A143658 -id:A143658 - cp's OEIS frontend

A101836 a(n) = 2^n - A143658(n).

0, 0, 1, 2, 5, 12, 25, 50, 99, 198, 400, 803, 1605, 3210, 6422, 12848, 25692, 51384, 102784, 205563, 411115, 822234, 1644470, 3288958, 6577915, 13155768, 26311537, 52623102, 105246259, 210492628, 420985102, 841970065, 1683940202, 3367880511, 6735761133, 13471522233

Offset: 0

Robert G. Wilson v, Aug 31 2008

nonn

The number of integers not exceeding 2^n that are not squarefree.

Amiram Eldar, Table of n, a(n) for n = 0..64 (calculated from the b-file at A143658)

Cf. A013929, A143658.

c = 0; k = 1; lst = {}; Do[ While[k <= 2^n, If[ !SquareFreeQ@ k, c++ ]; k++ ]; AppendTo[lst, c], {n, 0, 26}]; lst

a(0) inserted and more terms added by Amiram Eldar, Feb 09 2021

A061398 Number of squarefree integers between prime(n) and prime(n+1).

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Jun 07 2001

nonn

			Between 113 and 127 the 6 squarefree numbers are 114, 115, 118, 119, 122, 123, so a(30)=6.
From _Gus Wiseman_, Nov 06 2024: (Start)
The a(n) squarefree numbers for n = 1..16:
  1   2   3   4   5   6   7   8   9   10  11  12  13  14  15  16
  ---------------------------------------------------------------
  .   .   6   10  .   14  .   21  26  30  33  38  42  46  51  55
                      15      22          34  39              57
                                          35                  58
(End)

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

Cf. A000040, A005117, A013928.
Cf. A179211. [Reinhard Zumkeller, Jul 05 2010]
Counting all composite numbers (not just squarefree) gives A046933.
The version for nonsquarefree numbers is A061399.
Zeros are A068360.
The version for prime-powers is A080101.
Partial sums are A337030.
The version for non-prime-powers is A368748.
Excluding prime(n+1) from the range gives A373198.
Ones are A377430.
Positives are A377431.
The version for perfect-powers is A377432.
The version for non-perfect-powers is A377433 + 2.
For squarefree numbers (A005117) between primes:
- length is A061398 (this sequence)
- min is A112926
- max is A112925
- sum is A373197
For squarefree numbers between powers of two:
- length is A077643 (except initial terms), partial sums A143658
- min is A372683, difference A373125, indices A372540, firsts of A372475
- max is A372889, difference A373126
- sum is A373123
For primes between powers of two:
- length is A036378
- min is A104080 or A014210, indices A372684 (firsts of A035100)
- max is A014234, difference A013603
- sum is A293697 (except initial terms)
Cf. A001223, A049093, A049094, A076259, A077641, A377434, A377436.

p:= 2:
for n from 1 to 200 do
  q:= nextprime(p);
A[n]:= nops(select(numtheory:-issqrfree, [$p+1..q-1]));
p:= q;
od:
seq(A[i],i=1..200); # Robert Israel, Jan 06 2017

a[n_] := Count[Range[Prime[n]+1, Prime[n+1]-1], _?SquareFreeQ];
Array[a, 100] (* Jean-François Alcover, Feb 28 2019 *)
Count[Range[#[[1]]+1,#[[2]]-1],?(SquareFreeQ[#]&)]&/@Partition[ Prime[ Range[120]],2,1] (* _Harvey P. Dale, Oct 14 2021 *)

