A007674 -id:A007674 - cp's OEIS frontend

A076259 Gaps between squarefree numbers: a(n) = A005117(n+1) - A005117(n).

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

Offset: 1

Reinhard Zumkeller, Oct 03 2002

This sequence is unbounded, as a simple consequence of the Chinese remainder theorem. - Thomas Ordowski, Jul 22 2015
Conjecture: lim sup_{n->oo} a(n)/log(A005117(n)) = 1/2. - Thomas Ordowski, Jul 23 2015 [Note: this conjecture is equivalent to lim sup a(n)/log n = 1/2. - Charles R Greathouse IV, Dec 05 2024]
a(n) = 1 infinitely often since the density of the squarefree numbers, 6/Pi^2, is greater than 1/2. In particular, at least 2 - Pi^2/6 = 35.5...% of the terms are 1. - Charles R Greathouse IV, Jul 23 2015
From Amiram Eldar, Mar 09 2021: (Start)
The asymptotic density of the occurrences of 1 in this sequence is density(A007674)/density(A005117) =  A065474/A059956 = 0.530711... (A065469).
The asymptotic density of the occurrences of 2 in this sequence is (density(A069977)-density(A007675))/density(A005117) =  (A065474-A206256)/A059956 = 0.324294...  (End)

			As 24 = 3*2^3 and 25 = 5^2, the next squarefree number greater A005117(16) = 23 is A005117(17) = 26, therefore a(16) = 26-23 = 3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Mayank Pandey, Squarefree numbers in short intervals, arXiv preprint, arXiv:2401.13981 [math.NT], 2024.

Cf. A005117, A013661, A020753 (records), A020754, A076260.

Cf. A007674, A007675, A059956, A065469, A065474, A069977, A206256.

a076259 n = a076259_list !! (n-1)
a076259_list = zipWith (-) (tail a005117_list) a005117_list
-- Reinhard Zumkeller, Aug 03 2012

A076259 := proc(n) A005117(n+1)-A005117(n) ; end proc: # R. J. Mathar, Jan 09 2013

Select[Range[200], SquareFreeQ] // Differences (* Jean-François Alcover, Mar 10 2019 *)

t=1; for(n=2,1e3, if(issquarefree(n), print1(n-t", "); t=n)) \\ Charles R Greathouse IV, Jul 23 2015

from math import isqrt
from sympy import mobius
def A076259(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)
    r, k = n+1, f(n+1)+1
    while r != k:
        r, k = k, f(k)+1
    return int(r-m) # Chai Wah Wu, Aug 15 2024

Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = Pi^2/6 (A013661). - Amiram Eldar, Oct 21 2020

a(n) < n^(1/5) for large enough n by a result of Pandey. (The constant Pi^2/6 can be absorbed by any eta > 0.) - Charles R Greathouse IV, Dec 04 2024

A053797 Lengths of successive gaps between squarefree numbers.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Apr 07 2000

From Gus Wiseman, Jun 11 2024: (Start)
Also the length of the n-th maximal run of nonsquarefree numbers. These runs begin:
       4
     8   9
      12
      16
      18
      20
    24  25
    27  28
      32
      36
      40
    44  45
  48  49  50
(End)

			The first gap is at 4 and has length 1; the next starts at 8 and has length 2 (since neither 8 nor 9 are squarefree).

Peter Kagey, Table of n, a(n) for n = 1..10000
M. Filaseta and O. Trifonov, On Gaps between Squarefree Numbers. In Analytic Number Theory, Vol 85, 1990, Birkhäuser, Basel, pp. 235-253.
E. Fogels, On the average values of arithmetic functions, Proc. Cambridge Philos. Soc. 1941, 37: 358-372.
L. Marmet, First occurrences of squarefree gaps...
L. Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv preprint arXiv:1210.3829 [math.NT], 2012. - From _N. J. A. Sloane_, Jan 01 2013
K. F. Roth, On the gaps between squarefree numbers, J. London Math. Soc. 1951 (2) 26:263-268.
Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

Gaps between terms of A005117.
For squarefree runs we have A120992, antiruns A373127 (firsts A373128).
For composite runs we have A176246 (rest of A046933), antiruns A373403.
For prime runs we have A251092 (rest of A175632), antiruns A027833.
Position of first appearance of n is A373199(n).
For antiruns instead of runs we have A373409.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A007674, A020754, A045882, A053806, A061398, A061399.

