A013929 -id:A013929 - cp's OEIS frontend

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

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.

A376594 Inflection and undulation points in the sequence of nonsquarefree numbers (A013929).

Original entry on oeis.org

5, 11, 12, 13, 17, 19, 20, 25, 33, 37, 39, 40, 41, 47, 53, 57, 62, 70, 71, 76, 81, 82, 83, 88, 92, 93, 96, 98, 103, 109, 113, 118, 123, 130, 131, 133, 137, 139, 146, 149, 154, 155, 156, 161, 165, 168, 169, 174, 179, 180, 183, 187, 188, 189, 193, 201, 211, 213

Offset: 1

Gus Wiseman, Oct 04 2024

nonn

These are points at which the second differences (A376593) are zero.

			The nonsquarefree numbers (A013929) are:
  4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, ...
with first differences (A078147):
  4, 1, 3, 4, 2, 2, 4, 1, 2, 1, 4, 4, 4, 4, 1, 3, 1, 1, 2, 2, 2, 4, 3, 1, 4, 4, 3, ...
with first differences (A376593):
  -3, 2, 1, -2, 0, 2, -3, 1, -1, 3, 0, 0, 0, -3, 2, -2, 0, 1, 0, 0, 2, -1, -2, 3, ...
with zeros (A376594) at:
  5, 11, 12, 13, 17, 19, 20, 25, 33, 37, 39, 40, 41, 47, 53, 57, 62, 70, 71, 76, ...

Gus Wiseman, Inflection and undulation points in the nonsquarefree numbers.

The first differences were A078147.
These are the zeros of A376593.
The complement is A376595.
A000040 lists the prime numbers, differences A001223.
A005117 lists squarefree numbers, differences A076259.
A013929 lists nonsquarefree numbers, differences A078147.
A064113 lists positions of adjacent equal prime gaps.
A114374 counts partitions into nonsquarefree numbers.
For inflections and undulations: A064113 (prime), A376602 (composite), A376588 (non-perfect-power), A376597 (prime-power), A376600 (non-prime-power).
For nonsquarefree numbers: A013929 (terms), A078147 (first differences), A376593 (second differences), A376595 (nonzero curvature).
Cf. A007674, A053797, A053806, A061398, A112926, A120992, A251092, A375707, A376312, A376590, A376593.

Join@@Position[Differences[Select[Range[100],!SquareFreeQ[#]&],2],0]

A376595 Points of nonzero curvature in the sequence of nonsquarefree numbers (A013929).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 14, 15, 16, 18, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 38, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 72, 73, 74, 75, 77, 78, 79, 80, 84, 85, 86, 87, 89, 90, 91

Offset: 1

Gus Wiseman, Oct 04 2024

nonn

These are points at which the second differences (A376593) are nonzero.

			The nonsquarefree numbers (A013929) are:
  4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, ...
with first differences (A078147):
  4, 1, 3, 4, 2, 2, 4, 1, 2, 1, 4, 4, 4, 4, 1, 3, 1, 1, 2, 2, 2, 4, 3, 1, 4, 4, 3, ...
with first differences (A376593):
  -3, 2, 1, -2, 0, 2, -3, 1, -1, 3, 0, 0, 0, -3, 2, -2, 0, 1, 0, 0, 2, -1, -2, 3, ...
with nonzeros (A376594) at:
  1, 2, 3, 4, 6, 7, 8, 9, 10, 14, 15, 16, 18, 21, 22, 23, 24, 26, 27, 28, 29, 30, ...

Gus Wiseman, Points of nonzero curvature in the nonsquarefree numbers.

The first differences were A078147.
These are the nonzeros of A376593.
The complement is A376594.
A000040 lists the prime numbers, differences A001223.
A005117 lists squarefree numbers, differences A076259.
A013929 lists nonsquarefree numbers, differences A078147.
A114374 counts integer partitions into nonsquarefree numbers.
For points of nonzero curvature: A333214 (prime), A376603 (composite), A376589 (non-perfect-power), A376592 (squarefree), A376598 (prime-power), A376601 (non-prime-power).
For nonsquarefree numbers: A078147 (first differences), A376593 (second differences), A376594 (inflections and undulations).
Cf. A007674, A036263, A053797, A053806, A061398, A120992, A373198, A375707, A376306, A376312, A376590.

Join@@Position[Sign[Differences[Select[Range[100],!SquareFreeQ[#]&],2]],1|-1]

A046027 Smallest multiple prime factor of the n-th nonsquarefree number (A013929).

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 2, 2, 5, 3, 2, 2, 2, 2, 2, 3, 2, 7, 5, 2, 3, 2, 2, 3, 2, 2, 2, 5, 2, 2, 3, 2, 2, 3, 2, 2, 7, 3, 2, 2, 2, 2, 2, 3, 2, 11, 2, 5, 3, 2, 2, 3, 2, 2, 2, 7, 2, 5, 2, 3, 2, 2, 3, 2, 2, 13, 3, 2, 5, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 11, 3, 2, 7, 2, 5, 2, 2, 2, 3

Offset: 1

Eric W. Weisstein

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Least Prime Factor.
Eric Weisstein's World of Mathematics, Squareful.

