A071068 -id:A071068 - cp's OEIS frontend

A262992 Sum of the squarefree numbers among the partition parts of n into two parts.

0, 2, 3, 8, 6, 14, 17, 24, 24, 29, 34, 51, 45, 65, 72, 87, 87, 104, 104, 133, 123, 155, 166, 189, 189, 202, 215, 229, 215, 259, 274, 305, 305, 355, 372, 407, 407, 463, 482, 521, 521, 583, 604, 669, 647, 670, 693, 740, 740, 740, 740, 817, 791, 844, 844, 899

Offset: 1

Wesley Ivan Hurt, Oct 06 2015

			a(3)=3; there is one partition of 3 into two parts: (2,1). The sum of the squarefree parts of this partition is 2+1=3, so a(3)=3.
a(5)=6; there are two partitions of 5 into two parts: (4,1) and (3,2). The sum of the squarefree parts of these partitions is 3+2+1=6, so a(5)=6.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences related to partitions

Cf. A066779, A008683, A071068, A261985, A262351, A262868, A262870, A262871, A262991.

with(numtheory): A262992:=n->add(i*mobius(i)^2 + (n-i)*mobius(n-i)^2, i=1..floor(n/2)): seq(A262992(n), n=1..100);

Table[Sum[i*MoebiusMu[i]^2 + (n - i)*MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 100}]

vector(100, n, sum(k=1, n\2, k*moebius(k)^2 + (n-k)*moebius(n-k)^2)) \\ Altug Alkan, Oct 07 2015

a(n)=my(s, k2, m=n-1); forsquarefree(k=1, sqrtint(m), k2=k[1]^2; s+= k2*binomial(m\k2+1, 2)*moebius(k)); s + (n%4==2 && issquarefree(n/2))*n/2 \\ Charles R Greathouse IV, Jan 13 2018

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

a(n) = A262870(n) + A262871(n).

A294101 Number of partitions of n into two parts such that one is squarefree and the other is nonsquarefree.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 0, 2, 3, 3, 1, 2, 4, 4, 1, 3, 4, 6, 2, 5, 5, 7, 2, 6, 7, 7, 4, 7, 9, 9, 4, 8, 8, 10, 1, 9, 11, 11, 4, 10, 12, 10, 4, 9, 14, 14, 5, 11, 15, 17, 5, 12, 13, 19, 8, 14, 14, 18, 8, 15, 17, 17, 9, 17, 19, 19, 7, 18, 18, 22, 3, 19, 19, 21, 8

Offset: 1

Wesley Ivan Hurt, Oct 22 2017

Index entries for sequences related to partitions

Cf. A008683 (mu), A008966, A071068, A262868, A262869, A262991, A294100, A294232, A294233.

Table[Floor[n/2] - Sum[KroneckerDelta[MoebiusMu[k]^2, MoebiusMu[n - k]^2], {k, Floor[n/2]}], {n, 80}]

a(n) = floor(n/2) - Sum_{i=1..floor(n/2)} [mu(i)^2 = mu(n-i)^2], where [] is the Iverson bracket.
From Wesley Ivan Hurt, Jul 16 2025: (Start)
a(n) = A294232(n) + A294233(n).
a(n) = A262991(n) - 2*A071068(n). (End)

A347648 Number of partitions of n into at most 2 squarefree parts.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 3, 4, 4, 4, 4, 5, 5, 4, 4, 6, 5, 6, 5, 7, 5, 6, 5, 7, 6, 6, 6, 8, 7, 8, 7, 11, 8, 8, 8, 11, 9, 9, 10, 13, 10, 9, 9, 13, 10, 8, 8, 14, 11, 10, 8, 13, 11, 12, 10, 15, 12, 12, 11, 15, 12, 12, 12, 18, 13, 14, 12, 21, 14, 15, 13

Offset: 0

Ilya Gutkovskiy, Sep 09 2021

nonn

Cf. A005117, A071068, A073576, A347649.