SF:= select(numtheory:-issqrfree,[$1..1000]):
map(`-`,select(`>`,SF[2..-1]-SF[1..-2],1),1); # Robert Israel, Sep 22 2015

ReplaceAll[Differences[Select[Range@384, SquareFreeQ]] - 1, 0 -> Nothing] (* Michael De Vlieger, Sep 22 2015 *)

Offset set to 1 by Peter Kagey, Sep 29 2015

A112925 Largest squarefree integer < the n-th prime.

Original entry on oeis.org

1, 2, 3, 6, 10, 11, 15, 17, 22, 26, 30, 35, 39, 42, 46, 51, 58, 59, 66, 70, 71, 78, 82, 87, 95, 97, 102, 106, 107, 111, 123, 130, 134, 138, 146, 149, 155, 161, 166, 170, 178, 179, 190, 191, 195, 197, 210, 222, 226, 227, 231, 238, 239, 249, 255, 262, 267, 269, 274, 278

Offset: 1

Leroy Quet, Oct 06 2005

nonn

			6 is the largest squarefree less than the 4th prime, 7. So a(4) = 6.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

For prime powers instead of squarefree numbers we have A065514, opposite A345531.
Restriction of A070321 (differences A378085) to the primes; see A378619.
The opposite is A112926, differences A378037.
Subtracting each term from prime(n) gives A240473, opposite A240474.
For nonsquarefree numbers we have A378033, differences A378036, see A378034, A378032.
For perfect powers we have A378035.
First differences are A378038.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, differences A076259.
A013928 counts squarefree numbers up to n - 1.
A013929 lists the nonsquarefree numbers, differences A078147.
A061398 counts squarefree numbers between primes, zeros A068360.
A061399 counts nonsquarefree numbers between primes, zeros A068361.
A112929 counts squarefree numbers up to prime(n).
Cf. A061400, A112928, A112930.
Cf. A007674, A007947, A045882, A053797, A053806, A067535, A072284, A073247, A081217, A175216, A280892, A377783.

with(numtheory): a:=proc(n) local p,B,j: p:=ithprime(n): B:={}: for j from 1 to p-1 do if abs(mobius(j))>0 then B:=B union {j} else B:=B fi od: B[nops(B)] end: seq(a(m),m=1..75); # Emeric Deutsch, Oct 14 2005

With[{k = 120}, Table[SelectFirst[Range[Prime@ n - 1, Prime@ n - Min[Prime@ n - 1, k], -1], SquareFreeQ], {n, 60}]] (* Michael De Vlieger, Aug 16 2017 *)

a(n,p=prime(n))=while(!issquarefree(p--),); p \\ Charles R Greathouse IV, Aug 16 2017

a(n) = prime(n) - A240473(n). - Gus Wiseman, Jan 10 2025

More terms from Emeric Deutsch, Oct 14 2005

A068781 Lesser of two consecutive numbers each divisible by a square.

