, 38, 45, 15, 385, 10, 98, 1156, 30, 1938, 50, 121, 2187, 35, 125, 234256, 23, 315, 245, 6902, 72, 3496900, 120, 24, 1083, 9699690, 108, 81, 32, 15625, 1458, 65536, 75, 225, 231, 57, 2185, 106, 243, 8836, 771147, 150, 105, 143, 154, 14946, 68, 529, 162

Offset: 1

Antti Karttunen, Feb 15 2020

Cf. A003961, A005940, A156552, A193231, A332449, A332451, A366263.

Cf. A293448 (A057889 similarly conjugated).

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) }; \\ From A193231
A332450(n) = A005940(1+A193231(A156552(n)));

a(n) = A005940(1+A193231(A156552(n))).
a(A003961(n)) = A332451(a(n)); a(A332451(n)) = A003961(a(n)).
a(n) = A366263(A156552(n)). - Antti Karttunen, Oct 06 2023

A369043 LCM-transform of Blue code (A193231).

Original entry on oeis.org

1, 3, 2, 5, 2, 1, 7, 1, 1, 1, 13, 1, 11, 3, 2, 17, 2, 1, 19, 1, 1, 23, 1, 1, 31, 29, 1, 3, 1, 1, 5, 1, 1, 1, 7, 1, 1, 53, 1, 1, 61, 1, 1, 1, 1, 1, 59, 1, 1, 1, 2, 1, 1, 1, 37, 1, 1, 1, 47, 1, 41, 43, 1, 1, 1, 1, 1, 1, 3, 83, 1, 1, 1, 89, 1, 1, 1, 1, 1, 1, 1, 71, 1, 1, 2, 1, 67, 1, 1, 1, 73, 1, 79, 1, 1, 1, 103, 101

Offset: 1

Antti Karttunen, Jan 12 2024

nonn

Blue code, A193231, is a self-inverse permutation related to the binary expansion of n that keeps all the numbers of range [2^k, 2^(1+k)[ in the same range, i.e., for all n >= 1, A000523(A193231(n)) = A000523(n), from which it immediately follows that A193231 has the property S mentioned in the comments of A368900, and therefore this sequence is equal to A014963(A193231(n)), for n >= 1.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A014963, A193231, A368900, A369041, A369042, A369044, A369045.

up_to = 65537; \\ Checked up to 2^17;
LCMtransform(v) = { my(len = length(v), b = vector(len), g = vector(len)); b[1] = g[1] = 1; for(n=2,len, g[n] = lcm(g[n-1],v[n]); b[n] = g[n]/g[n-1]); (b); };
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) };
v369043 = LCMtransform(vector(up_to,i,A193231(i)));
A369043(n) = v369043[n];
A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };

a(n) = lcm {1..A193231(n)} / lcm {1..A193231(n-1)}.
a(n) = A014963(A193231(n)). [See comments.]
For n >= 1, Product_{d|n} a(A193231(d)) = n. [Implied by above.]

A234751 Self-inverse permutation of integers induced by the restriction of blue-code to irreducible polynomials over GF(2): a(n) = A091227(A193231(A014580(n))).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 12 2014

This permutation is also induced when A234747 is restricted to primes: a(n) = A000720(A234747(A000040(n))) because of the way A234747 has been defined.

Note that each subsequence a(1)..a(A062692(j)) is closed (i.e., no cycles are leaking out) because blue code (A193231) preserves the degree of polynomials over GF(2) it operates upon.

Antti Karttunen, Table of n, a(n) for n = 1..2538
Index entries for sequences that are permutations of the natural numbers

Cf. A014580, A062692, A193231, A234747, A234750.

(define (A234751 n) (A091227 (A234750 n)))

a(n) = A091227(A234750(n)) = A091227(A193231(A014580(n))).

A244987 Self-inverse permutation of natural numbers, A245703-conjugate of Blue code: a(n) = A245704(A193231(A245703(n))).

Original entry on oeis.org

1, 3, 2, 6, 5, 4, 13, 8, 21, 15, 23, 16, 7, 25, 10, 12, 41, 18, 19, 64, 9, 22, 11, 49, 14, 26, 77, 39, 37, 34, 263, 105, 38, 30, 88, 70, 29, 33, 28, 133, 17, 54, 73, 126, 51, 462, 53, 60, 24, 66, 45, 74, 47, 42, 78, 94, 156, 81, 239, 48, 97, 62, 100, 20, 155, 50, 79, 98, 84, 36, 167, 141, 43, 52, 129, 164, 27, 55

Offset: 1

Antti Karttunen, Aug 07 2014

nonn

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

Cf. A000040, A002808, A010051, A193231, A245703, A245704, A234747, A234748, A245450, A245454.

allocatemem(234567890);
default(primelimit, 2^22);
A014580 = vector(2^18);
A091226 = vector(2^22);
A091242 = vector(2^22);
A002808(n)={ my(k=-1); while( -n + n += -k + k=primepi(n), ); n}; \\ This function from M. F. Hasler
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, j++; A091242[j] = n; A091226[n] = A091226[n-1]); n++);
A091245(n) = ((n-A091226[n])-1);
A245703(n) = if(1==n, 1, if(isprime(n), A014580[A245703(primepi(n))], A091242[A245703(n-primepi(n)-1)]));
A245704(n) = if(1==n, 1, if(isA014580(n), prime(A245704(A091226[n])), A002808(A245704(A091245(n)))));
A193231(n) = {my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2)};
A244987(n) = A245704(A193231(A245703(n)));
for(n=1, 10001, write("b244987.txt", n, " ", A244987(n)));

(define (A244987 n) (A245704 (A193231 (A245703 n))))

a(n) = A245704(A193231(A245703(n))).
Other identities. For all n >= 1, the following holds:
A010051(a(n)) = A010051(n). [Maps primes to primes and composites to composites].

cp's OEIS Frontend

A293443 Multiplicative with a(p^e) = A019565(A193231(e)).

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A366263 Doudna sequence permuted by Blue code: a(n) = A005940(1+A193231(n)).

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A234024 Self-inverse permutation of nonnegative integers, A059893-conjugate of blue code: a(n) = A059893(A193231(A059893(n))).

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A234026 Permutation of nonnegative integers: a(n) = A193231(A054429(n)).

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A245454 Self-inverse permutation of nonnegative integers, A075158-conjugate of blue code: a(n) = 1 + A075157(A193231(A075158(n-1))).

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A280501 "Blue kernel" of n: a(n) = A091255(n, A193231(n)).

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A332450 Self-inverse permutation of natural numbers, "Blue code" conjugated by A156552 and its inverse: a(n) = A005940(1+A193231(A156552(n))).

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A369043 LCM-transform of Blue code (A193231).

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A234751 Self-inverse permutation of integers induced by the restriction of blue-code to irreducible polynomials over GF(2): a(n) = A091227(A193231(A014580(n))).

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