A285718 a(1) = 0, and for n > 1, a(n) = the least squarefree number x such that n-x is also squarefree.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 3, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 3, 5, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 3, 6, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1

Offset: 1

Antti Karttunen, May 02 2017

nonn

For any n > 1 there is at least one decomposition of n as a sum of two squarefree numbers (cf. A071068 and the Mathematics Stack Exchange link). Of all pairs (x,y) of positive squarefree numbers for which x <= y and x+y = n, sequences A285718 and A285719 give the unique pair for which the difference y-x is the largest possible.

Question: Are there arbitrarily large terms in this sequence?

			For n=51 we see that 50 (2*5*5), 49 (7*7) and 48 (2^4 * 3) are all nonsquarefree (A013929). 47 (a prime) is squarefree, but 51 - 47 = 4 is not. On the other hand, both 46 (2*23) and 5 are squarefree numbers, thus a(51) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Mathematics Stack Exchange, Sums of square free numbers, is this conjecture equivalent to Goldbach's conjecture? (See especially the answer of Aryabhata)
K. Rogers, The Schnirelmann density of the squarefree integers, Proc. Amer. Math. Soc. 15 (1964), pp. 515-516.

Cf. A005117, A008683, A013929, A071068, A285719, A285720, A285734.

Table[If[n == 1, 0, x = 1; While[Nand[SquareFreeQ@ x, SquareFreeQ[n - x]], x++]; x], {n, 120}] (* Michael De Vlieger, May 03 2017 *)

from sympy.ntheory.factor_ import core
def issquarefree(n): return core(n) == n
def a285718(n):
    if n==1: return 0
    x = 1
    while True:
        if issquarefree(x) and issquarefree(n - x):return x
        else: x+=1
print([a285718(n) for n in range(1, 121)]) # Indranil Ghosh, May 02 2017

(define (A285718 n) (if (= 1 n) 0 (let loop ((k 1)) (if (not (zero? (A008683 (- n (A005117 k))))) (A005117 k) (loop (+ 1 k))))))

a(n) = n - A285719(n).

A285719 a(1) = 1, and for n > 1, a(n) = the largest squarefree number k such that n-k is also squarefree.

Original entry on oeis.org

1, 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, 46, 51, 51, 53, 53, 55, 55, 57, 58, 59, 59, 61, 62, 62, 62, 65, 66, 67, 67, 69, 70, 71, 71, 73, 74, 74, 74, 77, 78, 79, 79, 79, 82, 83, 83, 85, 86, 87, 87, 89, 89, 91, 91

Offset: 1

Antti Karttunen, May 02 2017

nonn

For any n > 1 there is at least one decomposition of n as a sum of two squarefree numbers (cf. A071068 and the Mathematics Stack Exchange link). Of all pairs (x,y) of positive squarefree numbers for which x <= y and x+y = n, sequences A285718 and A285719 give the unique pair for which the difference y-x is the largest possible.

Note: a(n+1) differs from A070321(n) for the first time at n=50, with a(51) = 46, while A070321(50) = 47.

			For n=51 we see that 50 (2*5*5), 49 (7*7) and 48 (2^4 * 3) are all nonsquarefree (A013929). 47 (a prime) is squarefree, but 51 - 47 = 4 is not. On the other hand, both 46 (2*23) and 5 are squarefree numbers, thus a(51) = 46.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Mathematics Stack Exchange, Sums of square free numbers, is this conjecture equivalent to Goldbach's conjecture? (See especially the answer of Aryabhata)
K. Rogers, The Schnirelmann density of the squarefree integers, Proc. Amer. Math. Soc. 15 (1964), pp. 515-516.

Cf. A005117, A008683, A013928, A013929, A070321, A071068, A285718, A285735.

lsfn[n_]:=Module[{k=n-1},While[!SquareFreeQ[k]||!SquareFreeQ[n-k],k--];k]; Join[{1},Array[ lsfn,100,2]] (* Harvey P. Dale, Apr 27 2023 *)

from sympy.ntheory.factor_ import core
def issquarefree(n): return core(n) == n
def a285718(n):
    if n==1: return 0
    x = 1
    while True:
        if issquarefree(x) and issquarefree(n - x):return x
        else: x+=1
def a285719(n): return n - a285718(n)
print([a285719(n) for n in range(1, 121)]) # Indranil Ghosh, May 02 2017

(define (A285719 n) (- n (A285718 n)))
(define (A285719 n) (if (= 1 n) n (let loop ((k (A013928 n))) (if (not (zero? (A008683 (- n (A005117 k))))) (A005117 k) (loop (- k 1))))))

a(n) = n - A285718(n).