Original entry on oeis.org

8, 24, 27, 44, 48, 49, 63, 75, 80, 98, 99, 116, 120, 124, 125, 135, 147, 152, 168, 171, 175, 188, 207, 224, 242, 243, 244, 260, 275, 279, 288, 296, 315, 324, 332, 342, 343, 350, 351, 360, 363, 368, 375, 387, 404, 423, 424, 440, 459, 475, 476, 495, 507, 512

Offset: 1

Robert G. Wilson v, Mar 04 2002

nonn

Also numbers m such that mu(m)=mu(m+1)=0, where mu is the Moebius-function (A008683); A081221(a(n))>1. - Reinhard Zumkeller, Mar 10 2003
The sequence contains an infinite family of arithmetic progressions like {36a+8}={8,44,80,116,152,188,...} ={4(9a+2)}. {36a+9} provides 2nd nonsquarefree terms. Such AP's can be constructed to any term by solution of a system of linear Diophantine equation. - Labos Elemer, Nov 25 2002
1. 4k^2 + 4k is a member for all k; i.e., 8 times a triangular number is a member. 2. (4k+1) times an odd square - 1 is a member. 3. (4k+3) times odd square is a member. - Amarnath Murthy, Apr 24 2003
The asymptotic density of this sequence is 1 - 2/zeta(2) + Product_{p prime} (1 - 2/p^2) = 1 - 2 * A059956 + A065474 = 0.1067798952... (Matomäki et al., 2016). - Amiram Eldar, Feb 14 2021
Maximum of the n-th maximal anti-run of nonsquarefree numbers (A013929) differing by more than one. For runs instead of anti-runs we have A376164. For squarefree instead of nonsquarefree we have A007674. - Gus Wiseman, Sep 14 2024

			44 is in the sequence because 44 = 2^2 * 11 and 45 = 3^2 * 5.
From _Gus Wiseman_, Sep 14 2024: (Start)
Splitting nonsquarefree numbers into maximal anti-runs gives:
  (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)
The maxima are a(n). The corresponding pairs are (8,9), (24,25), (27,28), (44,45), etc.
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Kaisa Matomäki, Maksym Radziwiłł and Terence Tao, Sign patterns of the Liouville and Möbius functions, Forum of Mathematics, Sigma, Vol. 4. (2016), e14.

Cf. A068780, A068140, A068781, A068782, A068783, A068784, A068785.
Cf. A049535, A077647, A078143, A045882.
Subsequence of A261869.
Cf. A007674, A373409, A373410, A373412, A375707, A376164.
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.
Cf. A049094, A061399, A072284, A120992, A294242, A373573, A375709.

a068781 n = a068781_list !! (n-1)
a068781_list = filter ((== 0) . a261869) [1..]
-- Reinhard Zumkeller, Sep 04 2015

Select[ Range[2, 600], Max[ Transpose[ FactorInteger[ # ]] [[2]]] > 1 && Max[ Transpose[ FactorInteger[ # + 1]] [[2]]] > 1 &]
f@n_:= Flatten@Position[Partition[SquareFreeQ/@Range@2000,n,1], Table[False,{n}]]; f@2 (* Hans Rudolf Widmer, Aug 30 2022 *)
Max/@Split[Select[Range[100], !SquareFreeQ[#]&],#1+1!=#2&]//Most (* Gus Wiseman, Sep 14 2024 *)

isok(m) = !moebius(m) && !moebius(m+1); \\ Michel Marcus, Feb 14 2021

A261869(a(n)) = 0. - Reinhard Zumkeller, Sep 04 2015

A112926 Smallest squarefree integer > the n-th prime.

Original entry on oeis.org

3, 5, 6, 10, 13, 14, 19, 21, 26, 30, 33, 38, 42, 46, 51, 55, 61, 62, 69, 73, 74, 82, 85, 91, 101, 102, 105, 109, 110, 114, 129, 133, 138, 141, 151, 154, 158, 165, 170, 174, 181, 182, 193, 194, 199, 201, 213, 226, 229, 230, 235, 241, 246, 253, 258, 265, 271, 273

