A243071 -id:A243071 - cp's OEIS frontend

A292264 a(n) = n - A292944(n).

0, 1, 2, 2, 4, 4, 4, 5, 8, 8, 8, 9, 8, 8, 10, 11, 16, 16, 16, 17, 16, 16, 18, 19, 16, 16, 16, 17, 20, 20, 22, 23, 32, 32, 32, 33, 32, 32, 34, 35, 32, 32, 32, 33, 36, 36, 38, 39, 32, 32, 32, 33, 32, 32, 34, 35, 40, 40, 40, 41, 44, 44, 46, 47, 64, 64, 64, 65, 64, 64, 66, 67, 64, 64, 64, 65, 68, 68, 70, 71, 64, 64, 64, 65, 64, 64, 66, 67, 72, 72, 72, 73, 76, 76

Offset: 0

Antti Karttunen, Sep 30 2017

Because A292263(n) = a(A243071(n)), the sequence works as a "masking function" where the 1-bits in a(n) (always a subset of the 1-bits in binary expansion of n) indicate which numbers are of the form 6k+1 or 6k+5 in binary tree A163511 (or in its mirror image tree A005940) on that trajectory which leads from the root of the tree to the node containing A163511(n).

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A005940, A163511, A292263.

Cf. A048735, A292944, A292272 but also A292254, A292256, A292942, A292946 for similarly constructed sequences.

(define (A292264 n) (A292263 (A163511 n)))

a(n) = n - A292944(n).
a(n) = A292263(A163511(n)).
a(n) = A292942(n) + A292946(n).
a(n) = A292254(n) + A292256(n).

A292941 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 6)].

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 9, 4, 4, 8, 18, 8, 37, 18, 8, 8, 74, 8, 149, 16, 16, 36, 298, 16, 9, 74, 8, 36, 596, 16, 1193, 16, 36, 148, 16, 16, 2387, 298, 72, 32, 4774, 32, 9549, 72, 16, 596, 19098, 32, 19, 18, 148, 148, 38196, 16, 33, 72, 296, 1192, 76392, 32, 152785, 2386, 32, 32, 72, 72, 305571, 296, 596, 32, 611142, 32, 1222285, 4774, 16, 596, 32

Offset: 1

Antti Karttunen, Sep 28 2017

nonn

Base-2 expansion of a(n) encodes the steps where numbers of the form 6k+1 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n. An exception is the most significant bit of a(n) which corresponds with the final 1, but is shifted one bit-position towards right (less significant end).

The AND - XOR formulas just restate the fact that J(-3|n) = J(-1|n)*J(3|n), as the Jacobi-symbol is multiplicative (also) with respect to its upper argument.

Cf. A005940, A163511, A243071, A292942, A292943, A292945.

Cf. also A292253, A292263, A292381, A292385.

(define (A292941 n) (if (<= n 2) (- n 1) (+ (if (= 1 (modulo n 6)) 1 0) (* 2 (A292941 (A252463 n))))))

a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 6)], where the last part of the formula is Iverson bracket, giving 1 only if n is of the form 6k+1, and 0 otherwise.
Also, for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 2)]*[J(-3|n) = 1], where J is the Jacobi-symbol.
a(n) = A292263(n) AND (A292253(n) XOR A292383(n)), where AND is bitwise-and (A004198) and XOR is bitwise-XOR (A003987).
a(n) = A292263(n) AND (A292255(n) XOR A292385(n)). [See comments.]
For n >= 0, a(A163511(n)) = A292942(n).
For n >= 1, a(n) + A292943(n) + A292945(n) = A243071(n).

A292942 a(n) = A292941(A163511(n)).

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 9, 8, 8, 8, 9, 16, 16, 16, 16, 16, 16, 18, 19, 16, 16, 16, 16, 16, 16, 18, 18, 32, 32, 32, 33, 32, 32, 32, 33, 32, 32, 32, 32, 36, 36, 38, 39, 32, 32, 32, 33, 32, 32, 32, 32, 32, 32, 32, 33, 36, 36, 36, 37, 64, 64, 64, 64, 64, 64, 66, 67, 64, 64, 64, 64, 64, 64, 66, 66, 64, 64, 64, 65, 64, 64, 64, 65, 72, 72, 72, 72, 76, 76

Offset: 0

Antti Karttunen, Sep 28 2017

nonn

Because A292941(n) = a(A243071(n)), the sequence works as a "masking function" where the 1-bits in a(n) (always a subset of the 1-bits in binary expansion of n) indicate which numbers are of the form 6k+1 in binary tree A163511 (or its mirror image tree A005940) on that trajectory which leads from the root of the tree to the node containing A163511(n).

The AND - XOR formula is just a restatement of the fact that J(-3|n) = J(-1|n)*J(3|n), as the Jacobi-symbol is multiplicative (also) with respect to its upper argument.

Cf. A005940, A163511, A292941.

Cf. also A292247, A292248, A292254, A292256, A292264, A292271, A292274, A292592, A292593, A292944, A292946 (for similarly constructed sequences).

(define (A292942 n) (A292941 (A163511 n)))

a(n) = A292941(A163511(n)).
a(n) = A292264(n) AND (A292254(n) XOR A292274(n)), where AND is bitwise-and (A004198) and XOR is bitwise-XOR (A003987). [See comments.]
For all n >= 0, a(n) + A292944(n) + A292946(n) = n.

A292946 a(n) = A292945(A163511(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 4, 4, 4, 5, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 8, 8, 8, 8, 8, 8, 10, 10, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 4, 4, 4, 5, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0

Offset: 0

Antti Karttunen, Sep 28 2017

nonn

Because A292945(n) = a(A243071(n)), the sequence works as a "masking function" where the 1-bits in a(n) (always a subset of the 1-bits in binary expansion of n) indicate which numbers are of the form 6k+5 in binary tree A163511 (or its mirror image tree A005940) on that trajectory which leads from the root of the tree to the node containing A163511(n).

The AND - XOR formulas just restate the fact that J(-3|n) = J(-1|n)*J(3|n), as the Jacobi-symbol is multiplicative (also) with respect to its upper argument.

Cf. A005940, A163511, A292945.

Cf. also A292247, A292248, A292254, A292256, A292264, A292271, A292274, A292592, A292593, A292942, A292944 (for similarly constructed sequences).

(define (A292946 n) (A292945 (A163511 n)))

a(n) = A292945(A163511(n)).
a(n) = A292264(n) AND (A292256(n) XOR A292274(n)), where AND is bitwise-and (A004198) and XOR is bitwise-XOR (A003987).
a(n) = A292264(n) AND (A292254(n) XOR A292271(n)). [See comments.]
For all n >= 0, A292942(n) + A292944(n) + a(n) = n.

A292945 Base-2 expansion of a(n) encodes the steps where numbers of the form 6k+5 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 5, 0, 10, 4, 0, 0, 21, 0, 42, 4, 4, 10, 85, 0, 0, 20, 0, 8, 171, 0, 342, 0, 8, 42, 1, 0, 684, 84, 20, 8, 1369, 8, 2738, 20, 0, 170, 5477, 0, 0, 0, 40, 40, 10955, 0, 8, 16, 84, 342, 21911, 0, 43822, 684, 8, 0, 17, 16, 87644, 84, 168, 2, 175289, 0, 350578, 1368, 0, 168, 3, 40, 701156, 16, 0, 2738, 1402313, 16, 40, 5476, 340, 40

Offset: 1

Antti Karttunen, Sep 28 2017

nonn

The AND - XOR formulas are just a restatement of the fact that J(-3|n) = J(-1|n)*J(3|n), i.e., that Jacobi-symbol is multiplicative (also) with respect to its upper argument.

Cf. A005940, A163511, A243071, A292941, A292943, A292946.

Cf. also A292253, A292255.

(define (A292945 n) (if (<= n 1) 0 (+ (if (= 5 (modulo n 6)) 1 0) (* 2 (A292945 (A252463 n))))))

a(1) = 0, and for n > 1, a(n) = 2*a(A252463(n)) + [n == 5 (mod 6)], where the last part of the formula is Iverson bracket, giving 1 only if n is of the form 6k+5, and 0 otherwise.
Also, for n > 1, a(n) = 2*a(A252463(n)) + [n == 1 (mod 2)]*[J(-3|n) = -1], where J is the Jacobi-symbol.
a(n) = A292263(n) AND (A292255(n) XOR A292383(n)), where AND is bitwise-and (A004198) and XOR is bitwise-XOR (A003987).
a(n) = A292263(n) AND (A292253(n) XOR A292385(n)). [See comments.]
For n >= 0, a(A163511(n)) = A292946(n).
For n >= 1, A292941(n) + A292943(n) + a(n) = A243071(n).

