A013928 -id:A013928 - cp's OEIS frontend

A285720 Number of ways to write n as a sum of two unordered squarefree numbers so that their addition in base-2 does not produce carries.

0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 6, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 11, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 7, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 3, 0, 0, 0, 11, 0, 0, 0, 3, 0, 0, 0, 7, 0, 0, 0, 7, 0, 0, 0, 13, 0, 0, 0, 3, 0, 0, 0, 9, 0

Offset: 1

Antti Karttunen, May 02 2017

Antti Karttunen, Table of n, a(n) for n = 1..4095
Index entries for sequences related to binary expansion of n

Cf. A003987, A004198, A005117, A008683, A008966, A013928, A071068, A088512.

Table[Sum[Abs[MoebiusMu[i] MoebiusMu[n - i]] Boole[BitXor[i, n - i] == n], {i, Floor[n/2]}], {n, 120}] (* Michael De Vlieger, May 03 2017 *)

from sympy import mobius
def a003987(n, i): return i^(n - i) == n
def a(n): return sum([abs(mobius(i)*mobius(n - i))*(1*a003987(n, i)) for i in range(1, n//2 + 1)])
print([a(n) for n in range(1,121)]) # Indranil Ghosh, May 02 2017

(define (A285720 n) (let loop ((k (A013928 n)) (s 0)) (if (or (zero? k) (< (A005117 k) (- n (A005117 k)))) s (loop (- k 1) (+ s (if (and (= 1 (A008966 (- n (A005117 k)))) (zero? (A004198bi (A005117 k) (- n (A005117 k))))) 1 0)))))) ;; Where A004198bi implements bitwise-AND (A004198).

a(n) = Sum_{i=1..floor(n/2)} abs(mu(i)*mu(n-i))*[A003987(i,n-i) == n]. (Here [] is Iverson bracket, giving in this case 1 only if (i XOR (n-i)) is equal to n, and 0 otherwise. mu is Moebius mu function, A008683.)
a(n) <= A071068(n).
a(n) <= A088512(n).

A057639 First differences of zero-sites (A028442) of Mertens's function A002321.

Original entry on oeis.org

37, 1, 18, 7, 28, 8, 44, 4, 1, 9, 1, 3, 1, 2, 48, 17, 1, 3, 1, 2, 16, 75, 2, 1, 1, 20, 2, 1, 2, 4, 1, 1, 2, 27, 8, 2, 1, 1, 2, 1, 5, 1, 5, 1, 2, 1, 1, 1, 2, 1, 109, 4, 66, 1, 27, 1, 1, 144, 4, 8, 2, 1, 2, 13, 1, 2, 9, 1, 1, 24, 1, 3, 16, 8, 6, 1, 2, 3, 4, 2, 1, 2, 5, 1, 2, 4, 3, 2, 1, 3, 1, 82, 3, 5

Offset: 1

Labos Elemer, Oct 11 2000

nonn

Mertens's function (A002321) is oscillating. The width of its waves is given here.

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

Cf. A002321, A013928, A028442, A013940, A051400, A051401, A051402.

Differences[Position[Accumulate[Array[MoebiusMu,1500]],0]//Flatten] (* Harvey P. Dale, Nov 10 2016 *)

lista(kmax) = {my(s = 0, k1 = 2); for(k2 = 3, kmax, s += moebius(k2); if(s == 0, print1(k2 - k1, ", "); k1 = k2));} \\ Amiram Eldar, Jun 09 2024

a(n) = A028442(n+1) - A028442(n).

Offset corrected by Amiram Eldar, Jun 09 2024

A068576 Numbers k such that Sum_{j=1..k} mu(j)^2 = floor(6*k/Pi^2).