Offset: 1

Leroy Quet, Oct 06 2005

nonn

			10 is the smallest squarefree number greater than the 4th prime, 7. So a(4) = 10.
From _Gus Wiseman_, Dec 07 2024: (Start)
The first number line below shows the squarefree numbers. The second shows the primes:
--1--2--3-----5--6--7-------10-11----13-14-15----17----19----21-22-23-------26--
=====2==3=====5=====7==========11====13==========17====19==========23===========
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A061400, A112928, A112929, A112930.
Restriction of A067535, differences A378087.
The unrestricted opposite is A070321, differences A378085.
The opposite is A112925, differences A378038.
Subtracting prime(n) from each term gives A240474, opposite A240473.
For nonsquarefree we have A377783, restriction of A120327.
The nonsquarefree differences are A377784, restriction of A378039.
First differences are A378037.
For perfect power we have A378249, A378617, A378250, A378251.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A061398 counts squarefree numbers between primes, zeros A068360.
A061399 counts nonsquarefree numbers between primes, zeros A068361.
Cf. A007674, A013928, A053797, A053806, A072284, A073247, A175216, A280892, A345531, A378032, A378618.

with(numtheory): a:=proc(n) local p,B,j: p:=ithprime(n): B:={}: for j from p+1 to p+20 do if abs(mobius(j))>0 then B:=B union {j} else B:=B fi od: B[1] end: seq(a(m),m=1..75); # Emeric Deutsch, Oct 10 2005

Do[k = Prime[n] + 1; While[ !SquareFreeQ[k], k++ ]; Print[k], {n, 1, 100}] (* Ryan Propper, Oct 10 2005 *)
With[{k = 120}, Table[SelectFirst[Range[Prime@ n + 1, Prime@ n + k], SquareFreeQ], {n, 58}]] (* Michael De Vlieger, Aug 16 2017 *)

a(n,p=prime(n))=while(!issquarefree(p++),); p \\ Charles R Greathouse IV, Aug 16 2017

a(n) = prime(n) + A240474(n). - Gus Wiseman, Dec 07 2024

More terms from Ryan Propper and Emeric Deutsch, Oct 10 2005

A061399 Number of nonsquarefree integers between primes prime(n) and prime(n+1).

Original entry on oeis.org

0, 1, 0, 2, 1, 1, 1, 1, 4, 0, 2, 1, 0, 2, 4, 2, 1, 2, 1, 1, 2, 2, 2, 3, 3, 0, 1, 1, 1, 7, 1, 3, 0, 4, 1, 3, 2, 1, 4, 2, 1, 3, 1, 1, 1, 4, 3, 2, 1, 1, 2, 1, 6, 2, 2, 2, 1, 3, 2, 0, 4, 6, 1, 1, 2, 4, 3, 5, 1, 3, 1, 4, 3, 3, 1, 3, 2, 1, 3, 3, 1, 4, 1, 1, 2, 2, 3, 2, 0, 1, 5, 3, 2, 3, 1, 3, 4, 1, 9, 1, 5, 2, 3, 0, 3

Offset: 1

Labos Elemer, Jun 07 2001

nonn

			Between 113 and 127 the 7 numbers which are not squarefree are {116,117,120,121,124,125,126}, so a(30)=7.
From _Gus Wiseman_, Dec 07 2024: (Start)
The a(n) nonsquarefree numbers for n = 1..15:
   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  ----------------------------------------------------------
   .   4   .   8  12  16  18  20  24   .  32  40   .  44  48
               9                  25      36          45  49
                                  27                      50
                                  28                      52
(End)

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

Zeros are A068361.
First differences of A378086, restriction of A057627 to the primes.
Other classes (instead of nonsquarefree):
- For composite we have A046933, first differences of A065890.
- For squarefree see A061398, A068360, A071403, A373197, A373198, A377431.
- For prime power we have A080101.
- For non prime power we have A368748, see A378616.
- For perfect power we have A377432, zeros A377436.
- For non perfect power we have A377433, A029707.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147.
A120327 gives the least nonsquarefree number >= n.
Cf. A007674, A053806, A072284, A073247, A179211, A224363.
Cf. A013928, A067535, A070321, A112925, A112926, A240473, A240474.
Cf. A045882, A377049, A377783, A378032, A378618.

Count[Range[#[[1]]+1,#[[2]]-1],?(!SquareFreeQ[#]&)]&/@Partition[Prime[Range[120]],2,1] (* _Harvey P. Dale, Mar 31 2024 *)

{ n=0; q=2; forprime (p=3, prime(1001), a=0; for (i=q+1, p-1, a+=!issquarefree(i)); write("b061399.txt", n++, " ", a); q=p ) } \\ Harry J. Smith, Jul 22 2009

a(n) = my(p=prime(n)); sum(k=p, nextprime(p+1), ! issquarefree(k)); \\ Michel Marcus, Dec 09 2024

from sympy import mobius, prime
def A061399(n): return sum(not mobius(m) for m in range(prime(n)+1,prime(n+1))) # Chai Wah Wu, Jul 20 2024

A065474 Decimal expansion of Product_{p prime} (1 - 2/p^2).

Original entry on oeis.org

3, 2, 2, 6, 3, 4, 0, 9, 8, 9, 3, 9, 2, 4, 4, 6, 7, 0, 5, 7, 9, 5, 3, 1, 6, 9, 2, 5, 4, 8, 2, 3, 7, 0, 6, 6, 5, 7, 0, 9, 5, 0, 5, 7, 9, 6, 6, 5, 8, 3, 2, 7, 0, 9, 9, 6, 1, 8, 1, 1, 2, 5, 2, 4, 5, 3, 2, 5, 0, 0, 6, 3, 4, 8, 6, 2, 4, 4, 6, 0, 9, 8, 8, 4, 5, 2, 3, 4, 8, 1, 5, 6, 8, 5, 6, 3, 7, 5, 5, 2, 1, 7, 7, 2, 7, 3

Offset: 0

N. J. A. Sloane, Nov 19 2001

Density of A007674, squarefree n such that n + 1 is squarefree. - Charles R Greathouse IV, Aug 10 2011
Product_{k>=1} (1 - 2/k^2) = sin(sqrt(2)*Pi) / (sqrt(2)*Pi). - Vaclav Kotesovec, May 23 2020
The asymptotic probability that, for two integers k and m, 0 < k <= m, we have gcd(k*(k+1), m) = 1 (when k and m are chosen at random in the range 1..n and n->oo) (Tóth and Sándor, 1989). - Amiram Eldar, Apr 29 2023

			0.322634098939244670579531692548...

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

Henri Cohen, High precision computation of Hardy-Littlewood constants, (1998).
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 163 and 184.
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011, constant F_1^(2).
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
L. Tóth and J. Sándor, An asymptotic formula concerning a generalized Euler function, Fibonacci Quarterly, Vol. 27, No. 2 (1989), pp. 176-180.
Eric Weisstein's World of Mathematics, Squarefree.
Eric Weisstein's World of Mathematics, Prime Products.
Zack Wolske, Number Fields with Large Minimal Index, Ph.D. Thesis, University of Toronto, 2018.

Cf. A007674, A058026, A065493, A074893.

$MaxExtraPrecision = 800; digits = 98; terms = 800; P[n_] := PrimeZetaP[n]; LR = Join[{0, 0}, LinearRecurrence[{0, 2}, {-4, 0}, terms + 10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)

prodeulerrat(1 - 2/p^2) \\ Amiram Eldar, Mar 16 2021

Edited by Dean Hickerson, Sep 10 2002

More digits from Vaclav Kotesovec, Dec 18 2019

A070321 Greatest squarefree number <= n.

Original entry on oeis.org

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, 47, 51, 51, 53, 53, 55, 55, 57, 58, 59, 59, 61, 62, 62, 62, 65, 66, 67, 67, 69, 70, 71, 71

Offset: 1

Benoit Cloitre, May 11 2002

a(n) = Max( core(k) : k=1,2,3,...,n ) where core(x) is the squarefree part of x (the smallest integer such that x*core(x) is a square).

