A135141 -id:A135141 - cp's OEIS frontend

A233277 Permutation of nonnegative integers obtained by entangling complementary pair A005187 & A055938 with odd and even numbers.

0, 1, 2, 3, 5, 4, 6, 7, 11, 10, 9, 13, 8, 12, 14, 15, 23, 22, 21, 19, 20, 18, 27, 17, 26, 25, 29, 16, 24, 28, 30, 31, 47, 46, 45, 43, 44, 42, 39, 41, 38, 37, 55, 40, 36, 54, 35, 53, 34, 51, 59, 52, 50, 33, 49, 58, 57, 61, 32, 48, 56, 60, 62, 63, 95, 94, 93, 91

Offset: 0

Antti Karttunen, Dec 18 2013

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191
Indranil Ghosh, Python program to generate the sequence
Indranil Ghosh, PARI program to generate the sequence
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A233278.
Cf. also A079559, A213714, A234017, A054429.
Similarly constructed permutation pairs: A135141/A227413, A232751/A232752, A233275/A233276, A233279/A233280, A003188/A006068.

a(0)=0, a(1)=1, and thereafter, if A079559(n)=0, a(n) = 2*a(A234017(n)), else a(n) = 1+(2*a(A213714(n)-1)).

a(n) = A054429(A233275(n)). [Follows from the definitions of these sequences]

A233278 a(0)=0, a(1)=1, after which a(2n) = A055938(a(n)), a(2n+1) = A005187(1+a(n)).

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 7, 12, 10, 9, 8, 13, 11, 14, 15, 27, 23, 21, 19, 20, 18, 17, 16, 28, 25, 24, 22, 29, 26, 30, 31, 58, 53, 48, 46, 44, 41, 40, 38, 43, 39, 37, 35, 36, 34, 33, 32, 59, 54, 52, 49, 51, 47, 45, 42, 60, 56, 55, 50, 61, 57, 62, 63, 121, 113, 108

Offset: 0

Antti Karttunen, Dec 18 2013

This permutation is obtained by "entangling" even and odd numbers with complementary pair A055938 & A005187, meaning that it can be viewed as a binary tree. Each child to the left is obtained by applying A055938(n) to the parent node containing n, and each child to the right is obtained as A005187(n+1):
                                     0
                                     |
                  ...................1...................
                 2                                       3
       5......../ \........4                   6......../ \........7
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  12       10          9       8          13       11         14       15
27  23   21  19      20 18   17 16      28  25   24  22     29  26   30  31
etc.
For n >= 1, A256991(n) gives the contents of the immediate parent node of the node containing n, while A070939(n) gives the total distance to zero at the root from the node containing n, with A256478(n) telling how many of the terms encountered on that journey are terms of A005187 (including the penultimate 1 but not the final 0 in the count), while A256479(n) tells how many of them are terms of A055938.
Permutation A233276 gives the mirror image of the same tree.

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

Inverse permutation: A233277.
Cf. A005187, A054429, A055938, A256991, A256478, A256479.
Cf. also A070939 (the binary width of both n and a(n)).
Related arrays: A255555, A255557.
Similarly constructed permutation pairs: A005940/A156552, A135141/A227413, A232751/A232752, A233275/A233276, A233279/A233280, A003188/A006068.

a(0)=0, a(1)=1, and thereafter, a(2n) = A055938(a(n)), a(2n+1) = A005187(1+a(n)).
As a composition of related permutations:
a(n) = A233276(A054429(n)).

Name changed and the illustration of binary tree added by Antti Karttunen, Apr 19 2015

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).

A243343 a(1)=1; thereafter, if n is the k-th squarefree number (i.e., n = A005117(k)), a(n) = 1 + (2a(k-1)); otherwise, when n is k-th nonsquarefree number (i.e., n = A013929(k)), a(n) = 2a(k).