Original entry on oeis.org

28, 56, 153, 172, 173, 175, 176, 177, 178, 180, 181, 344, 351, 352, 353, 354, 356, 357, 361, 362, 363, 365, 366, 367, 368, 370, 371, 373, 374, 375, 383, 386, 391, 393, 394, 395, 396, 397, 400, 405, 408, 425, 428, 640, 752, 848, 849, 850, 851, 852, 853, 854

Offset: 1

Benoit Cloitre, Mar 26 2002

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

Cf. A008683, A013928, A059956.

seq[max_] := Flatten @ Position[Accumulate @ Array[Boole @ SquareFreeQ[#] &, max] - Floor[6*Range[max]/Pi^2], 0]; seq[1000] (* Amiram Eldar, Feb 17 2021 *)

isok(k) = sum(j=1, k, moebius(j)^2) == 6*k\Pi^2; \\ Michel Marcus, Feb 15 2021

A070544 Number of squarefree numbers s such that n < s < 2n.

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 3, 5, 6, 6, 6, 8, 7, 7, 7, 9, 9, 11, 11, 13, 13, 14, 13, 15, 15, 15, 16, 17, 17, 18, 18, 19, 19, 20, 20, 22, 22, 22, 22, 24, 23, 24, 24, 26, 27, 27, 27, 29, 30, 30, 30, 32, 32, 34, 34, 36, 36, 37, 36, 38, 37, 38, 38, 39, 39, 40, 40, 41, 41, 42, 42, 44, 44, 44, 45

Offset: 1

Benoit Cloitre, May 02 2002

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

Cf. A008683, A008966, A013928, A059956, A128076.

Table[Count[Range[n+1,2n-1],?SquareFreeQ],{n,80}] (* _Harvey P. Dale, Mar 23 2019 *)

a(n) = sum(i=n+1,2*n-1,issquarefree(i));

Limit_{n -> oo} a(n)/n = 6/Pi^2 (A059956).
From Wesley Ivan Hurt, Jan 08 2022: (Start)
a(n) = Sum_{k=1..n-1} mu(2n-k)^2.
a(n) = Sum_{k=n+1..2n-1} mu(k)^2.
a(n) = Sum_{k=(n^2-n+2)/2..(n^2+n-2)/2} mu(A128076(k))^2. (End)
a(n) = A013928(2*n) - A013928(n) - A008966(n). - Amiram Eldar, Apr 29 2025

A230490 Size of largest subset of [1..n] containing no three terms in a geometric progression with integer ratio.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13, 14, 15, 15, 16, 16, 17, 18, 19, 20, 21, 22, 23, 23, 24, 25, 26, 26, 27, 28, 29, 30, 31, 32, 33, 33, 34, 35, 36, 36, 37, 38, 39, 39, 40, 41, 42, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 52, 52, 53, 54, 55, 55, 56, 57, 58, 59, 60, 61, 62, 62, 63, 64, 65, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 76, 77, 78, 79, 79, 80, 81, 81, 81

Offset: 1

Nathan McNew, Oct 20 2013

nonn

Trivial lower bound: a(n) >= A013928(n+1). - Charles R Greathouse IV, Oct 20 2013

McNew proves that if n is sufficiently large, then the n-th term is between 0.818n and 0.820n. - Kevin O'Bryant, Aug 17 2015

			The integers [1..9] include the three geometric progressions (1,2,4) (2,4,8) and (1,3,9), which cannot all be precluded with any 1 exclusion, but 2 exclusions suffice. Thus the size of the largest subsets of [1..9] free of integer ratio geometric progressions is 7.

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..152
M. Beiglboeck, V. Bergelson, N. Hindman, and D. Strauss, Multiplicative structures in additively large sets, J. Combin. Theory Ser. A 113 (2006)
N. McNew, On sets of integers which contain no three terms in geometric progression, arXiv:1310.2277 [math.NT], 2013.
M. B. Nathanson and K. O'Bryant, A problem of Rankin on sets without geometric progressions, arXiv:1408.2880 [math.NT], 2014.
K. O'Bryant, Sets of natural numbers with proscribed subsets, arXiv:1410.4900 [math.NT], 2014-2015.

