A013928 -id:A013928 - cp's OEIS frontend

A243352 If n is k-th squarefree number [i.e., n = A005117(k)], a(n) = 2k-1; otherwise, when n is k-th nonsquarefree number [i.e., n = A013929(k)], a(n) = 2k.

1, 3, 5, 2, 7, 9, 11, 4, 6, 13, 15, 8, 17, 19, 21, 10, 23, 12, 25, 14, 27, 29, 31, 16, 18, 33, 20, 22, 35, 37, 39, 24, 41, 43, 45, 26, 47, 49, 51, 28, 53, 55, 57, 30, 32, 59, 61, 34, 36, 38, 63, 40, 65, 42, 67, 44, 69, 71, 73, 46, 75, 77, 48, 50, 79, 81, 83, 52, 85, 87, 89

Offset: 1

Antti Karttunen, Jun 04 2014

Odd numbers occur (in order) at the positions given by squarefree numbers, A005117, and even numbers occur (in order) at the positions given by their complement, nonsquarefree numbers, A013929.

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

Inverse: A088610. Cf. A243343, A072062.

(define (A243352 n) (if (zero? (A008966 n)) (* 2 (A057627 n)) (+ (* 2 (A013928 n)) 1)))

If mu(n) = 0, a(n) = 2*A057627(n), otherwise, a(n) = 1 + 2 * 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, A000035(a(n)) = A008966(n) = A008683(n)^2, or equally, a(n) = mu(n) modulo 2.

A285111 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(A005117(1+n)) = 2a(n), a(A065642(n)) = 1 + 2a(n).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 7, 5, 12, 16, 13, 14, 10, 24, 15, 32, 27, 26, 25, 28, 20, 48, 55, 9, 30, 11, 21, 64, 54, 52, 31, 50, 56, 40, 111, 96, 110, 18, 51, 60, 22, 42, 41, 49, 128, 108, 223, 17, 103, 104, 61, 62, 447, 100, 43, 112, 80, 222, 109, 192, 220, 57, 63, 36, 102, 120, 113, 44, 84, 82, 895, 98, 256, 99, 221, 216, 446, 34, 207, 23

Offset: 1

Antti Karttunen, Apr 17 2017

nonn

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

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

Inverse: A285112.
Cf. A005117, A008683, A013928, A065642, A285328.
Similar or related permutations: A243343, A243345, A277695, A284571.

from operator import mul
from sympy import primefactors
from sympy.ntheory.factor_ import core
from functools import reduce
def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))
def a285328(n):
    if core(n) == n: return 1
    k=n - 1
    while k>0:
        if a007947(k) == a007947(n): return k
        else: k-=1
def a013928(n): return sum([1 for i in range(1, n) if core(i) == i])
def a(n):
    if n<3: return n - 1
    if core(n)==n: return 2*a(a013928(n))
    else: return 1 + 2*a(a285328(n))
print([a(n) for n in range(1, 121)]) # Indranil Ghosh, Apr 17 2017

a(1) = 0, a(2) = 1, and for n > 2, if A008683(n) <> 0 [when n is squarefree], a(n) = 2*a(A013928(n)), otherwise a(n) = 1 + 2*a(A285328(n)).

A373119 Cardinality of the largest subset of {1,...,n} such that no four distinct elements of this subset multiply to a square.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 15, 15, 15, 15, 16, 16, 17, 17, 17, 17, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 20, 20, 21, 21, 22, 22, 22, 22, 23, 23, 24, 24, 24, 24, 25, 25, 26, 26, 26

Offset: 1

Terence Tao, May 26 2024

nonn

a(n) >= A000720(n).
a(n) ~ n/log n (Erdős-Sárközy-Sós). Best bounds currently are due to Pach-Vizer.
a(n+1)-a(n) is either 0 or 1 for any n. (Is equal to 1 when n+1 is prime.)
If "four" is replaced by "one", "two", "three", "five", or "any odd", one obtains A028391, A013928, A372306, A373178, and A373114 respectively.

			a(7)=6, because the set {1,2,3,4,5,7} has no four distinct elements multiplying to a square, but {1,2,3,4,5,6,7} has 1*2*3*6 = 6^2.

P. Erdős, A. Sárközy, and V. T. Sós, On Product Representations of Powers, I, Europ. J. Combinatorics 16 (1995), 567-588.
P. Pach and M. Vizer, Improved Lower Bounds for Multiplicative Square-Free Sequences, The Electronic Journal of Combinatorics, Volume 30, Issue 4 (2023), P4.31.
Terence Tao, On product representations of squares, arXiv:2405.11610 [math.NT], May 2024.