{ n=0; q=2; forprime (p=3, prime(1001), a=0; for (i=q+1, p-1, a+=issquarefree(i)); write("b061398.txt", n++, " ", a); q=p ) } \\ Harry J. Smith, Jul 22 2009

a(n) = my(pp=prime(n)+1); sum(k=pp, nextprime(pp)-1, issquarefree(k)); \\ Michel Marcus, Feb 28 2019

from math import isqrt
from sympy import mobius, prime, nextprime
def A061398(n):
    p = prime(n)
    q = nextprime(p)
    r = isqrt(p-1)+1
    return sum(mobius(k)*((q-1)//k**2) for k in range(r,isqrt(q-1)+1))+sum(mobius(k)*((q-1)//k**2-(p-1)//k**2) for k in range(1,r))-1 # Chai Wah Wu, Jun 01 2024

a(n) = A013928(A000040(n+1)) - A013928(A000040(n)) - 1. - Robert Israel, Jan 06 2017

a(n) = A373198(n) - 1. - Gus Wiseman, Nov 06 2024

A014234 Largest prime <= 2^n.

Original entry on oeis.org

2, 3, 7, 13, 31, 61, 127, 251, 509, 1021, 2039, 4093, 8191, 16381, 32749, 65521, 131071, 262139, 524287, 1048573, 2097143, 4194301, 8388593, 16777213, 33554393, 67108859, 134217689, 268435399, 536870909, 1073741789, 2147483647, 4294967291, 8589934583, 17179869143, 34359738337, 68719476731, 137438953447

Offset: 1

Jud McCranie

nonn

For n>1 largest prime factor of the denominator of A027611(2^n) = 2^n*(2^n)-th harmonic number. - Alexander Adamchuk, Aug 02 2006

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 390.

T. D. Noe, Table of n, a(n) for n=1..1000
Fred Curtis, C++ program for A014234
Harry J. Smith, PrimePi2 - Computes the Prime Pi(x) counting function [Broken link]
Harry J. Smith, PrimePi2 - Computes the Prime Pi(x) counting function [Cached copy]

Cf. A000079, A014210.
Cf. A013603 (2^n - a(n)).
See comment for the relationship to A027611.
These primes have indices A007053 = number of primes <= 2^n.
The opposite is A104080, delta A092131, indices A372684.
For squarefree instead of prime we have A372889, indices A143658.
A036378 counts primes between powers of 2, A293697 adds them up.
Cf. A035100, A130739, A211997.

a:= n-> prevprime(2^n+1):
seq(a(n), n=1..40);  # Alois P. Heinz, Apr 23 2020

PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; Table[ Abs[ PrevPrim[2^n]], {n, 1, 30} ]
Join[{2},NextPrime[2^Range[2,40],-1]] (* Harvey P. Dale, Jun 26 2011 *)

a(n) = precprime(2^n) \\ Michel Marcus, Aug 08 2013

Terms for n=31, n=32 added by Fred Curtis (fred(AT)f2.org), Dec 08 2009

A373198 Number of squarefree numbers from prime(n) to prime(n+1) - 1.

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 1, 3, 2, 2, 4, 3, 2, 2, 2, 4, 1, 4, 3, 1, 4, 2, 4, 5, 1, 2, 3, 1, 3, 7, 3, 3, 2, 6, 1, 3, 4, 3, 2, 4, 1, 7, 1, 3, 1, 8, 9, 2, 1, 3, 4, 1, 4, 4, 4, 4, 1, 3, 2, 2, 6, 8, 3, 1, 2, 10, 3, 5, 1, 1, 5, 4, 3, 3, 3, 3, 6, 3, 5, 7, 1, 6, 1, 5, 2, 4, 5

Offset: 1

Gus Wiseman, May 29 2024

nonn

			This is the sequence of row-lengths of A005117 treated as a triangle with row-sums A373197:
   2
   3
   5   6
   7  10
  11
  13  14  15
  17
  19  21  22
  23  26
  29  30
  31  33  34  35
  37  38  39
  41  42
  43  46
  47  51
  53  55  57  58

Counting all numbers (not just squarefree) gives A001223, sum A371201.
For composite instead of squarefree we have A046933.
For squarefree numbers (A005117) between primes:
- sum is A373197
- length is A373198 (this sequence) = A061398 - 1
- min is A000040
- max is A112925, opposite A112926
For squarefree numbers between powers of two:
- sum is A373123
- length is A077643, partial sums A143658
- min is A372683, delta A373125, indices A372540, firsts of A372475
- max is A372889, delta A373126
For primes between powers of two:
- sum is A293697 (except initial terms)
- length is A036378
- min is A104080 or A014210, indices A372684 (firsts of A035100)
- max is A014234, delta A013603
Cf. A372473 (firsts of A372472), A372541 (firsts of A372433).
Cf. A049093, A049094, A076259, A077641.

Table[Length[Select[Range[Prime[n],Prime[n+1]-1],SquareFreeQ]],{n,100}]

from math import isqrt
from sympy import prime, nextprime, mobius
def A373198(n):
    p = prime(n)
    q = nextprime(p)
    r = isqrt(p-1)+1
    return sum(mobius(k)*((q-1)//k**2) for k in range(r,isqrt(q-1)+1))+sum(mobius(k)*((q-1)//k**2-(p-1)//k**2) for k in range(1,r)) # Chai Wah Wu, Jun 01 2024

a(n) = A061398(n) + 1.

A104080 Smallest prime >= 2^n.

Original entry on oeis.org

2, 2, 5, 11, 17, 37, 67, 131, 257, 521, 1031, 2053, 4099, 8209, 16411, 32771, 65537, 131101, 262147, 524309, 1048583, 2097169, 4194319, 8388617, 16777259, 33554467, 67108879, 134217757, 268435459, 536870923, 1073741827, 2147483659

Offset: 0

Cino Hilliard, Mar 03 2005

Jinyuan Wang, Table of n, a(n) for n = 0..1000

Cf. A104081, A338475.
Except initial terms and offset, same as A014210 and A203074.
The opposite (greatest prime <= 2^n) is A014234, indices A007053.
The distance from 2^n is A092131, opposite A013603.
Counting zeros instead of both bits gives A372474, cf. A035103, A211997.
Counting ones instead of both bits gives A372517, cf. A014499, A061712.
For squarefree instead of prime we have A372683, cf. A143658, A372540.
The indices of these prime are given by A372684.
Cf. A007918, A007920, A029837, A035100, A049095, A130739.

Join[{2,2},NextPrime[#]&/@(2^Range[2,40])]  (* Harvey P. Dale, Jan 26 2011 *)
NextPrime[2^Range[0,50]-1]  (* Vladimir Joseph Stephan Orlovsky, Apr 11 2011 *)

g(n,b=2) = for(x=0,n,print1(nextprime(b^x)","))

a(n) = nextprime(2^n); \\ Michel Marcus, Nov 01 2020

a(n) = A014210(n), n <> 1. - R. J. Mathar, Oct 14 2008
Sum_{n >= 0} 1/a(n) = A338475 + 1/6 = 1.4070738... (because 1/6 = 1/2 - 1/3). - Bernard Schott, Nov 01 2020
From Gus Wiseman, Jun 03 2024: (Start)
a(n) = A007918(2^n).
a(n) = 2^n + A092131(n).
a(n) = prime(A372684(n)).
(End)

A077643 Number of squarefree integers in closed interval [2^n, -1 + 2*2^n], i.e., among 2^n consecutive numbers beginning with 2^n.

Original entry on oeis.org

1, 2, 3, 5, 9, 19, 39, 79, 157, 310, 621, 1246, 2491, 4980, 9958, 19924, 39844, 79672, 159365, 318736, 637457, 1274916, 2549816, 5099651, 10199363, 20398663, 40797299, 81594571, 163189087, 326378438, 652756861, 1305513511, 2611026987, 5222053970, 10444108084

Offset: 0

Labos Elemer, Nov 14 2002

nonn

Number of squarefree numbers with binary expansion of length n, or with n bits. The sum of these numbers is given by A373123. - Gus Wiseman, Jun 02 2024

			For n=4: among the 16 numbers of {16, ..., 31}, nine are squarefree [17, 19, 21, 22, 23, 26, 29, 30, 31], so a(4) = 9.

Amiram Eldar, Table of n, a(n) for n = 0..63 (calculated from the b-file at A143658)

Cf. A077641, A077642.
Partial sums (except first term) are A143658.
Run-lengths of A372475.
The minimum is A372683, delta A373125, indices A372540.
The maximum is A372889 (except at n=1), delta A373126, indices A143658.
Row-sums are A373123.
A005117 lists squarefree numbers, first differences A076259.
A053797 gives nonempty lengths of exclusive gaps between squarefree numbers.
A029837 counts bits, row-lengths of A030190 and A030308.
For primes between powers of 2:
- sum A293697
- length A036378 or A162145
- min A104080 or A014210, delta A092131, indices A372684
- max A014234, delta A013603, indices A007053
For squarefree numbers between primes:
- sum A373197
- length A373198 = A061398 - 1
- min A000040
- max A112925 (delta A240473), opposite A112926 (delta A240474)
Cf. A010036, A029931, A035100, A049093-A049096, A372473 (firsts of A372472), A372541 (firsts of A372433).

Table[Apply[Plus, Table[Abs[MoebiusMu[2^w+j]], {j, 0, 2^w-1}]], {w, 0, 15}]
(* second program *)
Length/@Split[IntegerLength[Select[Range[10000],SquareFreeQ],2]]//Most (* Gus Wiseman, Jun 02 2024 *)

{ a(n) = sum(m=1,sqrtint(2^(n+1)-1), moebius(m) * ((2^(n+1)-1)\m^2 - (2^n-1)\m^2) ) } \\ Max Alekseyev, Oct 18 2008

a(n) = Sum_{j=0..-1+2^n} abs(mu(2^n + j)).

a(n)/2^n approaches 1/zeta(2), so limiting sequence is floor(2^n/zeta(2)), n >= 0. - Wouter Meeussen, May 25 2003

More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 12 2003
More terms from Wouter Meeussen, May 25 2003
a(25)-a(32) from Max Alekseyev, Oct 18 2008
a(33)-a(34) from Amiram Eldar, Jul 17 2024

A373197 Sum of all squarefree numbers from prime(n) to prime(n+1) - 1.

Original entry on oeis.org

2, 3, 11, 17, 11, 42, 17, 62, 49, 59, 133, 114, 83, 89, 98, 223, 59, 254, 206, 71, 302, 161, 341, 462, 97, 203, 314, 107, 330, 824, 386, 398, 275, 856, 149, 460, 635, 494, 337, 702, 179, 1294, 191, 582, 197, 1635, 1950, 449, 227, 690, 943, 239, 983, 1013, 1036

Offset: 1

Gus Wiseman, May 29 2024

nonn

			This is the sequence of row sums of A005117 treated as a triangle with row-lengths A373198:
   2
   3
   5   6
   7  10
  11
  13  14  15
  17
  19  21  22
  23  26
  29  30
  31  33  34  35
  37  38  39
  41  42
  43  46
  47  51
  53  55  57  58

Counting all numbers (not just squarefree) gives A371201.
For the sectioning of A005117 (squarefree between prime):
- sum is A373197 (this sequence)
- length is A373198 = A061398 - 1
- min is A000040
- max is A112925, opposite A112926
For squarefree numbers between powers of two:
- sum is A373123
- length is A077643, partial sums A143658
- min is A372683, delta A373125, indices A372540, firsts of A372475
- max is A372889, delta A373126
For primes between powers of two:
- sum is A293697 (except initial terms)
- length is A036378
- min is A104080 or A014210, indices A372684 (firsts of A035100)
- max is A014234, delta A013603
Cf. A372473 (firsts of A372472), A372541 (firsts of A372433).
Cf. A046933, A049093, A049094, A076259, A077641.