Cf. A013929, A020639, A046028, A249739.

Select[ FactorInteger[#], #[[2]]>1&, 1][[1, 1]]& /@ Select[ Range[300], !SquareFreeQ[#]& ] (* Jean-François Alcover, Nov 06 2012 *)

lista(nn) = apply(x->factor(x)[1,1], apply(x->x/core(x), select(x->!issquarefree(x), [1..nn]))); \\ Michel Marcus, Jun 24 2025

from math import isqrt
from sympy import mobius, factorint
def A046027(n):
    def f(x): return n+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)
    s = factorint(m)
    return next(p for p in sorted(s) if s[p]>1) # Chai Wah Wu, Jul 22 2024

a(n) = A249739(A013929(n)). - Amiram Eldar, Feb 11 2021

A285320 If n == 0 or A008683(n) == 0, then a(n) = 0, otherwise a(n) = 1+a(A048675(n)); number of iterations of A048675 needed before the result is either zero or nonsquarefree number (A013929).

Original entry on oeis.org

0, 1, 2, 3, 0, 1, 4, 1, 0, 0, 2, 1, 0, 1, 1, 5, 0, 1, 0, 1, 0, 3, 2, 1, 0, 0, 2, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 3, 3, 0, 1, 2, 1, 0, 0, 2, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 3, 1, 1, 0, 1, 1, 0, 0, 1, 2, 1, 0, 3, 2, 1, 0, 1

Offset: 0

Antti Karttunen, Apr 18 2017

Conjecture: all terms are well-defined (finite). This implies also the conjecture I have made in A019565.

			a(38) = 3 because 38 = 2*19 (thus squarefree), A048675(38) = 129 (= 3*43), A048675(129) = 8194 (= 2*17*241) and A048675(8194) = 4503599627370561 (= 3^2 * 37 * 71 * 190483425427), so three steps were needed before nonsquarefree number was reached.
a(74) >= 3 as A048675(74) = 2049 (squarefree), A048675(2049) =  10633823966279326983230456482242756610 (squarefree), A048675(10633823966279326983230456482242756610) = ???

A left inverse of A109162.
Cf. A008683, A005117, A013929, A048675.
Cf. also A285319, A285331, A285332.

A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ Michel Marcus, Oct 10 2016
A285320(n) = if(!n || !moebius(n),0,1+A285320(A048675(n)));

(define (A285320 n) (if (or (zero? n) (zero? (A008683 n))) 0 (+ 1 (A285320 (A048675 n)))))

If n == 0 or A008683(n) == 0, then a(n) = 0, otherwise a(n) = 1+a(A048675(n)).

a(A109162(n)) = n.

A015051 Let m = A013929(n); then a(n) = smallest k such that m divides k^4.

Original entry on oeis.org

2, 2, 3, 6, 2, 6, 10, 6, 5, 3, 14, 4, 6, 10, 22, 15, 6, 7, 10, 26, 6, 14, 30, 21, 4, 34, 6, 15, 38, 10, 3, 42, 22, 30, 46, 12, 14, 33, 10, 26, 6, 14, 58, 39, 30, 11, 62, 5, 42, 4, 66, 15, 34, 70, 6, 21, 74, 30, 38, 51, 78, 20, 6, 82, 42, 13, 57, 86

Offset: 1

R. Muller

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Henry Ibstedt, Surfing on the Ocean of Numbers, Erhus Univ. Press, Vail, 1997.

Cf. A013929, A015050, A015049, A053166.

f[p_, e_] := p^Ceiling[e/4]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; s /@ Select[Range[200], !SquareFreeQ[#] &] (* Amiram Eldar, Feb 10 2021 *)

lista(kmax) = {my(f); for(k = 2, kmax, f = factor(k); if(!issquarefree(f), print1(prod(i = 1, #f~, f[i,1]^ceil(f[i,2]/4)), ", ")));} \\ Amiram Eldar, Jan 06 2024

a(n) = A053166(A013929(n)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2) * (zeta(2) * zeta(7) * Product_{p prime} (1-1/p^2+1/p^3-1/p^4+1/p^5-1/p^6) - 1)/(zeta(2)-1)^2 = 0.635465442379... . - Amiram Eldar, Jan 06 2024

Description corrected by Diego Torres (torresvillarroel(AT)hotmail.com), Jun 23 2002

Offset corrected by Amiram Eldar, Feb 10 2021

A162966 Union of 1 and nonsquarefree numbers (A013929).

