A005117 -id:A005117 - cp's OEIS frontend

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))).

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.]

A284311 Array T(n,k) read by antidiagonals (downward): T(1,k) = A005117(k+1) (squarefree numbers > 1); for n > 1, columns are nonsquarefree numbers (in ascending order) with exactly the same prime factors as T(1,k).

Original entry on oeis.org

2, 3, 4, 5, 9, 8, 6, 25, 27, 16, 7, 12, 125, 81, 32, 10, 49, 18, 625, 243, 64, 11, 20, 343, 24, 3125, 729, 128, 13, 121, 40, 2401, 36, 15625, 2187, 256, 14, 169, 1331, 50, 16807, 48, 78125, 6561, 512, 15, 28, 2197, 14641, 80, 117649, 54, 390625, 19683, 1024

Offset: 1

Bob Selcoe, Mar 24 2017

A permutation of the natural numbers > 1.

T(1,k)= A005117(m) with m > 1; terms in column k = T(1,k) * A162306(T(1,k)) only not bounded by T(1,k). Let T(1,k) = b. Terms in column k are multiples of b and numbers c such that c | b^e with e >= 0. Alternatively, terms in column k are multiples bc with c those numbers whose prime divisors p also divide b. - Michael De Vlieger, Mar 25 2017

			Array starts:
    2    3     5  6      7  10       11        13  14  15
    4    9    25 12     49  20      121       169  28  45
    8   27   125 18    343  40     1331      2197  56  75
   16   81   625 24   2401  50    14641    371293  98 135
   32  243  3125 36  16807  80   161051   4826809 112 225
   64  729 15625 48 117649 100  1771561  62748517 196 375
  128 2187 78125 54 823543 160 19487171 815730721 224 405
Column 6 is: T(1,6) = 2*5; T(2,6) = 2^2*5; T(3,6) = 2^3*5; T(4,6) = 2*5^2; T(5,6) = 2^4*5, etc.

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Cf. A005117 (squarefree numbers), A033845 (column 4), columns 1,2,3,5 are powers of primes, A033846 (column 6), A033847 (column 9), A033849 (column 10).
The columns that are powers of primes have indices A071403(n) - 1. - Michel Marcus, Mar 24 2017
See also A007947; the k-th column of the array corresponds to the numbers with radical A005117(k+1). - Rémy Sigrist, Mar 24 2017
Cf. A284457 (this sequence read by antidiagonals upwards), A285321 (a similar array, but columns come in different order).
Cf. A065642.
Cf. A008479 (index of the row where n is located), A285329 (of the column).

f[n_, k_: 1] := Block[{c = 0, sgn = Sign[k], sf}, sf = n + sgn; While[c < Abs[k], While[! SquareFreeQ@ sf, If[sgn < 0, sf--, sf++]]; If[sgn < 0, sf--, sf++]; c++]; sf + If[sgn < 0, 1, -1]] (* after Robert G. Wilson v at A005117 *); T[n_, k_] := T[n, k] = Which[And[n == 1, k == 1], 2, k == 1, f@ T[n - 1, k], PrimeQ@ T[n, 1], T[n, 1]^k, True, Module[{j = T[n, k - 1]/T[n, 1] + 1}, While[PowerMod[T[n, 1], j, j] != 0, j++]; j T[n, 1]]]; Table[T[n - k + 1, k], {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, Mar 25 2017 *)

(define (A284311 n) (A284311bi  (A002260 n) (A004736 n)))
(define (A284311bi row col) (if (= 1 row) (A005117 (+ 1 col)) (A065642 (A284311bi (- row 1) col))))
;; Antti Karttunen, Apr 17 2017

From Antti Karttunen, Apr 17 2017: (Start)
A(1,k) = A005117(1+k), A(n,k) = A065642(A(n-1,k)).
A(A008479(n), A285329(n)) = n for all n >= 2.
(End)

A328393 Numbers whose arithmetic derivative (A003415) is a squarefree number (A005117).

Original entry on oeis.org

2, 3, 5, 6, 7, 9, 10, 11, 13, 17, 18, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 37, 38, 41, 42, 43, 45, 47, 49, 53, 57, 58, 59, 61, 62, 63, 66, 67, 69, 70, 71, 73, 74, 75, 78, 79, 82, 83, 85, 89, 90, 93, 97, 98, 101, 102, 103, 105, 106, 107, 109, 110, 113, 114, 117, 118, 121, 126, 127, 129, 130, 131, 133, 134, 137, 139, 142

Offset: 1

Antti Karttunen, Oct 19 2019

nonn

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

Union of A000040 and A328234. Complement of A328303.
Cf. A005117, A008966, A003415.
Cf. A328252 (nonsquarefree terms), A157037, A192192, A327978 (other subsequences).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA328393(n) = issquarefree(A003415(n));

A062822 Sum of divisors of the squarefree numbers: sigma(A005117(n)).