A324200 a(n) = 2^(A000043(n)-1) * ((2^A059305(n)) - 1), where A059305 gives the prime index of the n-th Mersenne prime, while A000043 gives its exponent.

Original entry on oeis.org

6, 60, 32752, 137438953408

Offset: 1

Antti Karttunen, Feb 18 2019

nonn

If there are no odd perfect numbers then these are the positions of zeros in A324185.
The next term has 314 digits:
11781361728633673532894774498354952494238773929196300355071513798753168641589311119865182769801300280680127783231251635087526446289021607771691249214388576215221396663491984443067742263787264024212477244347842938066577043117995647400274369612403653814737339068225047641453182709824206687753689912418253153056583680.

Subsequence of A023758 and A324199.

Cf. A000043, A000396, A000668, A000720, A007814, A023758, A059305, A156552, A243071, A324185.

A324200(n) = (2^(A000043(n)-1))*((2^primepi(A000668(n)))-1);

a(n) = ((2^A000720(A000668(n)))-1) * 2^(A000043(n)-1) = ((2^A059305(n)) - 1) * 2^(A000043(n)-1).
a(n) = A243071(A156552(A324201(n))) = A243071(A156552(A062457(A000043(n)))).
If no odd perfect numbers exist, then a(n) = A243071(A000396(n)), and thus A007814(a(n)) = A007814(A000396(n)).

A364495 Odd numbers k such that k divides A163511(k).

Original entry on oeis.org

1, 3, 9, 105, 429, 1365, 1617, 3887, 4235, 10829, 14025, 17745, 21125, 22627, 38025, 54587, 70805, 100555, 115159, 147875, 168751, 169065, 175769, 181447, 181545, 291525, 297297, 303875, 338675, 350987, 501787, 513825, 518035, 549081, 560947, 566865, 594473, 624169, 676039, 735875, 745147, 831875, 869193, 957125

Offset: 1

Antti Karttunen, Jul 27 2023

nonn

			For n = 513825 = 3 * 5^2 * 13 * 17 * 31, A163511(n) = 13873275 = 3^4 * 5^2 * 13 * 17 * 31, so A163511(n)/n = 27 (which is an integer), and therefore 513825 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..801

Odd terms in A364494.
After 1, sequence A243071(A364965(n)), for n>=1, sorted into ascending order.
Cf. A163511.
Cf. also A364296, A364498, A364545, A364963.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A054429(n) = ((3<<#binary(n\2))-n-1);
A163511(n) = if(!n,1,A005940(1+A054429(n)))
A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
isA364495(n) = ((n%2)&&!(A163511(n)%n));

A246682 Permutation of natural numbers: a(1) = 0, a(2) = 1, and for n > 1, a(2n) = nthcomposite(a(n)), a(2n-1) = nthprime(a(A064989(2n-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

0, 1, 2, 4, 3, 6, 5, 9, 7, 8, 11, 12, 31, 10, 13, 16, 127, 14, 709, 15, 19, 20, 5381, 21, 17, 46, 23, 18, 52711, 22, 648391, 26, 29, 166, 41, 24, 9737333, 858, 71, 25, 174440041, 30, 3657500101, 32, 37, 6186

Offset: 1

Antti Karttunen, Sep 01 2014

nonn

Note the indexing: the domain starts from 1, while the range includes also zero.

Has an infinite number of infinite cycles. See comments at A246681.

Index entries for sequences that are permutations of the natural numbers

Inverse: A246681.
Similar or related permutations: A246376, A246378, A243071, A246368, A064216, A246380.
Cf. A000035, A000040, A002808, A007097, A010051, A064989, A049076, A078442.

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)};
A246682(n) = if(n < 3, n-1, if(!(n%2), A002808(A246682(n/2)), prime(A246682(A064989(n)))));
for(n=1, 46, write("b246682.txt", n, " ", A246682(n)));
(Scheme, two variants)
(definec (A246682 n) (cond ((<= n 2) (- n 1)) ((even? n) (A002808 (A246682 (/ n 2)))) (else (A000040 (A246682 (A064989 n))))))
(define (A246682 n) (A246378 (A243071 n)))

a(1) = 0, a(2) = 1, and for n > 1, a(2n) = nthcomposite(a(n)), a(2n-1) = nthprime(a(A064989(2n-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(A243071(n)).
Other identities.
For all n >= 1 the following holds:
a(A000040(n)) = A007097(n-1). [Maps primes to the iterates of primes].
A049076(a(A000040(n))) = n. [Follows from above].
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 & A246380 have the same property].