A307726 Number of partitions of n into 2 prime powers (not including 1).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 2, 4, 3, 4, 4, 4, 2, 4, 3, 4, 4, 4, 3, 5, 3, 6, 4, 7, 4, 7, 2, 5, 4, 6, 3, 5, 3, 5, 5, 6, 2, 7, 3, 7, 4, 6, 2, 8, 3, 7, 4, 6, 2, 7, 3, 6, 4, 7, 2, 9, 2, 7, 5, 7, 2, 9, 3, 7, 6, 7, 3, 9, 2, 8, 4, 6, 4, 10, 3, 9, 4, 7, 3, 11, 4, 8, 3, 7, 2, 10, 2, 8, 3, 8

Offset: 0

Ilya Gutkovskiy, Apr 24 2019

			a(10) = 3 because we have [8, 2], [7, 3] and [5, 5].

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for sequences related to partitions

Cf. A000961, A023894, A061358, A071068, A071330, A071331, A246655, A280242, A307727.

# note that this requires A246655 to be pre-computed
f:= proc(n, k, pmax) option remember;
  local t, p, j;
  if n = 0 then return `if`(k=0, 1, 0) fi;
  if k = 0 then return 0 fi;
  if n > k*pmax then return 0 fi;
  t:= 0:
  for p in A246655 do
    if p > pmax then return t fi;
    t:= t + add(procname(n-j*p, k-j, min(p-1, n-j*p)), j=1..min(k, floor(n/p)))
  od;
  t
end proc:
map(f, [$0..100]); # Robert Israel, Apr 29 2019

Array[Count[IntegerPartitions[#, {2}], _?(AllTrue[#, PrimePowerQ] &)] &, 101, 0]

a(n) = [x^n y^2] Product_{k>=1} 1/(1 - y*x^A246655(k)).

A280829 Number of partitions of n into two squarefree semiprimes.

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, Jan 08 2017

			a(20) = 2; there are 2 partitions of 20 into two squarefree semiprimes: (14,6) and (10,10).

Indranil Ghosh, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions

Cf. A006881, A071068, A280710.

with(numtheory): A280829:=n->add(floor(bigomega(i)*mobius(i)^2/2)*floor(2*mobius(i)^2/bigomega(i))*floor(bigomega(n-i)*mobius(i)^2/2)*floor(2*mobius(n-i)^2/bigomega(n-i)), i=2..floor(n/2)): seq(A280829(n), n=1..100);

Table[Sum[Floor[PrimeOmega[i] MoebiusMu[i]^2 / 2] Floor[2 MoebiusMu[i]^2 / PrimeOmega[i]] Floor[PrimeOmega[n - i] MoebiusMu[i]^2 / 2] Floor[2 MoebiusMu[n - i]^2 / PrimeOmega[n - i]], {i, 2, Floor[n/2]}], {n, 1, 90}] (* Indranil Ghosh, Mar 10 2017, translated from Maple code *)

for(n=1, 90, print1(sum(i=2, floor(n/2), floor(bigomega(i) * moebius(i)^2 / 2) * floor(2 * moebius(i)^2 / bigomega(i)) * floor(bigomega(n - i) * moebius(i)^2 / 2) * floor(2 * moebius(n - i)^2 / bigomega(n - i))),", ")) \\ Indranil Ghosh, Mar 10 2017

a(n) = Sum_{k=1..floor(n/2)} c(k) * c(n-k), where c = A280710.

A307815 Number of partitions of n into 3 squarefree parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 3, 5, 4, 5, 5, 7, 7, 9, 8, 11, 11, 13, 11, 15, 14, 18, 15, 20, 19, 23, 20, 24, 24, 27, 24, 30, 29, 34, 30, 37, 36, 42, 36, 45, 44, 50, 44, 54, 54, 59, 52, 62, 63, 68, 57, 69, 70, 78, 65, 78, 78, 88, 74, 86, 87, 98, 84, 98, 98, 107, 93, 109, 108, 120, 102, 124, 123

Offset: 0

Ilya Gutkovskiy, Apr 30 2019

nonn

			a(10) = 4 because we have [7, 2, 1], [6, 3, 1], [6, 2, 2] and [5, 3, 2].

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Rémy Sigrist)
Index entries for sequences related to partitions

Cf. A005117, A008683, A068307, A071068, A073576, A098235, A280210, A307727.

b:= proc(n, i) option remember; `if`(n=0, [1, 0$3], `if`(i<1, 0, b(n, i-1)+
      `if`(numtheory[issqrfree](i), [0, b(n-i, min(i, n-i))[1..3][]], 0)))
    end:
a:= n-> b(n$2)[4]:
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 30 2019

Array[Count[IntegerPartitions[#, {3}], _?(AllTrue[#, SquareFreeQ] &)] &, 75, 0]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, {1, 0, 0, 0}, If[i < 1, {0, 0, 0, 0}, b[n, i - 1] + If[SquareFreeQ[i], {0, Sequence @@ b[n - i, Min[i, n - i]][[1 ;; 3]]}, {0, 0, 0, 0}]]];
a[n_] := b[n, n][[4]];
a /@ Range[0, 80] (* Jean-François Alcover, Jun 06 2021, after Alois P. Heinz *)

a(n) = [x^n y^3] Product_{k>=1} 1/(1 - mu(k)^2*y*x^k).

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} mu(i)^2 * mu(k)^2 * mu(n-i-k)^2, where mu is the Mobius function. - Wesley Ivan Hurt, May 09 2019

A261927 Sum of the larger parts of the partitions of n into two squarefree parts.

Original entry on oeis.org

0, 1, 2, 5, 3, 8, 11, 18, 13, 12, 16, 34, 28, 31, 37, 63, 50, 56, 44, 88, 59, 83, 73, 129, 93, 91, 100, 138, 105, 103, 123, 195, 151, 173, 169, 303, 201, 199, 219, 345, 255, 256, 298, 442, 341, 274, 289, 482, 380, 294, 255, 525, 401, 410, 270, 539, 422, 487

Offset: 1

Wesley Ivan Hurt, Oct 02 2015

			a(4) = 5. There are two partitions of 4 into two squarefree parts: (3, 1) and (2, 2). The sum of the larger parts of these partitions is 3 + 2 = 5.
a(5) = 3. There is only one partition of 5 into two squarefree parts: (3, 2). The larger part is 3, thus a(5) = 3.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions

Cf. A008683, A071068, A261985, A262351.

with(numtheory): A261987:=n->add((n-i)*mobius(i)^2*mobius(n-i)^2, i=1..floor(n/2)): seq(A261987(n), n=1..70);

Table[Sum[(n - i) MoebiusMu[i]^2 * MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 70}]
Table[Total[Select[IntegerPartitions[n,{2}],AllTrue[#,SquareFreeQ]&][[All,1]]],{n,60}] (* Harvey P. Dale, Apr 26 2022 *)

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

a(n) = A262351(n) - A261985(n).

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.

Original entry on oeis.org

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).

cp's OEIS Frontend

A262992 Sum of the squarefree numbers among the partition parts of n into two parts.

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A294101 Number of partitions of n into two parts such that one is squarefree and the other is nonsquarefree.

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A347648 Number of partitions of n into at most 2 squarefree parts.

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A285718 a(1) = 0, and for n > 1, a(n) = the least squarefree number x such that n-x is also squarefree.

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A285719 a(1) = 1, and for n > 1, a(n) = the largest squarefree number k such that n-k is also squarefree.

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A307726 Number of partitions of n into 2 prime powers (not including 1).

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A280829 Number of partitions of n into two squarefree semiprimes.

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A307815 Number of partitions of n into 3 squarefree parts.

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