Table[Total[Select[Range[Prime[n],Prime[n+1]-1],SquareFreeQ]],{n,15}]

A375707 First differences minus 1 of nonsquarefree numbers.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 16 2024

Also the number of squarefree numbers between the nonsquarefree numbers A013929(n) and A013929(n+1).
Delete all 0's to get A120992.
The image is {0,1,2,3}.
Add 1 to all terms for A078147.

			The runs of squarefree numbers begin:
  (5,6,7)
  ()
  (10,11)
  (13,14,15)
  (17)
  (19)
  (21,22,23)
  ()
  (26)
  ()
  (29,30,31)
  (33,34,35)

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

Positions of 0, 1, 2, 3 are A375709, A375710, A375711, A375712. This is a set partition of the positive integers into four blocks.
For runs of squarefree numbers:
- length: A120992, anti A373127
- min: A072284, anti A373408
- max: A373415, anti A007674
- sum: A373413, anti A373411
For runs of nonsquarefree numbers:
- length: A053797, anti A373409
- min: A053806, anti A373410
- max: A376164, anti A068781
- sum: A373414, anti A373412
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A046933 counts composite numbers between consecutive primes.
A073784 counts primes between consecutive composite numbers.
A093555 counts non-prime-powers between consecutive prime-powers.
Cf. A061398, A061399, A077641, A077643, A143658, A173143, A251092, A294242, A373125, A373126, A373573.

Differences[Select[Range[100],!SquareFreeQ[#]&]]-1

lista(nmax) = {my(prev = 4); for (n = 5, nmax, if(!issquarefree(n), print1(n - prev - 1, ", "); prev = n));} \\ Amiram Eldar, Sep 17 2024

Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = 6/(Pi^2-6) = 1.550546... . - Amiram Eldar, Sep 17 2024

A071172 Number of squarefree integers <= 10^n.

Original entry on oeis.org

1, 7, 61, 608, 6083, 60794, 607926, 6079291, 60792694, 607927124, 6079270942, 60792710280, 607927102274, 6079271018294, 60792710185947, 607927101854103, 6079271018540405, 60792710185403794, 607927101854022750, 6079271018540280875, 60792710185402613302, 607927101854026645617

Offset: 0

Robert G. Wilson v, Jun 10 2002

nonn

The limit of a(n)/10^n is 6/Pi^2 (see A059956). - Gerard P. Michon, Apr 30 2009

J. Pawlewicz, Table of n, a(n) for n = 0..36
W. Hürlimann, A First Digit Theorem for Square-Free Integer Powers, Pure Mathematical Sciences, Vol. 3, 2014, no. 3, 129 - 139 HIKARI Ltd.
G. P. Michon, On the number of squarefree integers not exceeding N. - _Gerard P. Michon_, Apr 30 2009
J. Pawlewicz, Counting square-free numbers, arXiv preprint arXiv:1107.4890 [math.NT], 2011.
Eric Weisstein's World of Mathematics, Squarefree

Cf. A005117, A013928.
Apart from first two terms, same as A053462.
Binary counterpart is A143658. - Gerard P. Michon, Apr 30 2009

f[n_] := Sum[ MoebiusMu[i]Floor[n/i^2], {i, Sqrt@ n}]; Table[ f[10^n], {n, 0, 14}] (* Robert G. Wilson v, Aug 04 2012 *)

a(n)=sum(d=1,sqrtint(n=10^n),moebius(d)*n\d^2) \\ Charles R Greathouse IV, Nov 14 2012

a(n)=my(s); forsquarefree(d=1,sqrtint(n=10^n), s += n\d[1]^2 * moebius(d)); s \\ Charles R Greathouse IV, Jan 08 2018

from math import isqrt
from sympy import mobius
def A071172(n): return sum(mobius(k)*(10**n//k**2) for k in range(1,isqrt(10**n)+1)) # Chai Wah Wu, May 10 2024

a(n) = Sum_{i=1..10^(n/2)} A008683(i)*floor(10^n/i^2). - Gerard P. Michon, Apr 30 2009

Extended by Eric W. Weisstein, Sep 14 2003
3 more terms from Jud McCranie, Sep 01 2005
4 more terms from Gerard P. Michon, Apr 30 2009

A372683 Least squarefree number >= 2^n.