Cf. A003002, A013928, A208746 is similar but also allows progressions with rational ratio, like (4,6,9).

ok(v)=for(i=3,#v,my(k=v[i]);fordiv(core(k,1)[2],d,if(d>1 && setsearch(v,k/d) && setsearch(v,k/d^2), return(0)))); 1
a(n)=my(v=select(k->4*k>n&&issquarefree(k),vector(n,i,i)), u=setminus(vector(n, i,i),v),r,H);for(i=1,2^#u-1,H=hammingweight(i); if(H>r && ok(vecsort(concat(v,vecextract(u,i)),,8)),r=H));#v+r \\ Charles R Greathouse IV, Oct 20 2013

A243284 a(n) = the number of distinct ways of writing such products m = k^2 * j, 0 < j <= k, (j and k natural numbers) that m is in range [1,n]; Partial sums of A102354.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 17

Offset: 1

Antti Karttunen, Jun 02 2014

nonn

a(n) = the number of distinct ways of writing such products m = k^2 * j, 0 < j <= k, (j and k natural numbers) that m is in range [1,n].
Different ways to write product for the same m are counted separately, e.g. for 64, both 8^2 * 1 and 4^2 * 4 are counted, so a(64) = a(63)+2 = 13+2 = 15.
Differs from A243283 for the first time at n=48, where a(48)=11, while A243283(48)=10. This is because 48 = 2*2*2*2*3 is the first integer which can be represented in the form k^2 * j, 0 < j <= k (namely as 48 = 4^2 * 3), even though it is not a member of A070003.

Partial sums of A102354.

Cf. A243283, A013928, A057627.

A246419 Numbers n such that n^6 - 2 is not squarefree.

Original entry on oeis.org

10, 22, 50, 143, 204, 267, 311, 312, 423, 455, 461, 479, 506, 556, 579, 600, 649, 818, 845, 889, 987, 1008, 1104, 1108, 1134, 1178, 1258, 1273, 1333, 1343, 1416, 1423, 1467, 1537, 1610, 1637, 1712, 1756, 1779, 2001, 2005, 2045, 2065, 2066, 2104, 2166, 2205

Offset: 1

Mark E. Shoulson, Aug 25 2014

nonn

			10^6 - 2 = 999998, which is divisible by 16129 = 127^2, so 10 is in the sequence.
11^6 - 2 = 1771559, which is prime, and 12^6 - 2 = 2985982 = 2 * 17 * 31 * 2833 (all prime), so neither of these numbers is in the sequence.

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

Cf. A005117, A013928.

[n: n in [0..3000] | not IsSquarefree(n^6-2)]; // Vincenzo Librandi, Oct 16 2014

remove(t -> numtheory:-issqrfree(t^6-2), [$1..3000]); # Robert Israel, Aug 25 2014

Select[Range[2000], MoebiusMu[#^6 - 2] == 0 &] (* Alonso del Arte, Aug 25 2014 *)

isok(n) = ! issquarefree(n^6-2); \\ Michel Marcus, Oct 11 2014

from sympy import factorint
[i for i in range(1, 500) if any(v>1 for v in factorint(i**6-2).values())]

A285881 Number of even squarefree numbers <= n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18

Offset: 1

Ilya Gutkovskiy, Apr 27 2017

nonn

Eric Weisstein's World of Mathematics, Squarefree

Cf. A005117, A008683, A013928, A039956, A285879.

Table[Sum[Boole[EvenQ[k] && SquareFreeQ[k]], {k, 1, n}], {n, 85}]
nmax = 85; Rest[CoefficientList[Series[Sum[Boole[EvenQ[k] && MoebiusMu[k]^2 == 1] x^k/(1 - x), {k, 1, nmax}], {x, 0, nmax}], x]]

from sympy.ntheory.factor_ import core
def a(n): return sum([1 for k in range(1, n + 1) if k%2==0 and core(k)==k]) # Indranil Ghosh, Apr 28 2017

G.f.: Sum_{k>=1} x^A039956(k)/(1 - x).

a(n) ~ 2*n/Pi^2.

A294062 Sum of the differences of the larger and smaller parts in the partitions of 2n into two parts with the smaller part squarefree.