Original entry on oeis.org

1, 3, 4, 6, 12, 8, 18, 12, 14, 24, 24, 18, 20, 32, 36, 24, 42, 30, 72, 32, 48, 54, 48, 38, 60, 56, 42, 96, 44, 72, 48, 72, 54, 72, 80, 90, 60, 62, 96, 84, 144, 68, 96, 144, 72, 74, 114, 96, 168, 80, 126, 84, 108, 132, 120, 90, 112, 128, 144, 120, 98, 102, 216, 104, 192

Offset: 1

Jason Earls, Jul 20 2001

nonn

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

Cf. A000203, A005117, A002117, A152649, A265668, A001221.

a062822 1 = 1
a062822 n = product $ map (+ 1) $ a265668_row n
-- Reinhard Zumkeller, Dec 13 2015

DivisorSigma[1,#]&/@Select[Range[150],SquareFreeQ] (* Harvey P. Dale, May 18 2014 *)

j=[]; for(n=1,200, if(issquarefree(n),j=concat(j, sigma(n)))); j

from math import isqrt
from sympy import mobius, divisor_sigma
def A062822(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return divisor_sigma(m) # Chai Wah Wu, Aug 12 2024

a(n) = Product_{k=1..A001221(n)} (A265668(n,k) + 1). - Reinhard Zumkeller, Dec 13 2015
From Amiram Eldar, Nov 21 2022: (Start)
a(n) = A000203(A005117(n)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/(72*zeta(3)) = A152649 / A002117 = 1.1254908... . (End)

A072048 Number of divisors of the squarefree numbers: tau(A005117(n)).

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 4, 2, 8, 2, 4, 4, 4, 2, 4, 4, 2, 8, 2, 4, 2, 4, 2, 4, 4, 4, 2, 2, 4, 4, 8, 2, 4, 8, 2, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 2, 4, 4, 4, 4, 2, 2, 8, 2, 8, 4, 2, 2, 8, 4, 2, 8, 4, 4, 4, 4, 4, 2, 4, 8, 2, 4, 4, 2, 8, 2, 4, 4, 4, 4, 4, 2, 2, 8, 4, 2

Offset: 1

Reinhard Zumkeller, Jun 09 2002

nonn

Also the number of cubefree numbers with the same squarefree kernel as the n-th squarefree number, see A073245.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
B. Gordon and K. Rogers, Sums of the divisor function, Canadian Journal of Mathematics, Vol. 16 (1964), pp. 151-158.

Cf. A000005, A001620, A062822, A065473.

Cf. A000079, A001221, A005117, A072047.

a072048 = (2 ^) . a072047  -- Reinhard Zumkeller, Dec 13 2015

A072048:=n->`if`(numtheory[issqrfree](n) = true, numtheory[tau](n), NULL); seq(A072048(k), k=1..100); # Wesley Ivan Hurt, Oct 13 2013

DivisorSigma[0, Select[Range[200], SquareFreeQ]] (* Amiram Eldar, Oct 29 2022 *)

from math import isqrt
from sympy import mobius, divisor_count
def A072048(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return divisor_count(m) # Chai Wah Wu, Aug 12 2024

a(n) = A000005(A005117(n)).
a(n) = 2^A072047(n) = 2^A001221(A005117(n)).
Sum_{k=1..n} a(k) ~ A * n * log(n) + B * n + O(n^(1/2+eps)), where  A = A065473, B = A * ((2*gamma-1) + 6 * Sum_{p prime} (p-1)*log(p)/(p^2*(p+2)) = 0.236184..., and gamma = A001620 (Gordon and Rogers, 1964). - Amiram Eldar, Oct 29 2022

A111059 a(n) = Product_{k=1..n} A005117(k), the product of the first n squarefree positive integers.

Original entry on oeis.org

1, 2, 6, 30, 180, 1260, 12600, 138600, 1801800, 25225200, 378378000, 6432426000, 122216094000, 2566537974000, 56463835428000, 1298668214844000, 33765373585944000, 979195833992376000, 29375875019771280000

Offset: 1

Leroy Quet, Oct 07 2005

nonn

Do all terms belong to A242031 (weakly decreasing prime signature)? - Gus Wiseman, May 14 2021

			Since the first 6 squarefree positive integers are 1, 2, 3, 5, 6, 7, the 6th term of the sequence is 1*2*3*5*6*7 = 1260.
From _Gus Wiseman_, May 14 2021: (Start)
The sequence of terms together with their prime signatures begins:
             1: ()
             2: (1)
             6: (1,1)
            30: (1,1,1)
           180: (2,2,1)
          1260: (2,2,1,1)
         12600: (3,2,2,1)
        138600: (3,2,2,1,1)
       1801800: (3,2,2,1,1,1)
      25225200: (4,2,2,2,1,1)
     378378000: (4,3,3,2,1,1)
    6432426000: (4,3,3,2,1,1,1)
  122216094000: (4,3,3,2,1,1,1,1)
(End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..417

A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes.
A072047 applies Omega to each squarefree number.
A246867 groups squarefree numbers by Heinz weight (row sums: A147655).
A261144 groups squarefree numbers by smoothness (row sums: A054640).
A319246 gives the sum of prime indices of each squarefree number.
A329631 lists prime indices of squarefree numbers (reversed: A319247).
Cf. A001221, A002110, A014466, A070826, A338901, A339360.