Original entry on oeis.org

1, 2, 5, 10, 17, 33, 65, 129, 257, 514, 1027, 2049, 4097, 8193, 16385, 32770, 65537, 131073, 262145, 524289, 1048577, 2097154, 4194305, 8388609, 16777217, 33554433, 67108865, 134217730, 268435457, 536870913, 1073741826, 2147483649, 4294967297, 8589934594

Offset: 0

Gus Wiseman, May 26 2024

nonn

			The terms together with their binary expansions and binary indices begin:
       1:                    1 ~ {1}
       2:                   10 ~ {2}
       5:                  101 ~ {1,3}
      10:                 1010 ~ {2,4}
      17:                10001 ~ {1,5}
      33:               100001 ~ {1,6}
      65:              1000001 ~ {1,7}
     129:             10000001 ~ {1,8}
     257:            100000001 ~ {1,9}
     514:           1000000010 ~ {2,10}
    1027:          10000000011 ~ {1,2,11}
    2049:         100000000001 ~ {1,12}
    4097:        1000000000001 ~ {1,13}
    8193:       10000000000001 ~ {1,14}
   16385:      100000000000001 ~ {1,15}
   32770:     1000000000000010 ~ {2,16}
   65537:    10000000000000001 ~ {1,17}
  131073:   100000000000000001 ~ {1,18}
  262145:  1000000000000000001 ~ {1,19}
  524289: 10000000000000000001 ~ {1,20}

For primes instead of powers of two we have A112926, opposite A112925, sum A373197, length A373198.
Counting zeros instead of all bits gives A372473, firsts of A372472.
These are squarefree numbers at indices A372540, firsts of A372475.
Counting ones instead of all bits gives A372541, firsts of A372433.
The opposite (greatest squarefree number <= 2^n) is A372889.
The difference from 2^n is A373125.
For prime instead of squarefree we have:
- bits A372684, firsts of A035100
- zeros A372474, firsts of A035103
- ones A372517, firsts of A014499
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A005117 lists squarefree numbers.
A030190 gives binary expansion, reversed A030308, length A070939 or A029837.
A061398 counts squarefree numbers between primes (exclusive).
A077643 counts squarefree terms between powers of 2, run-lengths of A372475.
A143658 counts squarefree numbers up to 2^n.
Cf. A029931, A048793, A049095, A049096, A059015, A069010, A076259, A077641, A211997, A230877.

Table[NestWhile[#+1&,2^n,!SquareFreeQ[#]&],{n,0,10}]

a(n) = my(k=2^n); while (!issquarefree(k), k++); k; \\ Michel Marcus, May 29 2024

from itertools import count
from sympy import factorint
def A372683(n): return next(i for i in count(1<Chai Wah Wu, Aug 26 2024

a(n) = A005117(A372540(n)).

a(n) = A067535(2^n). - R. J. Mathar, May 31 2024

cp's OEIS Frontend

A101836 a(n) = 2^n - A143658(n).

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A061398 Number of squarefree integers between prime(n) and prime(n+1).

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A014234 Largest prime <= 2^n.

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Maple

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A373198 Number of squarefree numbers from prime(n) to prime(n+1) - 1.

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A104080 Smallest prime >= 2^n.

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A077643 Number of squarefree integers in closed interval [2^n, -1 + 2*2^n], i.e., among 2^n consecutive numbers beginning with 2^n.

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Mathematica

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A373197 Sum of all squarefree numbers from prime(n) to prime(n+1) - 1.

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Mathematica