A269386 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2a(n), a(2n+1) = 1 + 2a(A269380(2n+1)).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 15, 4, 5, 14, 31, 12, 63, 30, 13, 8, 127, 10, 11, 28, 9, 62, 255, 24, 511, 126, 29, 60, 1023, 26, 23, 16, 25, 254, 27, 20, 2047, 22, 61, 56, 4095, 18, 8191, 124, 17, 510, 16383, 48, 19, 1022, 21, 252, 32767, 58, 47, 120, 57, 2046, 55, 52, 65535, 46, 125, 32, 59, 50, 131071, 508, 49, 54, 262143, 40, 95, 4094, 253, 44

Offset: 1

Antti Karttunen, Mar 01 2016

nonn

Note the indexing: Domain starts from 1, range from 0.

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A269385.
Cf. A269380.
Related permutations: A260742, A269172, A269388.
Cf. also A252756, A269376.
Differs from A243071 and A252756 for the first time at n=19, which here a(19) = 11, instead of 255.

a(1) = 0, a(2) = 1, a(2n) = 2*a(n), a(2n+1) = 1 + 2*a(A269380(2n+1)).
As a composition of related permutations:
a(n) = A252756(A269172(n)).
a(n) = A269376(A260742(n)).

A364297 a(n) = A348717(A163511(n)).

Original entry on oeis.org

1, 2, 4, 2, 8, 4, 6, 2, 16, 8, 18, 4, 12, 6, 10, 2, 32, 16, 54, 8, 36, 18, 50, 4, 24, 12, 30, 6, 20, 10, 14, 2, 64, 32, 162, 16, 108, 54, 250, 8, 72, 36, 150, 18, 100, 50, 98, 4, 48, 24, 90, 12, 60, 30, 70, 6, 40, 20, 42, 10, 28, 14, 22, 2, 128, 64, 486, 32, 324, 162, 1250, 16, 216, 108, 750, 54, 500, 250, 686, 8, 144

Offset: 0

Antti Karttunen, Aug 15 2023

nonn

For all i, j: a(i) = a(j) => A278531(i) = A278531(j).
As the underlying sequence A163511 can be represented as a binary tree, so can this be also:
                                       1
                                       |
                    ...................2...................
                   4                                       2
         8......../ \........4                   6......../ \........2
        / \                 / \                 / \                 / \
       /   \               /   \               /   \               /   \
      /     \             /     \             /     \             /     \
    16       8          18       4          12       6          10       2
  32  16   54 8       36  18   50 4       24  12   30 6       20  10   14 2
etc.
Each rightward leaning branch stays constant, because a(2n+1) = a(n).
Conjecture: Mersenne primes (A000668) gives all such odd numbers k for which a(k) = A348717(k). If true, then it immediately implies that map n -> A163511(n) [or equally: map n ->  A243071(n)] has no other fixed points than those given by A007283. But see also A364959. - Edited Sep 03 2023

Cf. A000265, A000668, A007283, A163511, A243071, A348717, A364949, A364956, A365423.

Cf. also A245611, A245612, A278531, A286531, A364959, A364963.

A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));
A364297(n) = A348717(A163511(n));

a(0) = 1, a(1) = 2, a(2n) = A163511(2n) = 2*A163511(n), and for n > 0, a(2n+1) = a(n).

cp's OEIS Frontend

A292264 a(n) = n - A292944(n).

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A292941 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 6)].

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A292942 a(n) = A292941(A163511(n)).

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A292946 a(n) = A292945(A163511(n)).

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A292945 Base-2 expansion of a(n) encodes the steps where numbers of the form 6k+5 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

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A324200 a(n) = 2^(A000043(n)-1) * ((2^A059305(n)) - 1), where A059305 gives the prime index of the n-th Mersenne prime, while A000043 gives its exponent.

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A364495 Odd numbers k such that k divides A163511(k).

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A246682 Permutation of natural numbers: a(1) = 0, a(2) = 1, and for n > 1, a(2n) = nthcomposite(a(n)), a(2n-1) = nthprime(a(A064989(2n-1))), where nthprime = A000040, nthcomposite = A002808, and A064989(n) shifts the prime factorization of n one step towards smaller primes.

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A269386 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2*a(n), a(2n+1) = 1 + 2*a(A269380(2n+1)).

A269386 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2a(n), a(2n+1) = 1 + 2a(A269380(2n+1)).