Cf. A028391, A013928, A372306, A373114, A373178, A373195.

Lower bounded by A000720.

from math import isqrt
def is_square(n):
    return isqrt(n) ** 2 == n
def valid_subset(A):
    length = len(A)
    for i in range(length):
        for j in range(i + 1, length):
            for k in range(j + 1, length):
                for l in range(k + 1, length):
                    if is_square(A[i] * A[j] * A[k] * A[l]):
                        return False
    return True
def largest_subset_size(N):
    from itertools import combinations
    for size in reversed(range(1, N + 1)):
        for subset in combinations(range(1, N + 1), size):
            if valid_subset(subset):
                return size
for N in range(1, 23):
    print(largest_subset_size(N))

from math import prod
from functools import lru_cache
from itertools import combinations
from sympy.ntheory.primetest import is_square
@lru_cache(maxsize=None)
def A373119(n):
    if n==1: return 1
    i = A373119(n-1)+1
    if sum(1 for p in combinations(range(1,n),3) if is_square(n*prod(p))) > 0:
        a = [set(p) for p in combinations(range(1,n+1),4) if is_square(prod(p))]
        for q in combinations(range(1,n),i-1):
            t = set(q)|{n}
            if not any(s<=t for s in a):
                return i
        else:
            return i-1
    else:
        return i # Chai Wah Wu, May 30 2024

a(22)-a(37) from Michael S. Branicky, May 26 2024
a(38)-a(63) from Martin Ehrenstein, May 27 2024
a(64)-a(69) from Jinyuan Wang, Dec 30 2024

A373178 Cardinality of the largest subset of {1,...,n} such that no five distinct elements of this subset multiply to a square.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 6, 7, 7, 7, 8, 8, 9, 10, 10, 10, 11, 11, 12, 12, 13, 13, 14, 15, 15, 16, 17, 18, 19, 19, 20, 20, 20, 21, 21, 21, 22, 23, 23, 24, 25, 26, 27, 28, 29, 30, 31, 31, 31, 31, 32, 33, 34, 34, 34, 35, 36, 37, 38, 39, 40, 41, 42, 42, 42, 43, 44, 45, 46, 46

Offset: 1

Terence Tao, May 26 2024

a(n) >= A373114(n).
The limiting value of a(n)/n is unknown, but (if it exists), it is strictly less than 1, and at least A246849 ~ 0.828499... (see cited paper of Tao).
a(n+1)-a(n) is either 0 or 1 for any n.
If "five" is replaced by "one", "two", "three", "four", or "odd number of", one obtains A028391, A013928, A372306, A373119, A373114 respectively.

			a(8)=7, because the set {1,2,3,4,5,7,8} has no five distinct elements multiplying to a square, but {1,2,3,4,5,6,7,8} has 1*2*3*4*6 = 12^2.

Thomas Bloom, Problem 121, Erdős Problems.
Terence Tao, On product representations of squares, arXiv:2405.11610 [math.NT], May 2024.
Terence Tao, Erdős problem database, see no. 121.

Similar to A028391, A013928, A372306, A373119. Lower bounded by A373114.

from math import isqrt
def is_square(n):
    return isqrt(n) ** 2 == n
def precompute_tuples(N):
    tuples = []
    for i in range(1, N + 1):
        for j in range(i + 1, N + 1):
            for k in range(j + 1, N + 1):
                for l in range(k + 1, N + 1):
                    for m in range(l + 1, N + 1):
                        if is_square(i * j * k * l * m):
                            tuples.append((i, j, k, l, m))
    return tuples
def valid_subset(A, tuples):
    set_A = set(A)
    for i, j, k, l, m in tuples:
        if i in set_A and j in set_A and k in set_A and l in set_A and m in set_A:
            return False
    return True
def largest_subset_size(N, tuples):
    from itertools import combinations
    for size in reversed(range(1, N + 1)):
        for subset in combinations(range(1, N + 1), size):
            if valid_subset(subset, tuples):
                return size
for N in range(1, 26):
    print(largest_subset_size(N, precompute_tuples(N)))

from math import prod
from functools import lru_cache
from itertools import combinations
from sympy.ntheory.primetest import is_square
@lru_cache(maxsize=None)
def A373178(n):
    if n==1: return 1
    i = A373178(n-1)+1
    if sum(1 for p in combinations(range(1,n),4) if is_square(n*prod(p))) > 0:
        a = [set(p) for p in combinations(range(1,n+1),5) if is_square(prod(p))]
        for q in combinations(range(1,n),i-1):
            t = set(q)|{n}
            if not any(s<=t for s in a):
                return i
        else:
            return i-1
    else:
        return i # Chai Wah Wu, May 30 2024

a(26)-a(38) from Michael S. Branicky, May 27 2024
a(39)-a(47) from Michael S. Branicky, May 30 2024
a(48)-a(70) from Martin Ehrenstein, May 31 2024

A081239 #{(i,j): mu(i)*mu(j) = 0, 1<=i,j<=n}, where mu=A008683 (Moebius function).

