A243285 -id:A243285 - cp's OEIS frontend

A244982 Permutation of natural numbers: a(n) = A243285(A122111(2*n)).

1, 2, 4, 3, 8, 6, 18, 5, 11, 14, 38, 10, 79, 30, 22, 7, 164, 15, 337, 20, 47, 64, 694, 16, 35, 134, 26, 43, 1419, 32, 2888, 9, 100, 279, 73, 24, 5850, 575, 208, 34, 11822, 67, 23836, 92, 56, 1177, 47975, 19, 112, 50, 428, 193, 96431, 42, 152, 71, 877, 2395, 193614, 52

Offset: 1

Antti Karttunen, Jul 20 2014

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse: A244981.

Cf. A122111, A243285, A244983-A244984, A243505-A243506, A243065-A243066.

(define (A244982 n) (A243285 (A122111 (* 2 n))))

a(n) = A243285(A122111(2*n)).

A243287 a(1)=1, and for n > 1, if n is k-th number divisible by the square of its largest prime factor (i.e., n = A070003(k)), a(n) = 1 + (2a(k)); otherwise, when n = A102750(k), a(n) = 2a(k).

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 16, 5, 9, 12, 32, 10, 18, 24, 64, 7, 20, 17, 36, 48, 128, 14, 40, 34, 13, 72, 33, 96, 256, 28, 80, 11, 68, 26, 144, 19, 66, 192, 512, 56, 160, 22, 136, 52, 288, 38, 132, 384, 25, 65, 1024, 112, 320, 21, 44, 272, 104, 576, 76, 264, 768, 50, 130, 37, 2048

Offset: 1

Antti Karttunen, Jun 02 2014

This is an instance of "entanglement permutation", where two pairs of complementary subsets of natural numbers are interwoven with each other. In this case complementary pair A070003/A102750 (numbers which are divisible/not divisible by the square of their largest prime factor) is entangled with complementary pair odd/even numbers (A005408/A005843).

Thus this shares with the permutation A122111 the property that each term of A102750 is mapped to a unique even number and likewise each term of A070003 is mapped to a unique odd number.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A243288.
Cf. A005843, A005408, A070003, A102750, A243282, A243285, A241917, A122111.
Similarly constructed permutations: A243343-A243346, A135141-A227413, A237126-A237427, A193231.

a(1) = 1, and thereafter, if A241917(n) = 0 (i.e., n is a term of A070003), a(n) = 1 + (2*a(A243282(n))); otherwise a(n) = 2*a(A243285(n)) (where A243282 and A243285 give the number of integers <= n divisible/not divisible by the square of their largest prime factor).

A243288 Permutation of natural numbers: a(1)=1, a(2n) = A102750(a(n)), a(2n+1) = A070003(a(n)).

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 16, 5, 9, 12, 32, 10, 25, 22, 81, 7, 18, 13, 36, 17, 54, 42, 242, 14, 49, 34, 150, 30, 128, 99, 882, 11, 27, 24, 100, 19, 64, 46, 256, 23, 98, 68, 490, 55, 338, 279, 4624, 20, 72, 62, 432, 44, 245, 178, 2209, 40, 216, 154, 1800, 119, 1200, 966

Offset: 1

Antti Karttunen, Jun 02 2014

nonn

This is an instance of "entanglement permutation", where two pairs of complementary subsets of natural numbers are interwoven with each other. In this case complementary pair odd/even numbers (A005408/A005843) is entangled with complementary pair A070003/A102750 (numbers which are divisible/not divisible by the square of their largest prime factor).

Thus this shares with the permutation A122111 the property that each even number is mapped to a unique term of A102750 and each odd number (larger than 1) to a unique term of A070003.

Antti Karttunen, Table of n, a(n) for n = 1..94
Index entries for sequences that are permutations of the natural numbers

Inverse of A243287.
Cf. A005843, A005408, A070003, A102750, A243282, A243285, A241917, A122111.
Similarly constructed permutations: A243343-A243346, A135141-A227413, A237126-A237427, A193231.

a(1)=1, and for n > 1, if n=2k, a(n) = A102750(a(k)), otherwise, when n = 2k+1, a(n) = A070003(a(k)).

A243282 Partial sums of the characteristic function for A070003.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16

Offset: 1

Antti Karttunen, Jun 02 2014

nonn

a(n) tells how many natural numbers <= n there are which are divisible by the square of their largest prime divisor. (This definition excludes 1 as it has no prime divisors.)

For all n, a(A070003(n)) = n, thus this sequence works also as an inverse function for the injection A070003.

			A070003(402) = 10000, thus a(10000) = 402.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

One less than A243283.

Cf. A070003, A243285.