Rest[FoldList[Times,1,Select[Range[40],SquareFreeQ]]] (* Harvey P. Dale, Jun 14 2011 *)

m=30;k=1;for(n=1,m,if(issquarefree(n),print1(k=k*n,",")))

More terms from Klaus Brockhaus, Oct 08 2005

A243344 a(1) = 1, a(2n) = A013929(a(n)), a(2n+1) = A005117(1+a(n)).

Original entry on oeis.org

1, 4, 2, 12, 6, 8, 3, 32, 19, 18, 10, 24, 13, 9, 5, 84, 53, 50, 31, 49, 30, 27, 15, 63, 38, 36, 21, 25, 14, 16, 7, 220, 138, 136, 86, 128, 82, 81, 51, 126, 79, 80, 47, 72, 42, 44, 23, 162, 103, 99, 62, 96, 59, 54, 34, 64, 39, 40, 22, 45, 26, 20, 11, 564, 365

Offset: 1

Antti Karttunen, Jun 03 2014

nonn

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

Antti Karttunen, Table of n, a(n) for n = 1..4095
Index entries for sequences that are permutations of the natural numbers

Inverse: A243343.
Cf. A005843, A005408, A008966, A005117, A013929, A013928, A057627.
Similar permutations: A088610 (simple variant), A243345-A243346, A243347, A243287-A243288, A135141-A227413, A237126-A237427, A193231.

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

For all n, A008966(a(n)) = A000035(n), or equally, mu(a(n)) = n modulo 2, where mu is Moebius mu (A008683). [The same property holds for A088610.]

A243345 a(1)=1; thereafter, if n is k-th squarefree number [i.e., n = A005117(k)], a(n) = 2a(k-1); otherwise, when n is k-th nonsquarefree number [i.e., n = A013929(k)], a(n) = 2a(k)+1.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 16, 5, 9, 12, 32, 7, 10, 18, 24, 17, 64, 13, 14, 33, 20, 36, 48, 11, 19, 34, 25, 65, 128, 26, 28, 15, 66, 40, 72, 21, 96, 22, 38, 37, 68, 50, 130, 49, 35, 256, 52, 129, 27, 29, 56, 67, 30, 41, 132, 73, 80, 144, 42, 97, 192, 44, 23, 39, 76, 74, 136, 69, 100

Offset: 1

Antti Karttunen, Jun 03 2014

Any other fixed points than 1, 2, 6, 9, 135, 147, 914, ... ?

Any other points than 4, 21, 39, 839, 4893, 12884, ... where a(n) = n-1 ?

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

Inverse: A243346.
Cf. A005843, A005408, A008966, A005117, A013929, A013928, A057627.
Similar permutations: A243343-A243344, A243347, A243287-A243288, A135141-A227413, A237126-A237427, A193231.

a(1) = 1, and for n>1, if mu(n) = 0, a(n) = 1 + 2*a(A057627(n)), otherwise a(n) = 2*a(A013928(n)), where mu is Moebius mu function (A008683).

For all n > 1, A000035(a(n)+1) = A008966(n) = A008683(n)^2, or equally, a(n) = mu(n) + 1 modulo 2.

A243348 Difference between the n-th squarefree number and n: a(n) = A005117(n) - n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 3, 4, 4, 4, 5, 6, 7, 7, 7, 9, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 16, 16, 19, 20, 21, 22, 22, 22, 23, 23, 25, 25, 25, 26, 26, 26, 27, 27, 29, 29, 29, 31, 31, 32, 32, 32, 33, 34, 35, 35, 35, 36, 39, 39, 39, 40, 40, 40, 41, 41, 41, 42, 42, 42

Offset: 1

Antti Karttunen, Jun 04 2014

nonn

a(n) <= n, as A243351(n) = 2n - A005117(n) goes never negative (please see the plot A005117(n)/n given in the links section).
No runs longer than three appear, because there must be at least one gap (cf. A053806) in each range [4k+1 .. 4(k+1)] where no term(s) of A005117 appear.
See also A120992 which gives the run lengths.
Record values of first differences: a(2) - a(1) = 0, a(4) - a(3) = 1, a(7) - a(6) = 2, a(32) - a(31) = 3, a(151) - a(150) = 4, a(516) - a(515) = 5, a(13392) - a(13391) = 6, a(131965) - a(131964) = 7, a(664314) - a(664313) = 8, a(5392319) - a(5392318) = 9, and a(134453712) - a(134453711) = 11. - Charles R Greathouse IV, Nov 05 2017

Antti Karttunen, Table of n, a(n) for n = 1..10001
Antti Karttunen, Sequence plotted together with A243351 showing how their ratio develops.
Antti Karttunen, Ratio A005117(n)/n plotted in the same way, converging to Pi^2/6.