Original entry on oeis.org

1, 3, 7, 2, 15, 5, 31, 6, 14, 11, 63, 4, 13, 29, 23, 30, 127, 10, 9, 62, 27, 59, 47, 12, 28, 61, 22, 126, 255, 21, 19, 8, 125, 55, 119, 26, 95, 25, 57, 58, 123, 45, 253, 46, 60, 511, 43, 254, 20, 18, 39, 124, 17, 54, 251, 118, 111, 239, 53, 94, 191, 51, 24, 56

Offset: 1

Antti Karttunen, Jun 03 2014

nonn

This is an instance of an "entanglement permutation", where two pairs of complementary subsets of natural numbers are interwoven with each other. In this case complementary pair A005117/A013929 (numbers which are squarefree/not squarefree) is entangled with complementary pair odd/even numbers (A005408/A005843).
Thus this shares with permutation A243352 the property that each term of A005117 is mapped bijectively to a unique odd number and likewise each term of A013929 is mapped (bijectively) to a unique even number. However, instead of placing terms into those positions in monotone order this sequence recursively permutes the order of both subsets with the emerging permutation itself.
Are there any other fixed points than 1, 13, 54, 120, 1389, 3183, ... ?

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

Inverse: A243344.
Cf. A005843, A005408, A008966, A005117, A013929, A013928, A057627.
Similar permutations: A243352 (simple variant), A243345-A243346, A243347, A243287-A243288, A135141-A227413, A237126-A237427, A193231.

a(1) = 1; thereafter, if A008966(n) = 0 (i.e., n is a term of A013929, not squarefree), a(n) = 2*a(A057627(n)); otherwise a(n) = 2*a(A013928(n+1)-1)+1 (where A057627 and A013928(n+1) give the number of integers <= n divisible/not divisible by a square greater than one).

For all n, A000035(a(n)) = A008966(n) = A008683(n)^2, or equally, a(n) = mu(n) modulo 2. The same property holds for A243352.

A245605 Permutation of natural numbers: a(1) = 1, a(2n) = 2 * a(A064989(2n-1)), a(2n-1) = 1 + (2 * a(A064989(2n-1)-1)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 10, 7, 8, 13, 18, 17, 26, 11, 12, 37, 34, 25, 74, 15, 16, 69, 50, 21, 14, 19, 20, 33, 138, 41, 66, 35, 52, 53, 22, 277, 82, 31, 32, 45, 554, 65, 90, 27, 36, 1109, 130, 101, 42, 43, 28, 73, 2218, 149, 30, 71, 104, 57, 146, 209, 114, 51, 148, 133, 70, 293, 418, 555, 164, 141, 586, 329, 282, 75, 68, 105, 106, 1173, 658, 23, 24

Offset: 1

Antti Karttunen, Jul 29 2014

nonn

The even bisection halved gives A245607. The odd bisection incremented by one and halved gives A245707.

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

Inverse: A245606.
Cf. A048673, A064216, A064989, A245607, A245705, A245707.
Similar entanglement permutations: A244319, A005940, A163511, A243287, A243343, A243345, A244321, A245603, A245613, A135141, A237427.

A064989(n) = my(f = factor(n)); for(i=1, #f~, if((2 == f[i,1]),f[i,1] = 1,f[i,1] = precprime(f[i,1]-1))); factorback(f);
A245605(n) = if(1==n, 1, if(0==(n%2), 2*A245605(A064989(n-1)), 1+(2*A245605(A064989(n)-1))));
for(n=1, 10001, write("b245605.txt", n, " ", A245605(n)));

;; With memoization-macro definec.
(definec (A245605 n) (cond ((= 1 n) 1) ((even? n) (* 2 (A245605 (A064989 (- n 1))))) (else (+ 1 (* 2 (A245605 (-1+ (A064989 n))))))))

a(1) = 1, a(2n) = 2 * a(A064989(2n-1)), a(2n-1) = 1 + (2 * a(A064989(2n-1)-1)).
a(1) = 1, a(2n) = 2 * a(A064216(n)), a(2n-1) = 1 + (2 * a(A064216(n)-1)).
As a composition of related permutations:
a(n) = A245607(A048673(n)).

A236854 Self-inverse permutation of natural numbers: a(1)=1, then a(p_n)=c_{a(n)}, a(c_n)=p_{a(n)}, where p_n = n-th prime, c_n = n-th composite.

