A071068 -id:A071068 - cp's OEIS frontend

A282380 Number of ways to write n as a sum of two unordered nonsquarefree numbers.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 2, 1, 1, 0, 3, 1, 1, 1, 3, 2, 1, 1, 4, 2, 2, 1, 6, 2, 1, 1, 5, 2, 1, 2, 5, 3, 1, 1, 6, 3, 2, 1, 7, 4, 4, 1, 7, 4, 4, 2, 7, 4, 3, 3, 8, 4, 3, 3, 9, 4, 4, 2, 12, 4, 4, 3, 10, 5, 3, 4, 10, 6, 3, 3, 11, 5, 3, 3, 12, 5, 6, 3, 11, 6, 5, 4, 12, 5, 5, 7, 14, 5, 6, 5, 14, 5, 6

Offset: 1

Altug Alkan, Feb 13 2017

nonn

			a(16) = 2 because 16 = 4 + 12 and 16 = 8 + 8 are only corresponding solutions.

Altug Alkan, Table of n, a(n) for n = 1..10000

Cf. A013929, A071068.

a(n) = sum(k=1, n\2, !issquarefree(k) && !issquarefree(n-k));

A290136 Positive numbers that are not the sum of two nonprime squarefree numbers (A000469).

Original entry on oeis.org

1, 3, 4, 5, 6, 8, 9, 10, 13, 14, 17, 18, 19, 26, 33, 38, 46, 62, 82

Offset: 1

Ilya Gutkovskiy, Jul 20 2017

The sequence is conjectured to be complete.

Cf. A000469, A014092, A071068, A071331, A098235, A239508, A239509.

nmax = 82; f[x_] := Sum[Boole[SquareFreeQ[k] && PrimeNu[k] != 1] x^k, {k, 1, nmax}]^2; b = Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, nmax}]]; c = Complement[Range[nmax], b][[1 ;; 19]]

A345128 Number of squarefree products s*t from all positive integer pairs (s,t), such that s + t = n, s <= t.

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, Jun 08 2021

nonn

From Lei Zhou, Dec 19 2024: (Start)
a(n) is also the total number of appearance of n in A379049, by definition.
Conjecture: a(n) > 0 for all n > 1. (End)
A simple case of Zhou's conjecture: a(p) > 0 where p is prime. With some work this can be extended to a(n) > 0 for n with sum_{p | n} 1/p < 1/10 or so (the limit of the method is 6/Pi^2 - 1/2, so it can't prove the full conjecture). See my comment in A071068. - Charles R Greathouse IV, Dec 20 2024

			a(13) = 3; The partitions of 13 into two positive integer parts (s,t) where s <= t are (1,12), (2,11), (3,10), (4,9), (5,8), (6,7). The corresponding products are 1*12, 2*11, 3*10, 4*9, 5*8, and 6*7; 3 of which are squarefree.

Cf. A008683 (mu), A379049, A071068.

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

a(n)=my(s); forsquarefree(k=1,n\2, gcd(n,k[1])==1 && issquarefree(n-k[1]) && s++); s \\ Charles R Greathouse IV, Dec 20 2024

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

a(n) <= A071068(n) and hence a(n) < 0.303967n for n > 3. - Charles R Greathouse IV, Dec 20 2024

A100699 Number of ways to partition n into two squarefree numbers that are not prime.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Dec 09 2004

nonn

a(n) <= A071068(n).

			a(36) = #{1+35, 6+30, 10+26, 14+22, 15+21} = 5.

Cf. A000469, A061358, A005117.

cp's OEIS Frontend

A282380 Number of ways to write n as a sum of two unordered nonsquarefree numbers.

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A290136 Positive numbers that are not the sum of two nonprime squarefree numbers (A000469).

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A345128 Number of squarefree products s*t from all positive integer pairs (s,t), such that s + t = n, s <= t.

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A100699 Number of ways to partition n into two squarefree numbers that are not prime.

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