A243282 -id:A243282 - cp's OEIS frontend

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).

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)).

A243285 Number of integers 1 <= k <= n which are not divisible by the square of their largest noncomposite divisor.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 02 2014

nonn

a(n) tells how many natural numbers <= n there are which are not divisible by the square of their largest noncomposite divisor.
The largest noncomposite divisor of 1 is 1 itself, and 1 is divisible by 1^2, thus 1 is not included in the count, and a(1)=0.
The "largest noncomposite divisor" for any integer > 1 means the same thing as the largest prime divisor, and thus we are counting the terms of A102750 (Numbers n such that square of largest prime dividing n does not divide n).
Thus this is the partial sums of the characteric function for A102750.

			For n = 9, there are numbers 2, 3, 5, 6 and 7 which are not divisible by the square of their largest prime factor, while 1 is excluded (no prime factors) and 4 and 8 are divisible both by 2^2 and 9 is divisible by 3^2. Thus a(9) = 5.

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

Cf. A243282, A243283, A102750, A070003, A013928, A057627.

ndsQ[n_]:=Mod[n,Max[Select[Divisors[n],!CompositeQ[#]&]]^2]!=0; Accumulate[Table[If[ ndsQ[n],1,0],{n,80}]] (* Harvey P. Dale, Oct 14 2023 *)

from sympy import primefactors
def a243285(n): return 0 if n==1 else sum([1 for k in range(2, n + 1) if k%(primefactors(k)[-1]**2)!=0]) # Indranil Ghosh, Jun 15 2017

Scheme
```
(define (A243285 n) (- n (A243283 n)))
```

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

For all n, a(A102750(n)) = n, thus this sequence works also as an inverse function for the injection A102750.

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.

cp's OEIS Frontend

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).

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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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Formula

A243285 Number of integers 1 <= k <= n which are not divisible by the square of their largest noncomposite divisor.

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Formula

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

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cp's OEIS Frontend

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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Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A243285 Number of integers 1 <= k <= n which are not divisible by the square of their largest noncomposite divisor.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Scheme

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

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).