Original entry on oeis.org

1, 4, 9, 2, 16, 7, 6, 23, 3, 53, 26, 17, 14, 13, 83, 5, 12, 241, 35, 101, 59, 43, 8, 41, 431, 11, 37, 1523, 75, 149, 39, 547, 277, 191, 19, 179, 27, 3001, 31, 157, 24, 12763, 22, 379, 859, 167, 114, 3943, 1787, 1153, 67, 1063, 10, 103, 27457, 127, 919, 89, 21

Offset: 1

Antti Karttunen, Feb 01 2014, based on Katarzyna Matylla's original but misplaced definition for A135044 from Feb 11 2008

nonn

Shares with A026239 the property that the prime-positions 2, 3, 5, 7, ... contain only composite values and the composite-positions 4, 6, 8, 9, ..., etc. contain only prime values. However, instead of placing terms in those subsets in monotone order this sequence recursively permutes the order of both subsets with the emerging permutation itself, so this is a kind of "deep" variant of A026239. Alternatively, this can be viewed as yet another "entanglement permutation", where two pairs of complementary subsets of natural numbers are entangled with each other. In this case a complementary pair primes/composites (A000040/A002808) is entangled with a complementary pair composites/primes.

Maps A006508 to A007097 and vice versa.

			a(5)=c(a(3))=c(9)=16, because 5=prime(3), and the 9th composite number is c(9)=16.
Thus a(10)=prime(a(5))=prime(16)=53 (since 10 is the 5th composite), a(18)=prime(a(10))=prime(53)=241 (since 18 is the 10th composite), a(28)=prime(a(18))=prime(241)=1523.
A significant record value is a(198) = prime(a(152)) = prime(563167303) since 198=c(152); a(152)=prime(a(115)) since 152=c(115); a(115)=prime(a(84)); a(84)=prime(a(60)); a(60)=prime(a(42)); a(42)=prime(a(28)).

Chai Wah Wu, Table of n, a(n) for n = 1..735 (n = 1..150 from Alois P. Heinz)
Index entries for sequences that are permutations of the natural numbers

Differs from A135044 for the first time at n=8, where A135044(8)=13, while here a(8)=23.

Cf. A026239, A135141/A227413, A006508, A007097, A000040, A002808, A000720, A065855.

terms = 150; cc = Select[Range[4, 2 terms^2(*empirical*)], CompositeQ]; compositePi[k_?CompositeQ] := FirstPosition[cc, k][[1]]; a[1] = 1; a[p_?PrimeQ] := a[p] = cc[[a[PrimePi[p]]]]; a[k_] := a[k] = Prime[a[ compositePi[k]]]; Array[a, terms] (* Jean-François Alcover, Mar 02 2016 *)

A236854(n)={if(isprime(n), A002808(A236854(primepi(n))), n==1&&return(1);prime(A236854(n-primepi(n)-1)))} \\ without memoization: not much slower. - M. F. Hasler, Feb 03 2014

a236854=vector(999);a236854[1]=1;A236854(n)={a236854[n]&&return(a236854[n]); a236854[n]=if(isprime(n), A002808(A236854(primepi(n))), prime(A236854(n-primepi(n)-1)))} \\ Version with memoization. - M. F. Hasler, Feb 03 2014

from sympy import primepi, prime, isprime
def a002808(n):
    m, k = n, primepi(n) + 1 + n
    while m != k: m, k = k, primepi(k) + 1 + n
    return m # this function from Chai Wah Wu
def a(n): return n if n<2 else a002808(a(primepi(n))) if isprime(n) else prime(a(n - primepi(n) - 1))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 07 2017

a(1)=1, a(p_i) = A002808(a(i)) for primes with index i, a(c_j) = A000040(a(j)) for composites with index j (where 4 has index 1, 6 has index 2, and so on).

Values double-checked by M. F. Hasler, Feb 03 2014

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)).

A232751 Permutation of natural numbers obtained by entangling even and odd numbers with Hofstadter's complementary pair A005228 & A030124; inverse permutation to A232752.