Original entry on oeis.org

1, 4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160

Offset: 1

Jaroslav Krizek, Jul 19 2009

Or, the natural numbers that are not products of distinct primes. Also, number of prime divisors of n < number of nonprime divisors of n. - Juri-Stepan Gerasimov, Nov 10 2009

Cf. A013929.

t = {1}; Do[If[Max[Transpose[FactorInteger[n]][[2]]] > 1, AppendTo[t, n]], {n, 2, 160}]; t (* Jayanta Basu, Apr 30 2013 *)

is(n)=!issquarefree(n)||n==1 \\ Charles R Greathouse IV, Apr 04 2014

A378373 Number of composite numbers (A002808) between consecutive nonsquarefree numbers (A013929), exclusive.

Original entry on oeis.org

1, 0, 1, 2, 0, 0, 2, 0, 1, 0, 1, 3, 2, 1, 0, 1, 0, 0, 1, 0, 1, 2, 1, 0, 2, 2, 1, 0, 2, 0, 1, 3, 0, 1, 3, 0, 0, 0, 1, 2, 2, 2, 0, 2, 0, 2, 0, 0, 0, 2, 2, 0, 1, 3, 2, 0, 0, 0, 0, 2, 2, 1, 0, 2, 0, 1, 0, 1, 0, 2, 2, 3, 0, 1, 2, 0, 0, 3, 2, 0, 2, 3, 3, 2, 0, 1, 2

Offset: 1

Gus Wiseman, Dec 02 2024

nonn

All terms are 0, 1, 2, or 3 (cf. A078147).
The inclusive version is a(n) + 2.
The nonsquarefree numbers begin: 4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, ...

			The composite numbers counted by a(n) form the following set partition of A120944:
{6}, {}, {10}, {14,15}, {}, {}, {21,22}, {}, {26}, {}, {30}, {33,34,35}, {38,39}, ...

For prime (instead of nonsquarefree) we have A046933.
For squarefree (instead of nonsquarefree) we have A076259(n)-1.
For prime power (instead of nonsquarefree) we have A093555.
For prime instead of composite we have A236575.
For nonprime prime power (instead of nonsquarefree) we have A378456.
For perfect power (instead of nonsquarefree) we have A378614, primes A080769.
A002808 lists the composite numbers.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147.
A073247 lists squarefree numbers with nonsquarefree neighbors.
A120944 lists squarefree composite numbers.
A377432 counts perfect-powers between primes, zeros A377436.
A378369 gives distance to the next nonsquarefree number (A120327).
Cf. A065890, A067535, A067871, A080101, A081221, A151800, A243348, A366833, A377046, A378033, A378039, A378086.

v=Select[Range[100],!SquareFreeQ[#]&];
Table[Length[Select[Range[v[[i]]+1,v[[i+1]]-1],CompositeQ]],{i,Length[v]-1}]

A015049 Let m = A013929(n); then a(n) = smallest k such that m divides k^2.

Original entry on oeis.org

2, 4, 3, 6, 4, 6, 10, 12, 5, 9, 14, 8, 6, 20, 22, 15, 12, 7, 10, 26, 18, 28, 30, 21, 8, 34, 12, 15, 38, 20, 9, 42, 44, 30, 46, 24, 14, 33, 10, 52, 18, 28, 58, 39, 60, 11, 62, 25, 42, 16, 66, 45, 68, 70, 12, 21, 74, 30, 76, 51, 78, 40, 18, 82, 84, 13, 57, 86

Offset: 1

R. Muller

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Henry Ibstedt, Surfing on the Ocean of Numbers, Erhus Univ. Press, Vail, 1997.

Cf. A013929, A015050, A015051, A019554.

f[p_, e_] := p^Ceiling[e/2]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; s /@ Select[Range[200], !SquareFreeQ[#] &] (* Amiram Eldar, Feb 10 2021 *)

lista(kmax) = {my(f); for(k = 2, kmax, f = factor(k); if(!issquarefree(f), print1(k/core(f, 1)[2], ", ")));} \\ Amiram Eldar, Jan 06 2024

a(n) = A019554(A013929(n)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2)*(zeta(3)-1)/(zeta(2)-1)^2 = 0.799082... . - Amiram Eldar, Jan 06 2024

Description corrected by Diego Torres (torresvillarroel(AT)hotmail.com), Jun 23 2002

Offset corrected by Amiram Eldar, Feb 10 2021

cp's OEIS Frontend

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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A376594 Inflection and undulation points in the sequence of nonsquarefree numbers (A013929).

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A376595 Points of nonzero curvature in the sequence of nonsquarefree numbers (A013929).

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A046027 Smallest multiple prime factor of the n-th nonsquarefree number (A013929).

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A285320 If n == 0 or A008683(n) == 0, then a(n) = 0, otherwise a(n) = 1+a(A048675(n)); number of iterations of A048675 needed before the result is either zero or nonsquarefree number (A013929).

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A015051 Let m = A013929(n); then a(n) = smallest k such that m divides k^4.

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A162966 Union of 1 and nonsquarefree numbers (A013929).

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A378373 Number of composite numbers (A002808) between consecutive nonsquarefree numbers (A013929), exclusive.

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.