Cf. A005117, A053797, A053806, A243351, A243289, A243347.

A120992 gives the lengths of runs.

do(x)=my(v=List([0])); forfactored(n=2,x\1, if(vecmax(n[2][,2])==1, listput(v,n[1]-#v-1))); Vec(v) \\ Charles R Greathouse IV, Nov 05 2017

from math import isqrt
from sympy import mobius
def A243348(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m-n # Chai Wah Wu, Aug 12 2024

Scheme
```
(define (A243348 n) (- (A005117 n) n))
```

a(n) = A005117(n) - n.
a(n) = A243349(n) - A243289(n).
a(n) = n - A243351(n).
Limit_{n->oo} a(n)/A243351(n) = (Pi^2 - 6)/(12 - Pi^2) = 1.81637833.... - Charles R Greathouse IV, Jun 04 2014
a(n) ~ kn where k = Pi^2/6 - 1 = 0.644934.... - Charles R Greathouse IV, Nov 05 2017

A328244 Numbers whose second arithmetic derivative (A068346) is a squarefree number (A005117).

Original entry on oeis.org

6, 9, 10, 14, 18, 21, 22, 25, 30, 34, 38, 42, 46, 50, 57, 58, 62, 65, 66, 69, 70, 77, 78, 82, 85, 86, 93, 94, 99, 105, 114, 118, 121, 122, 125, 126, 130, 133, 134, 138, 142, 145, 146, 150, 154, 161, 165, 166, 169, 170, 174, 177, 182, 185, 186, 198, 201, 202, 206, 207, 209, 213, 214, 217, 221, 222, 230, 231, 237, 238, 242, 246, 253, 254, 255

Offset: 1

Antti Karttunen, Oct 11 2019

nonn

Numbers n for which A008966(A003415(A003415(n))) = 1.

Numbers whose first, second or third arithmetic is prime (A157037, A192192, A328239) are all included in this sequence, because: (1) taking arithmetic derivative of a prime gives 1, which is squarefree, (2) primes themselves are squarefree, and (3) only squarefree numbers may have arithmetic derivative that is a prime.

			For n=6, its first arithmetic derivative is A003415(6) = 5, and its second derivative is A003415(5) = 1, and 1 is a squarefree number (in A005117), thus 6 is included in this sequence.
For n=9, A003415(9) = 6, A003415(6) = 5, and 5, like all prime numbers, is squarefree, thus 9 is included in this sequence.
For n=14, A003415(14) = 9, A003415(9) = 6 = 2*3, and as 6 is squarefree, 14 is included in this sequence.

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

Cf. A003415, A005117, A008966, A068346, A328234, A328246.

Cf. A157037, A192192, A328239, A328245, A328253 (subsequences).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA328244(n) = { my(u=A003415(A003415(n))); (u>0 && issquarefree(u)); };

cp's OEIS Frontend

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))).

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A284311 Array T(n,k) read by antidiagonals (downward): T(1,k) = A005117(k+1) (squarefree numbers > 1); for n > 1, columns are nonsquarefree numbers (in ascending order) with exactly the same prime factors as T(1,k).

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A328393 Numbers whose arithmetic derivative (A003415) is a squarefree number (A005117).

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A062822 Sum of divisors of the squarefree numbers: sigma(A005117(n)).

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A072048 Number of divisors of the squarefree numbers: tau(A005117(n)).

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A111059 a(n) = Product_{k=1..n} A005117(k), the product of the first n squarefree positive integers.

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A243344 a(1) = 1, a(2n) = A013929(a(n)), a(2n+1) = A005117(1+a(n)).

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A243345 a(1)=1; thereafter, if n is k-th squarefree number [i.e., n = A005117(k)], a(n) = 2*a(k-1); otherwise, when n is k-th nonsquarefree number [i.e., n = A013929(k)], a(n) = 2*a(k)+1.

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A243345 a(1)=1; thereafter, if n is k-th squarefree number [i.e., n = A005117(k)], a(n) = 2a(k-1); otherwise, when n is k-th nonsquarefree number [i.e., n = A013929(k)], a(n) = 2a(k)+1.