Original entry on oeis.org

0, 1, 3, 2, 7, 5, 15, 6, 11, 31, 13, 23, 4, 63, 27, 47, 9, 127, 14, 55, 95, 19, 255, 29, 111, 191, 10, 39, 511, 59, 223, 383, 21, 79, 1023, 30, 119, 447, 767, 43, 159, 2047, 61, 239, 895, 12, 1535, 87, 319, 4095, 123, 479, 1791, 25, 3071, 175, 22, 639, 8191

Offset: 0

Antti Karttunen, Nov 30 2013

The permutation A135141 was obtained in analogous way by entangling even and odd numbers with primes and composites.
Note how all even numbers occur in positions given by A005228 from its second term 3 onward: 3, 7, 12, 18, 26, 35, 45, ... .
Note how all odd numbers occur in positions given by A030124: 2, 4, 5, 6, 8, 9, 10, 11, 13, 14, 15, ... .
See also the comments in A232752.
Interesting observation: For all numbers of form (2^n)-1, from 7 onward, the next term in the sequence which has that (2^n)-1 as its proper prefix (in decimal notation), appears to be 10*((2^n)-1)+9. For example, a(4)=7 and a(33)=79 is the first term of more than one decimal digits beginning with 7. For the higher values of A000225, we have examples of a(6)=15 & a(40)=159, a(9)=31 & a(48)=319, a(13)=63 & a(57)=639, a(17)=127 & a(66)=1279, a(22)=255 & a(76)=2559, a(28)=511 & a(87)=5119, a(34)=1023 & a(99)=10239, a(41)=2047 & a(111)=20479, a(49)=4095 & a(124)=40959, a(58)=8191 & a(138)=81919, a(67)=16383 & a(153)=163839, a(77)=32767 & a(168)=327679, a(88)=65535 & a(184)=655359.
So while each A000225(n) occurs at positions given by sequence 1, 2, 4, 6, 9, 13, 17, 22, 28, 34, 41, 49, 58, 67, 77, 88, 100, 112, 125, 139, 154, 169, 185, 202, 220, 239, 258, 278, 299, 321, 344, 367, ... (which from 2 onward are A232739, the iterates of A030124, cf. comment at A232752), each (10*A000225(n))+9 occurs at positions given by sequence 21, 27, 33, 40, 48, 57, 66, 76, 87, 99, 111, 124, 138, 153, 168, 184, 201, 219, 238, 257, 277, 298, 320, 343, 366, 390, 415, 441, 468, 496, 524, 553, ... Note how these seem to be one less than the previous sequence shifted 7 steps left.

Inverse permutation: A232752.
Cf. A135141, A005228, A030124, A005843 & A005408, A000079, A000225, A232747, A232749, A232739.
Cf. also the permutation pair A167151 & A225850.

a(0)=0, a(1)=1; for n > 1, when A232747(n)>0 (when n is in A005228), a(n) = 2*a(A232747(n)-1), otherwise (when n is in A030124) a(n) = (2*a(A232749(n))) + 1.

For all n >= 1, a(A232739(n)) = A000225(n+1).

A243346 a(1) = 1, a(2n) = A005117(1+a(n)), a(2n+1) = A013929(a(n)), where A005117 are squarefree and A013929 are nonsquarefree numbers.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 12, 5, 9, 13, 24, 10, 18, 19, 32, 7, 16, 14, 25, 21, 36, 38, 63, 15, 27, 30, 49, 31, 50, 53, 84, 11, 20, 26, 45, 22, 40, 39, 64, 34, 54, 59, 96, 62, 99, 103, 162, 23, 44, 42, 72, 47, 80, 79, 126, 51, 81, 82, 128, 86, 136, 138, 220, 17, 28, 33, 52, 41, 68, 73, 120

Offset: 1

Antti Karttunen, Jun 03 2014

nonn

This permutation entangles complementary pair A005843/A005408 (even/odd numbers) with complementary pair A005117/A013929 (numbers which are squarefree/are not squarefree).

Antti Karttunen, Table of n, a(n) for n = 1..8189
Index entries for sequences that are permutations of the natural numbers

Inverse: A243345.
Cf. A005843, A005408, A008966, A005117, A013929, A013928, A057627.
Similar permutations: A075378, A243343-A243344, A243347, A243287-A243288, A135141-A227413, A237126-A237427, A193231.

a(1) = 1, a(2n) = A005117(1+a(n)), a(2n+1) = A013929(a(n)).