Accumulate[Join[{0},Table[If[Divisible[n,Last[Select[Divisors[n],PrimeQ]]^2],1,0],{n,2,90}]]] (* Harvey P. Dale, Sep 05 2018 *)

a(n) = A243283(n)-1.

A243283 One more than the partial sums of the characteristic function of A070003.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 16

Offset: 1

Antti Karttunen, Jun 02 2014

nonn

a(n) tells how many positive integers <= n are divisible by the square of their largest noncomposite divisor. (This definition includes 1 as it is divisible by 1^2.)
a(n) = n - A243285(n).
a(1) = 1 and for all n > 1, a(A070003(n-1)) = n, thus this sequence works as an inverse function for the injection {a(1) = 1, a(n>1) = A070003(n-1)} (a sequence which is the union of {1} and A070003).

Antti Karttunen, Table of n, a(n) for n = 1..10000

One more than A243282.
Differs from A243284 for the first time at n=48. Here a(48)=10.
Cf. A070003, A102354, A243285, A057627.

A243286 Self-inverse permutation of natural numbers induced by the restriction of A243057 (or A243059 or A242420) to the union of {1} and A102750.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 12, 8, 9, 10, 27, 7, 13, 14, 36, 16, 62, 18, 19, 121, 148, 22, 23, 24, 43, 191, 11, 28, 283, 75, 113, 32, 87, 34, 481, 15, 388, 38, 39, 160, 1456, 42, 25, 795, 213, 602, 47, 74, 49, 818, 51, 52, 339, 54, 2699, 345, 57, 58, 59, 1053, 5219, 17, 914, 64

Offset: 1

Antti Karttunen, Jun 02 2014

nonn

			With n=7, the seventh number in the complement of A070003 (i.e. in the union of {1} and A102750) is A102750(6) = 10. When A243057 (or A243059) is applied to it, the result is another number that is a member of A102750, in this case 15, which occurs there as A102750(11). Thus a(7) = 11+1 = 12.

Antti Karttunen, Table of n, a(n) for n = 1..126
Index entries for sequences that are permutations of the natural numbers

Cf. A243057, A243059, A102750, A243285, A242420.

(define (A243286 n) (if (<= n 1) n (+ 1 (A243285 (A243057 (A102750 (- n 1)))))))

a(1) = 1, and for n>1, a(n) = 1 + A243285(A243057(A102750(n-1))). [Note: instead of A243057 one can also use A243059 or A242420.]

A253568 Even bisection of A122111: a(n) = A122111(2*n).

Original entry on oeis.org

2, 3, 6, 5, 12, 10, 24, 7, 15, 20, 48, 14, 96, 40, 30, 11, 192, 21, 384, 28, 60, 80, 768, 22, 45, 160, 35, 56, 1536, 42, 3072, 13, 120, 320, 90, 33, 6144, 640, 240, 44, 12288, 84, 24576, 112, 70, 1280, 49152, 26, 135, 63, 480, 224, 98304, 55, 180, 88, 960, 2560, 196608, 66, 393216, 5120, 140, 17, 360, 168, 786432, 448, 1920, 126, 1572864, 39

Offset: 1

Antti Karttunen, Jan 04 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024

Even bisection of A122111.
Topmost row of A253561.
Permutation of A102750.
Cf. A243285, A244982.

Scheme
```
(define (A253568 n) (A122111 (* 2 n)))
```

a(n) = A122111(2*n).
Other identities. For all n >= 1:
A244982(n) = A243285(a(n)).

cp's OEIS Frontend

A244982 Permutation of natural numbers: a(n) = A243285(A122111(2*n)).

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A243287 a(1)=1, and for n > 1, if n is k-th number divisible by the square of its largest prime factor (i.e., n = A070003(k)), a(n) = 1 + (2*a(k)); otherwise, when n = A102750(k), a(n) = 2*a(k).

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A243288 Permutation of natural numbers: a(1)=1, a(2n) = A102750(a(n)), a(2n+1) = A070003(a(n)).

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A243282 Partial sums of the characteristic function for A070003.

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Mathematica

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A243283 One more than the partial sums of the characteristic function of A070003.

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A243286 Self-inverse permutation of natural numbers induced by the restriction of A243057 (or A243059 or A242420) to the union of {1} and A102750.

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Author

Keywords

Examples

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Programs

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Formula

A253568 Even bisection of A122111: a(n) = A122111(2*n).

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Scheme

Formula

A243287 a(1)=1, and for n > 1, if n is k-th number divisible by the square of its largest prime factor (i.e., n = A070003(k)), a(n) = 1 + (2a(k)); otherwise, when n = A102750(k), a(n) = 2a(k).