			From _Gus Wiseman_, Dec 10 2024: (Start)
The squarefree numbers <= n are the following columns, with maxima a(n):
  1  2  3  3  5  6  7  7  7  10  11  11  13  14  15  15
     1  2  2  3  5  6  6  6  7   10  10  11  13  14  14
        1  1  2  3  5  5  5  6   7   7   10  11  13  13
              1  2  3  3  3  5   6   6   7   10  11  11
                 1  2  2  2  3   5   5   6   7   10  10
                    1  1  1  2   3   3   5   6   7   7
                             1   2   2   3   5   6   6
                                 1   1   2   3   5   5
                                         1   2   3   3
                                             1   2   2
                                                 1   1
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Mayank Pandey, Squarefree numbers in short intervals, arXiv preprint (2024). arXiv:2401.13981 [math.NT]

Cf. A007947, A076260.
Cf. A081217, A081218, A081210.
The distinct terms are A005117 (the squarefree numbers).
The opposite version is A067535, differences A378087.
The run-lengths are A076259.
Restriction to the primes is A112925; see A378038, A112926, A378037.
For nonsquarefree we have A378033; see A120327, A378036, A378032, A377783.
First differences are A378085.
Subtracting each term from n gives A378619.
A013929 lists the nonsquarefree numbers, differences A078147.
A061398 counts squarefree numbers between primes, zeros A068360.
A061399 counts nonsquarefree numbers between primes, zeros A068361.
Cf. A007674, A013928, A053797, A053806, A072284, A073247, A112929, A240473.

A070321 := proc(n)
    local a;
    for a from n by -1 do
        if issqrfree(a) then
            return a;
        end if;
    end do:
end proc:
seq(A070321(n),n=1..100) ; # R. J. Mathar, May 25 2023

a[n_] :=For[ k = n, True, k--, If[ SquareFreeQ[k], Return[k]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 27 2013 *)
gsfn[n_]:=Module[{k=n},While[!SquareFreeQ[k],k--];k]; Array[gsfn,80] (* Harvey P. Dale, Mar 27 2013 *)

a(n) = while (! issquarefree(n), n--); n; \\ Michel Marcus, Mar 18 2017

from itertools import count
from sympy import factorint
def A070321(n): return next(m for m in count(n,-1) if max(factorint(m).values(),default=0)<=1) # Chai Wah Wu, Dec 04 2024

a(n) = n - o(n^(1/5)) by a result of Pandey. - Charles R Greathouse IV, Dec 04 2024

a(n) = A005117(A013928(n+1)). - Ridouane Oudra, Jul 26 2025

New description from Reinhard Zumkeller, Oct 03 2002

A068361 Numbers n such that the number of squarefree numbers between prime(n) and prime(n+1) = prime(n+1)-prime(n)-1.

Original entry on oeis.org

1, 3, 10, 13, 26, 33, 60, 89, 104, 113, 116, 142, 148, 201, 209, 212, 234, 265, 268, 288, 313, 320, 332, 343, 353, 384, 398, 408, 477, 484, 498, 542, 545, 551, 577, 581, 601, 625, 636, 671, 719, 723, 726, 745, 794, 805, 815, 862, 864, 884, 944, 964, 995, 1054

Offset: 1

Benoit Cloitre, Feb 28 2002

nonn

Also numbers k such that all numbers from prime(k) to prime(k+1) are squarefree. All such primes are twins, so this is a subset of A029707. The other twin primes are A061368. - Gus Wiseman, Dec 11 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

A subset of A029707 (lesser index of twin primes).
Prime index of each (prime) term of A061351.
Positions of zeros in A061399.
For perfect power instead of squarefree we have A377436, zeros of A377432.
Positions of zeros in A377784.
The rest of the twin primes are at A378620, indices of A061368.
A000040 lists the primes, differences A001223, (run-lengths A333254, A373821).
A005117 lists the squarefree numbers, differences A076259.
A006562 finds balanced primes.
A013929 lists the nonsquarefree numbers, differences A078147.
A014574 is the intersection of A006093 and A008864.
A038664 locates the first prime gap of size 2n.
A046933 counts composite numbers between primes.
A061398 counts squarefree numbers between primes, zeros A068360.
A120327 gives the least nonsquarefree number >= n.
Cf. A000720, A007674, A013928, A057627, A070321, A071403, A072284, A073247, A112925, A122535, A155752, A224363, A240473, A251092.

Select[Range[100],And@@SquareFreeQ/@Range[Prime[#],Prime[#+1]]&] (* Gus Wiseman, Dec 11 2024 *)

isok(n) = for (k=prime(n)+1, prime(n+1)-1, if (!issquarefree(k), return (0))); 1; \\ Michel Marcus, Apr 29 2016

n such that A061398(n) = prime(n+1)-prime(n)-1.

prime(a(n)) = A061351(n). - Gus Wiseman, Dec 11 2024

A070552 Semiprimes k such that k+1 is also a semiprime.

Original entry on oeis.org

9, 14, 21, 25, 33, 34, 38, 57, 85, 86, 93, 94, 118, 121, 122, 133, 141, 142, 145, 158, 177, 201, 202, 205, 213, 214, 217, 218, 253, 298, 301, 302, 326, 334, 361, 381, 393, 394, 445, 446, 453, 481, 501, 514, 526, 537, 542, 553, 565, 622, 633, 634, 694, 697

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 03 2002

nonn

A064911(a(n))*A064911(a(n)+1) = 1. - Reinhard Zumkeller, Dec 03 2009

D. W. Wilson, Table of n, a(n) for n = 1..10000

Cf. A001358, A007674, A039832, A100484.

IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [4..700] | IsSemiprime(n) and IsSemiprime(n+1) ]; // Vincenzo Librandi, Jan 22 2016

f[n_]:=Last/@FactorInteger[n]=={1,1}||Last/@FactorInteger[n]=={2};lst={};Do[If[f[n]&&f[n+1],AppendTo[lst,n]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 25 2010 *)
Flatten[Position[Partition[Table[If[PrimeOmega[n]==2,1,0],{n,700}],2,1],{1,1}]] (* Harvey P. Dale, Feb 04 2015 *)
Select[Range[700], PrimeOmega[#] == PrimeOmega[# + 1] == 2 &] (* Vincenzo Librandi, Jan 22 2016 *)

forprime(p=3,1e3,if(bigomega(2*p-1)==2,print1(2*p-1", "));if(bigomega(2*p+1)==2,print1(2*p", "))) \\ Charles R Greathouse IV, Nov 09 2011

is(n)=if(n%2, isprime((n+1)/2) && bigomega(n)==2, isprime(n/2) && bigomega(n+1)==2) \\ Charles R Greathouse IV, Sep 08 2015

from sympy import factorint
def is_semiprime(n): return sum(e for e in factorint(n).values()) == 2
def ok(n): return is_semiprime(n) and is_semiprime(n+1)
print(list(filter(ok, range(698)))) # Michael S. Branicky, Sep 14 2021

a(n) >> n log n since either n or n+1 is in A100484. - Charles R Greathouse IV, Jul 21 2015

a(n) = A109373(n) - 1. - Zak Seidov Dec 19 2018

More terms from Vladeta Jovovic, May 03 2002

cp's OEIS Frontend

A076259 Gaps between squarefree numbers: a(n) = A005117(n+1) - A005117(n).

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A053797 Lengths of successive gaps between squarefree numbers.

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A112925 Largest squarefree integer < the n-th prime.

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A068781 Lesser of two consecutive numbers each divisible by a square.

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A112926 Smallest squarefree integer > the n-th prime.

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A061399 Number of nonsquarefree integers between primes prime(n) and prime(n+1).

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A065474 Decimal expansion of Product_{p prime} (1 - 2/p^2).