For all n > 1, A008966(a(n)) = A000035(n+1), or equally, mu(a(n)) + 1 = n modulo 2, where mu is Moebius mu (A008683). [A property shared with a simpler variant A075378].

A243347 a(1)=1, and for n>1, if mu(n) = 0, a(n) = A005117(1+a(A057627(n))), otherwise, a(n) = A013929(a(A013928(n))).

Original entry on oeis.org

1, 4, 12, 2, 32, 8, 84, 6, 19, 24, 220, 3, 18, 50, 63, 53, 564, 13, 9, 138, 49, 128, 162, 10, 31, 136, 38, 365, 1448, 36, 25, 5, 351, 126, 332, 30, 414, 27, 81, 82, 348, 99, 931, 103, 86, 3699, 96, 929, 21, 14, 64, 223, 16, 79, 892, 210, 325, 847, 80, 265, 1056, 72, 15, 51, 208, 212, 884, 221, 256

Offset: 1

Antti Karttunen, Jun 03 2014

nonn

Self-inverse permutation of natural numbers.

Shares with A088609 the property that after 1, positions indexed by squarefree numbers larger than one, A005117(n+1): 2, 3, 5, 6, 7, 10, 11, 13, 14, ... contain only nonsquarefree numbers A013929: 4, 8, 9, 12, 16, 18, 20, 24, ..., and vice versa. However, instead of placing terms in those subsets in monotone order this sequence recursively permutes the order of both subsets with the emerging permutation itself, thus implementing a kind of "deep" variant of A088609. Alternatively, 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 complementary pair A005117/A013929 is entangled with complementary pair A013929/A005117.

Antti Karttunen, Table of n, a(n) for n = 1..520
Index entries for sequences that are permutations of the natural numbers

Cf. A008966, A005117, A013929, A013928, A057627, A088609, A243348.

Similar permutations: A236854, A235491, A243343-A243346, A243347, A243287-A243288, A135141-A227413, A237126-A237427, A193231.

a(1), and for n>1, if mu(n) = 0, a(n) = A005117(1+a(A057627(n))), otherwise, a(n) = A013929(a(A013928(n))). [Here mu is Moebius mu-function, A008683, which is zero only when n is a nonsquarefree number, one of the numbers in A013929.]

For all n > 1, A008966(a(n)) = 1 - A008966(n), or equally, mu(a(n)) + 1 = mu(n) modulo 2, where mu is Moebius mu (A008683). [Note: Permutation A088609 satisfies the same condition.]

cp's OEIS Frontend

A233277 Permutation of nonnegative integers obtained by entangling complementary pair A005187 & A055938 with odd and even numbers.

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A233278 a(0)=0, a(1)=1, after which a(2n) = A055938(a(n)), a(2n+1) = A005187(1+a(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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A243343 a(1)=1; thereafter, if n is the k-th squarefree number (i.e., n = A005117(k)), a(n) = 1 + (2*a(k-1)); otherwise, when n is k-th nonsquarefree number (i.e., n = A013929(k)), a(n) = 2*a(k).

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A245605 Permutation of natural numbers: a(1) = 1, a(2n) = 2 * a(A064989(2n-1)), a(2n-1) = 1 + (2 * a(A064989(2n-1)-1)).

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A236854 Self-inverse permutation of natural numbers: a(1)=1, then a(p_n)=c_{a(n)}, a(c_n)=p_{a(n)}, where p_n = n-th prime, c_n = n-th composite.

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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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A232751 Permutation of natural numbers obtained by entangling even and odd numbers with Hofstadter's complementary pair A005228 & A030124; inverse permutation to A232752.

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A243346 a(1) = 1, a(2n) = A005117(1+a(n)), a(2n+1) = A013929(a(n)), where A005117 are squarefree and A013929 are nonsquarefree numbers.

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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 + (2a(k)); otherwise, when n = A102750(k), a(n) = 2a(k).

A243343 a(1)=1; thereafter, if n is the k-th squarefree number (i.e., n = A005117(k)), a(n) = 1 + (2a(k-1)); otherwise, when n is k-th nonsquarefree number (i.e., n = A013929(k)), a(n) = 2a(k).