Original entry on oeis.org

0, 0, 0, 7, 9, 11, 13, 28, 45, 51, 57, 80, 88, 96, 104, 135, 145, 180, 192, 231, 245, 259, 273, 320, 369, 387, 440, 495, 517, 539, 561, 624, 648, 672, 696, 767, 793, 819, 845, 924, 952, 980, 1008, 1095, 1184, 1216, 1248, 1343, 1440, 1539, 1577, 1680, 1720

Offset: 1

Reinhard Zumkeller, Mar 11 2003

nonn

A081238(n) + a(n) + A081240(n) = n^2;

a(n) = a(n-1) + 2*n + 1 iff mu(n) = 0.

			n mu(n) ... n: 1 2 3 4 5 6 7 8
- ------ .... |----------------
1 .. +1 ..... | + - - 0 - + - 0
2 .. -1 ..... | - + + 0 + - + 0
3 .. -1 ..... | - + + 0 + - + 0
4 ... 0 ..... | 0 0 0 0 0 0 0 0
5 .. -1 ..... | - + + 0 + - + 0 a(8)=28, as there are
6 .. +1 ..... | + - - 0 - + - 0 28 '0's in the 8x8-square
7 .. -1 ..... | - + + 0 + - + 0
8 ... 0 ..... | 0 0 0 0 0 0 0 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..500

Cf. A057627.

a081239 n = length [() | u <- [1..n], v <- [1..n],
                         a008683 u * a008683 v == 0]
-- Reinhard Zumkeller, Aug 03 2012

a(n) = n^2 - A013928(n+1)^2. - Vladeta Jovovic, Mar 12 2003

A107079 Minimal number of squared primes in a squarefree gap of length n.

Original entry on oeis.org

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

Offset: 1

Paul Barry, May 10 2005

nonn

Antti Karttunen, Table of n, a(n) for n = 1..100000
Louis Marmet, First occurrences of squarefree gaps and an algorithm for their computation
Louis Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv preprint arXiv:1210.3829 [math.NT], 2012. - _N. J. A. Sloane_, Jan 01 2013

One more than A013928. A left inverse of A005117.

Cf. A045882, A107078.

a[n_] := Sum[Boole[SquareFreeQ[k]], {k, 1, n-1}] + 1;
Array[a, 100] (* Jean-François Alcover, Sep 11 2018, from A013928 *)

A107079(n)=1+sum(k=1,n-1,bitand(moebius(k),1)) \\ Charles R Greathouse IV, Sep 22 2008

from math import isqrt
from sympy import mobius
def A107079(n): return 1+sum(mobius(k)*((n-1)//k**2) for k in range(1,isqrt(n-1)+1)) # Chai Wah Wu, Jan 03 2024

a(n) = sum{k=0..n-1, moebius_mu(n-k-1) mod 2}.
a(n) = A013928(n+1) + A107078(n).
From Antti Karttunen, Oct 07 2016: (Start)
a(n) = 1 + A013928(n). [Cf. Charles R Greathouse IV's PARI-program.]
For all n >= 1, a(A005117(n)) = n. (End)

New definition from Charles R Greathouse IV, Sep 22 2008

A246353 If n = Sum 2^e_i, e_i distinct, then a(n) = Position of (product prime_{e_i+1}) among squarefree numbers (A005117).

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 11, 19, 6, 10, 14, 28, 23, 44, 65, 129, 8, 15, 21, 41, 34, 69, 101, 203, 48, 94, 144, 283, 233, 470, 703, 1405, 9, 17, 26, 49, 40, 80, 120, 236, 57, 111, 168, 334, 279, 554, 833, 1661, 89, 176, 261, 521, 438, 873, 1304, 2610, 609, 1217, 1827, 3650, 3046, 6091, 9131

Offset: 0

Antti Karttunen, Aug 23 2014

nonn

This is an inverse function to A048672. Note the indexing: here the domain starts from 0, but the range starts from 1, while in A048672 it is the opposite.