Original entry on oeis.org

0, 2, 6, 12, 18, 26, 36, 48, 60, 72, 86, 102, 118, 136, 156, 178, 200, 224, 248, 274, 300, 328, 358, 390, 422, 454, 488, 522, 556, 592, 630, 670, 710, 752, 796, 842, 888, 936, 986, 1038, 1090, 1144, 1200, 1258, 1316, 1374, 1434, 1496, 1558, 1620, 1682, 1746

Offset: 1

Wesley Ivan Hurt, Oct 22 2017

Sum of the slopes of the tangent lines along the left side of the parabola b(x) = 2*n*x-x^2 at squarefree values of x for x in 0 < x <= n. For example, d/dx 2*n*x-x^2 = 2n-2x. So for a(6), the squarefree values of x are x=1,2,3,5,6 and so a(6) = 12-2*1 + 12-2*2 + 12-2*3 + 12-2*5 + 12-2*6 = 10 + 8 + 6 + 2 = 26. - Wesley Ivan Hurt, Mar 25 2018

			For n = 4, 8 can be partitioned into two parts with the smaller part squarefree in three ways: 7 + 1, 6 + 2, and 5 + 3, so a(4) = (7 - 1) + (6 - 2) + (5 - 3) = 12. - _Michael B. Porter_, Mar 27 2018

Index entries for sequences related to partitions

Cf. A008683 (mu), A008966, A013928, A066779.

Table[2*Sum[(n - i) MoebiusMu[i]^2, {i, n}], {n, 80}]

a(n) = 2 * sum(i=1, n, (n-i)*issquarefree(i)); \\ Michel Marcus, Mar 26 2018

a(n) = 2 * Sum_{i=1..n} (n - i) * mu(i)^2, where mu is the Möbius function (A008683).

a(n) = 2*(n*A013928(n) - A066779(n)). - Wesley Ivan Hurt, Jul 08 2025

A351520 Number of numbers <= n that are either squarefree, a divisor of n, or both.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 6, 8, 7, 7, 8, 10, 9, 10, 11, 14, 12, 14, 13, 15, 14, 15, 16, 20, 17, 17, 19, 19, 18, 19, 20, 24, 21, 22, 23, 28, 24, 25, 26, 30, 27, 28, 29, 31, 31, 30, 31, 37, 32, 33, 32, 34, 33, 37, 34, 38, 35, 36, 37, 41, 38, 39, 41, 44, 40, 41, 42, 44, 43, 44, 45, 53, 46, 47, 49

Offset: 1

Wesley Ivan Hurt, Feb 12 2022

nonn

			a(10) = 7; There are 7 numbers less than or equal to 10 that are either squarefree, a divisor of 10, or both. The numbers are 1,2,3,5,6,7,10.

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

Cf. A000005 (tau), A013928, A034444.

f:= proc(n) nops(select(t -> n mod t = 0 or numtheory:-issqrfree(t), [$1..n])) end proc:
map(f, [$1..100]); # Robert Israel, Dec 09 2024

Module[{nn=80,sf},sf=Select[Range[nn],SquareFreeQ[#]&];Table[Length[Union[Select[sf,#<= n&],Divisors[n]]],{n,nn}]] (* Harvey P. Dale, Jul 03 2023 *)

a(n) = tau(n) + Sum_{k=1..n} mu(k)^2 - Sum_{d|n} mu(d)^2.

a(n) = A000005(n) + A013928(n+1) - A034444(n).

cp's OEIS Frontend

A285720 Number of ways to write n as a sum of two unordered squarefree numbers so that their addition in base-2 does not produce carries.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Scheme

Formula

A057639 First differences of zero-sites (A028442) of Mertens's function A002321.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A068576 Numbers k such that Sum_{j=1..k} mu(j)^2 = floor(6*k/Pi^2).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A070544 Number of squarefree numbers s such that n < s < 2n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A230490 Size of largest subset of [1..n] containing no three terms in a geometric progression with integer ratio.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A243284 a(n) = the number of distinct ways of writing such products m = k^2 * j, 0 < j <= k, (j and k natural numbers) that m is in range [1,n]; Partial sums of A102354.

Views

Author

Keywords

Comments

Crossrefs

A246419 Numbers n such that n^6 - 2 is not squarefree.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

A285881 Number of even squarefree numbers <= n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica