A013929 -id:A013929 - cp's OEIS frontend

A015050 Let m = A013929(n); then a(n) = smallest k such that m divides k^3.

2, 2, 3, 6, 4, 6, 10, 6, 5, 3, 14, 4, 6, 10, 22, 15, 12, 7, 10, 26, 6, 14, 30, 21, 4, 34, 6, 15, 38, 20, 9, 42, 22, 30, 46, 12, 14, 33, 10, 26, 6, 28, 58, 39, 30, 11, 62, 5, 42, 8, 66, 15, 34, 70, 12, 21, 74, 30, 38, 51, 78, 20, 18, 82, 42, 13, 57, 86

Offset: 1

R. Muller

nonn

R. J. Mathar, Table of n, a(n) for n = 1..10491
Henry Ibstedt, Surfing on the Ocean of Numbers, Erhus Univ. Press, Vail, 1997.

Cf. A013929, A015049, A015051, A019555.

isA013929 := proc(n)
    not numtheory[issqrfree](n) ;
end proc:
A013929 := proc(n)
    option remember;
    local a;
    if n = 1 then
        4;
    else
        for a from procname(n-1)+1 do
            if isA013929(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
A015050 := proc(n)
    local m ;
    m := A013929(n) ;
    for k from 1 do
        if modp(k^3,m) = 0 then
            return k;
        end if;
    end do:
end proc:

f[p_, e_] := p^Ceiling[e/3]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; s /@ Select[Range[200], !SquareFreeQ[#] &] (* Amiram Eldar, Feb 09 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]/3)), ", ")));} \\ Amiram Eldar, Jan 06 2024

a(n) = A019555(A013929(n)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2) * (zeta(2) * zeta(5) * Product_{p prime} (1-1/p^2+1/p^3-1/p^4) - 1)/(zeta(2)-1)^2 = 0.6611256641303... . - Amiram Eldar, Jan 06 2024

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

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.

Original entry on oeis.org

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.

A375710 Numbers k such that A013929(k+1) - A013929(k) = 2. In other words, the k-th nonsquarefree number is 2 less than the next nonsquarefree number.

Original entry on oeis.org

5, 6, 9, 19, 20, 21, 33, 34, 36, 49, 57, 58, 62, 63, 66, 76, 77, 88, 89, 91, 96, 97, 103, 104, 113, 114, 118, 119, 130, 131, 132, 136, 142, 149, 150, 161, 162, 174, 175, 187, 188, 189, 190, 201, 202, 206, 215, 217, 218, 225, 226, 231, 232, 245, 246, 249, 253

Offset: 1

Gus Wiseman, Sep 09 2024

nonn

The difference of consecutive nonsquarefree numbers is at least 1 and at most 4, so there are four disjoint sequences of this type:
- A375709 (difference 1)
- A375710 (difference 2)
- A375711 (difference 3)
- A375712 (difference 4)

			The initial nonsquarefree numbers are 4, 8, 9, 12, 16, 18, 20, 24, 25, which first increase by 2 after the fifth and sixth terms.

Positions of 2's in A078147.
For prime numbers we have A029707.
For nonprime numbers we appear to have A014689.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A007674, A049094, A061399, A068781, A072284, A110969, A120992, A294242, A373409, A373573, A375927.

Join@@Position[Differences[Select[Range[1000], !SquareFreeQ[#]&]],2]

Complement of A375709 U A375711 U A375712.

A375711 Numbers k such that A013929(k+1) - A013929(k) = 3. In other words, the k-th nonsquarefree number is 3 less than the next nonsquarefree number.

Original entry on oeis.org

3, 16, 23, 27, 31, 44, 46, 51, 55, 60, 68, 74, 79, 86, 95, 101, 105, 107, 112, 116, 121, 126, 129, 146, 147, 152, 159, 164, 167, 172, 177, 182, 185, 191, 195, 199, 204, 209, 220, 223, 229, 234, 237, 242, 244, 257, 262, 270, 275, 285, 286, 291, 299, 305, 312

Offset: 1

Gus Wiseman, Sep 09 2024

nonn

The difference of consecutive nonsquarefree numbers is at least 1 and at most 4, so there are four disjoint sequences of this type:
- A375709 (difference 1)
- A375710 (difference 2)
- A375711 (difference 3)
- A375712 (difference 4)

			The initial nonsquarefree numbers are 4, 8, 9, 12, 16, 18, 20, 24, 25, which first increase by 3 after the third term.

Positions of 3's in A078147.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A007674, A014689, A049094, A061399, A068781, A072284, A110969, A120992, A294242, A373409, A375927.

Join@@Position[Differences[Select[Range[1000],!SquareFreeQ[#]&]],3]

Complement of A375709 U A375710 U A375712.

A375712 Numbers k such that A013929(k+1) - A013929(k) = 4. In other words, the k-th nonsquarefree number is 4 less than the next nonsquarefree number.

Original entry on oeis.org

1, 4, 7, 11, 12, 13, 14, 22, 25, 26, 29, 32, 35, 39, 40, 41, 42, 50, 53, 54, 61, 64, 70, 71, 72, 75, 78, 81, 82, 83, 84, 87, 90, 98, 99, 102, 109, 110, 117, 120, 123, 124, 127, 135, 139, 140, 144, 151, 154, 155, 156, 157, 160, 163, 168, 169, 170, 173, 176, 179

Offset: 1

Gus Wiseman, Sep 09 2024

nonn

The difference of consecutive nonsquarefree numbers is at least 1 and at most 4, so there are four disjoint sequences of this type:
- A375709 (difference 1)
- A375710 (difference 2)
- A375711 (difference 3)
- A375712 (difference 4)

			The initial nonsquarefree numbers are 4, 8, 9, 12, 16, 18, 20, 24, 25, which first increase by 4 after the first, fourth, and seventh terms.

For prime numbers we have A029709.
Positions of 4's in A078147.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A007674, A014689, A029707, A049094, A061399, A068781, A072284, A110969, A120992, A294242, A375927.

Join@@Position[Differences[Select[Range[100],!SquareFreeQ[#]&]],4]

Complement of A375709 U A375710 U A375711.

A376264 Run-sums of first differences (A078147) of nonsquarefree numbers (A013929).

Original entry on oeis.org

4, 1, 3, 4, 4, 4, 1, 2, 1, 16, 1, 3, 2, 6, 4, 3, 1, 8, 3, 1, 4, 1, 3, 4, 4, 4, 2, 2, 16, 1, 3, 1, 3, 2, 2, 4, 3, 1, 8, 3, 1, 4, 1, 3, 4, 4, 4, 1, 2, 1, 3, 1, 12, 1, 3, 4, 4, 4, 3, 1, 16, 1, 3, 4, 4, 4, 2, 3, 3, 4, 8, 1, 3, 4, 4, 3, 1, 3, 1, 8, 1, 3, 4, 1, 3, 4

Offset: 1

Gus Wiseman, Sep 26 2024

nonn

Does the image include all positive integers? I have only verified this up to 8.

			The sequence of nonsquarefree numbers (A013929) is:
  4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, ...
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, ...
with runs:
  (4),(1),(3),(4),(2,2),(4),(1),(2),(1),(4,4,4,4),(1),(3),(1,1),(2,2,2), ...
with sums (A376264):
  4, 1, 3, 4, 4, 4, 1, 2, 1, 16, 1, 3, 2, 6, 4, 3, 1, 8, 3, 1, 4, 1, 3, 4, ...

Before taking run-sums we had A078147.
For nonprime instead of nonsquarefree numbers we have A373822.
Positions of first appearances are A376265, sorted A376266.
For run-lengths instead of run-sums we have A376267.
For squarefree instead of nonsquarefree we have A376307.
For prime-powers instead of nonsquarefree numbers we have A376310.
For compression instead of run-sums we have A376312.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259 (ones A375927).
A013929 lists nonsquarefree numbers, differences A078147.
Cf. A053797, A053806, A061398, A072284, A120992, A373197, A373413, A375707, A376305, A376306, A376311.

Total/@Split[Differences[Select[Range[1000],!SquareFreeQ[#]&]]]//Most

A376267 Run-lengths of first differences (A078147) of nonsquarefree numbers (A013929).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 3, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2

Offset: 1

Gus Wiseman, Sep 27 2024

nonn

			The sequence of nonsquarefree numbers (A013929) is:
  4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, ...
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, ...
with runs:
  (4),(1),(3),(4),(2,2),(4),(1),(2),(1),(4,4,4,4),(1),(3),(1,1),(2,2,2), ...
with lengths (A376267):
  1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, ...

Robert Israel, Table of n, a(n) for n = 1..10000

For prime instead of nonsquarefree numbers we have A333254.
For run-sums instead of run-lengths we have A376264.
For squarefree instead of nonsquarefree we have A376306.
For prime-powers instead of nonsquarefree numbers we have A376309.
For compression instead of run-lengths we have A376312.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A005117 lists squarefree numbers, differences A076259 (ones A375927).
A013929 lists nonsquarefree numbers, differences A078147.
Cf. A007674, A053797, A053806, A072284, A112925, A120992, A373198, A375707, A376305, A376307, A376311, A380595.

nsf:= remove(numtheory:-issqrfree, [$4..1000]):
S:= nsf[2..-1]-nsf[1..-2]:
R:= NULL: x:= 4: t:= 1:
for i from 2 to nops(S) do
  if S[i] = x then t:= t+1
  else R:= R,t; x:= S[i]; t:= 1
  fi
od:
R; # Robert Israel, Jan 27 2025

Length/@Split[Differences[Select[Range[1000], !SquareFreeQ[#]&]]]//Most

A046028 Largest 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, 3, 2, 2, 3, 2, 7, 5, 2, 3, 2, 2, 3, 2, 2, 3, 5, 2, 2, 3, 2, 2, 3, 2, 2, 7, 3, 5, 2, 3, 2, 2, 3, 2, 11, 2, 5, 3, 2, 2, 3, 2, 2, 3, 7, 2, 5, 2, 3, 2, 2, 3, 2, 2, 13, 3, 2, 5, 2, 3, 2, 2, 3, 2, 7, 3, 5, 2, 3, 2, 2, 3, 2, 2, 5, 2, 2, 3, 2, 2, 11, 3, 2, 7, 2, 5, 3, 2, 2, 3

Offset: 1

Eric W. Weisstein

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Greatest Prime Factor.
Eric Weisstein's World of Mathematics, Squareful.

Cf. A006530, A046027, A013929, A249740.

Cf. A027748, A124010, A212177.

a046028 n = a046028_list !! (n-1)
a046028_list = f 1 where
   f x | null zs   = f (x + 1)
       | otherwise = (fst $ head zs) : f (x + 1)
       where zs = reverse $ filter ((> 1) . snd) $
                  zip (a027748_row x) (a124010_row x)
-- Reinhard Zumkeller, Dec 29 2012

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

from math import isqrt
from sympy import mobius, factorint
def A046028(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,reverse=True) if s[p]>1) # Chai Wah Wu, Jul 22 2024

a(n) = A249740(A013929(n)). - Amiram Eldar, Feb 11 2021

A062320 Nonsquarefree numbers squared. A013929(n)^2.

Original entry on oeis.org

16, 64, 81, 144, 256, 324, 400, 576, 625, 729, 784, 1024, 1296, 1600, 1936, 2025, 2304, 2401, 2500, 2704, 2916, 3136, 3600, 3969, 4096, 4624, 5184, 5625, 5776, 6400, 6561, 7056, 7744, 8100, 8464, 9216, 9604, 9801, 10000, 10816, 11664, 12544, 13456

Offset: 1

Jason Earls, Jul 05 2001

A008966(A037213(a(n))) = 0. - Reinhard Zumkeller, Sep 03 2015

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A013929, A008966, A037213, A320965.

Squares in A046099.

a062320 = (^ 2) . a013929 -- Reinhard Zumkeller, Sep 03 2015

for(n=1,55, if(issquarefree(n), n+1,print(n^2)))

n=-1; for (m=1, 10^9, if (!issquarefree(m), write("b062320.txt", n++, " ", m^2); if (n==1000, break))) \\ Harry J. Smith, Aug 04 2009

is(n)=issquare(n,&n) && !issquarefree(n) \\ Charles R Greathouse IV, Sep 18 2015

from math import isqrt
from sympy import mobius
def A062320(n):
    def f(x): return n+1+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f)**2 # Chai Wah Wu, Aug 31 2024

Sum_{n>=1} 1/a(n) = Pi^2/6 - 15/Pi^2. - Amiram Eldar, Jul 16 2020

More terms from Larry Reeves (larryr(AT)acm.org), Jul 11 2001

Offset corrected by Andrew Howroyd, Sep 18 2024

A063035 Number of integers m <= 10^n that contain a square factor (i.e., belong to A013929).

Original entry on oeis.org

3, 39, 392, 3917, 39206, 392074, 3920709, 39207306, 392072876, 3920729058, 39207289720, 392072897726, 3920728981706, 39207289814053, 392072898145897, 3920728981459595

Offset: 1

Robert G. Wilson v, Aug 02 2001

nonn

Note that "containing a square factor" (A013929) is different from "squareful" (A001694).

Robert G. Wilson v, Table of n, a(n) for n = 1..36

Cf. A001694, A013929, A229099.

For the complementary counts see A053462 and A071172.

f[n_] := Sum[-MoebiusMu[i]Floor[n/i^2], {i, 2, Sqrt@ n}]; Table[ f[10^n], {n, 0, 14}]

{ default(realprecision, 50); for (n=1, 100, t=10^n - 1; a=10^n - sum(k=1, sqrt(t), moebius(k)*floor(t/k^2)); write("b063035.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 16 2009

from math import isqrt
from sympy import mobius
def A063035(n): return (m:=10**n)-sum(mobius(k)*(m//k**2) for k in range(1,isqrt(m)+1)) # Chai Wah Wu, Jul 20 2024

Limit_{n->oo} a(n)/10^n = A229099. - Robert G. Wilson v, Aug 12 2014

a(n) = 10^n - A071172(n). - Amiram Eldar, Mar 10 2024

More terms from Harry J. Smith, Aug 16 2009

Edited (with a more precise definition and a new value for a(1)) by N. J. A. Sloane, Aug 06 2012. As a result of this change, the programs probably now give the wrong value for a(1). The source of the trouble was the ambiguous meaning of squareful - the official definition of squareful is A001694.

cp's OEIS Frontend

A015050 Let m = A013929(n); then a(n) = smallest k such that m divides k^3.

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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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A375710 Numbers k such that A013929(k+1) - A013929(k) = 2. In other words, the k-th nonsquarefree number is 2 less than the next nonsquarefree number.

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A375711 Numbers k such that A013929(k+1) - A013929(k) = 3. In other words, the k-th nonsquarefree number is 3 less than the next nonsquarefree number.

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A375712 Numbers k such that A013929(k+1) - A013929(k) = 4. In other words, the k-th nonsquarefree number is 4 less than the next nonsquarefree number.

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A376264 Run-sums of first differences (A078147) of nonsquarefree numbers (A013929).

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A376267 Run-lengths of first differences (A078147) of nonsquarefree numbers (A013929).

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A046028 Largest multiple prime factor of the n-th nonsquarefree number (A013929).

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