Sequence is obtained when the range of A019565 is compacted so that it becomes surjective on N, thus the logarithmic scatter plots look very similar. (Same applies to A064273). Compare also to the plot of A005940.

Antti Karttunen, Table of n, a(n) for n = 0..478
Index entries for sequences that are permutations of the natural numbers

Cf. A000040, A005940, A013928, A019565, A048672, A064273.

allocatemem(234567890);
default(primelimit, 2^22)
uplim_for_13928 = 13123111;
v013928 = vector(uplim_for_13928); A013928(n) = v013928[n];
v013928[1]=0; n=1; while((n < uplim_for_13928), if(issquarefree(n), v013928[n+1] = v013928[n]+1, v013928[n+1] = v013928[n]); n++);
A019565(n) = {factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A246353(n) = 1+A013928(A019565(n));
for(n=0, 478, write("b246353.txt", n, " ", A246353(n)));

from math import prod, isqrt
from sympy import prime, mobius
def A246353(n):
    m = prod(prime(i) for i,j in enumerate(bin(n)[-1:1:-1],1) if j=='1')
    return int(sum(mobius(k)*(m//k**2) for k in range(1, isqrt(m)+1))) # Chai Wah Wu, Feb 22 2025

(definec (A246353 n) (let loop ((n n) (i 1) (p 1)) (cond ((zero? n) (A013928 (+ 1 p))) ((odd? n) (loop (/ (- n 1) 2) (+ 1 i) (* p (A000040 i)))) (else (loop (/ n 2) (+ 1 i) p)))))

a(n) = A013928(1+A019565(n)) = 1 + A013928(A019565(n)).
a(n) = A064273(n) + 1.
For all n >= 0, A048672(a(n)) = n.
For all n >= 1, a(A048672(n)) = n.

A278100 Number of squarefree positive integers less than n^2.

Original entry on oeis.org

0, 3, 6, 11, 16, 23, 31, 39, 50, 61, 75, 89, 103, 120, 139, 157, 177, 199, 219, 243, 269, 297, 323, 351, 381, 412, 444, 477, 513, 547, 584, 624, 660, 703, 745, 789, 835, 882, 928, 977, 1025, 1073, 1124, 1174, 1230, 1285, 1342, 1400, 1460, 1523, 1582, 1645, 1708

Offset: 1

Jason Kimberley, Nov 12 2016

Jason Kimberley, Table of n, a(n) for n = 1..5000

Cf. A005117, A013928, A059956.

This is the row length sequence of A277648 and A278101.

A278100:=func;
[A278100(n):n in[1..53]]; // in cubic time

Table[Count[Range[n^2], k_ /; SquareFreeQ@ k], {n, 53}] (* Michael De Vlieger, Nov 24 2016 *)
Module[{nn=60,sf},sf=Accumulate[Table[If[SquareFreeQ[n],1,0],{n,0,nn^2}]];Table[sf[[k^2]],{k,nn}]] (* Harvey P. Dale, Nov 14 2020 *)

a(n) = #select(x->issquarefree(x), vector(n^2-1, k, k)); \\ Michel Marcus, Nov 12 2016

a(n) = A013928(n^2).

a(n) ~ 6*n^2/Pi^2 + O(n). - Amiram Eldar, Mar 09 2021

A284571 Permutation of natural numbers: a(1) = 1, a(A005117(1+n)) = 2a(n), a(A065642(1+n)) = 1 + 2a(n).

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 16, 9, 5, 12, 32, 17, 18, 10, 24, 33, 64, 65, 34, 11, 36, 20, 48, 129, 7, 66, 19, 37, 128, 130, 68, 49, 22, 72, 40, 97, 96, 258, 14, 69, 132, 38, 74, 73, 21, 256, 260, 81, 13, 29, 136, 15, 98, 521, 44, 39, 144, 80, 194, 257, 192, 516, 23, 137, 28, 138, 264, 45, 76, 148, 146, 197, 42, 512, 147, 193, 520, 162, 26, 27

Offset: 1

Antti Karttunen, Apr 17 2017

nonn

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

Inverse: A284572.
Cf. A005117, A008683, A013928, A065642, A285328.
Similar or related permutations: A243343, A243345, A277695, A285111.

from operator import mul
from sympy import primefactors
from sympy.ntheory.factor_ import core
def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))
def a285328(n):
    if core(n) == n: return 1
    k=n - 1
    while k>0:
        if a007947(k) == a007947(n): return k
        else: k-=1
def a013928(n): return sum(1 for i in range(1, n) if core(i) == i)
def a(n):
    if n==1: return 1
    if core(n)==n: return 2*a(a013928(n))
    else: return 1 + 2*a(a285328(n) - 1)
[a(n) for n in range(1, 121)] # Indranil Ghosh, Apr 17 2017

a(1) = 1, for n > 1, if A008683(n) <> 0 [when n is squarefree], a(n) = 2*a(A013928(n)), otherwise a(n) = 1 + 2*a(A285328(n)-1).

A285719 a(1) = 1, and for n > 1, a(n) = the largest squarefree number k such that n-k is also squarefree.

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 6, 7, 7, 7, 10, 11, 11, 13, 14, 15, 15, 17, 17, 19, 19, 21, 22, 23, 23, 23, 26, 26, 26, 29, 30, 31, 31, 33, 34, 35, 35, 37, 38, 39, 39, 41, 42, 43, 43, 43, 46, 47, 47, 47, 46, 51, 51, 53, 53, 55, 55, 57, 58, 59, 59, 61, 62, 62, 62, 65, 66, 67, 67, 69, 70, 71, 71, 73, 74, 74, 74, 77, 78, 79, 79, 79, 82, 83, 83, 85, 86, 87, 87, 89, 89, 91, 91

Offset: 1

Antti Karttunen, May 02 2017

nonn

For any n > 1 there is at least one decomposition of n as a sum of two squarefree numbers (cf. A071068 and the Mathematics Stack Exchange link). Of all pairs (x,y) of positive squarefree numbers for which x <= y and x+y = n, sequences A285718 and A285719 give the unique pair for which the difference y-x is the largest possible.

Note: a(n+1) differs from A070321(n) for the first time at n=50, with a(51) = 46, while A070321(50) = 47.

			For n=51 we see that 50 (2*5*5), 49 (7*7) and 48 (2^4 * 3) are all nonsquarefree (A013929). 47 (a prime) is squarefree, but 51 - 47 = 4 is not. On the other hand, both 46 (2*23) and 5 are squarefree numbers, thus a(51) = 46.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Mathematics Stack Exchange, Sums of square free numbers, is this conjecture equivalent to Goldbach's conjecture? (See especially the answer of Aryabhata)
K. Rogers, The Schnirelmann density of the squarefree integers, Proc. Amer. Math. Soc. 15 (1964), pp. 515-516.

Cf. A005117, A008683, A013928, A013929, A070321, A071068, A285718, A285735.

lsfn[n_]:=Module[{k=n-1},While[!SquareFreeQ[k]||!SquareFreeQ[n-k],k--];k]; Join[{1},Array[ lsfn,100,2]] (* Harvey P. Dale, Apr 27 2023 *)

from sympy.ntheory.factor_ import core
def issquarefree(n): return core(n) == n
def a285718(n):
    if n==1: return 0
    x = 1
    while True:
        if issquarefree(x) and issquarefree(n - x):return x
        else: x+=1
def a285719(n): return n - a285718(n)
print([a285719(n) for n in range(1, 121)]) # Indranil Ghosh, May 02 2017

(define (A285719 n) (- n (A285718 n)))
(define (A285719 n) (if (= 1 n) n (let loop ((k (A013928 n))) (if (not (zero? (A008683 (- n (A005117 k))))) (A005117 k) (loop (- k 1))))))

a(n) = n - A285718(n).

cp's OEIS Frontend

A243352 If n is k-th squarefree number [i.e., n = A005117(k)], a(n) = 2k-1; otherwise, when n is k-th nonsquarefree number [i.e., n = A013929(k)], a(n) = 2k.

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A285111 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(A005117(1+n)) = 2*a(n), a(A065642(n)) = 1 + 2*a(n).

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A373119 Cardinality of the largest subset of {1,...,n} such that no four distinct elements of this subset multiply to a square.

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A373178 Cardinality of the largest subset of {1,...,n} such that no five distinct elements of this subset multiply to a square.

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A081239 #{(i,j): mu(i)*mu(j) = 0, 1<=i,j<=n}, where mu=A008683 (Moebius function).

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A107079 Minimal number of squared primes in a squarefree gap of length n.

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PARI

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A246353 If n = Sum 2^e_i, e_i distinct, then a(n) = Position of (product prime_{e_i+1}) among squarefree numbers (A005117).

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PARI

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Scheme

Formula

A285111 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(A005117(1+n)) = 2a(n), a(A065642(n)) = 1 + 2a(n).

A284571 Permutation of natural numbers: a(1) = 1, a(A005117(1+n)) = 2a(n), a(A065642(1+n)) = 1 + 2a(n).