A098235 -id:A098235 - cp's OEIS frontend

A071068 Number of ways to write n as a sum of two unordered squarefree numbers.

0, 1, 1, 2, 1, 2, 2, 3, 2, 2, 2, 4, 3, 3, 3, 5, 4, 4, 3, 6, 4, 5, 4, 7, 5, 5, 5, 7, 5, 5, 5, 8, 6, 7, 6, 11, 7, 7, 7, 11, 8, 8, 9, 13, 10, 8, 8, 13, 10, 8, 7, 14, 10, 10, 7, 13, 10, 11, 9, 15, 11, 11, 11, 15, 11, 11, 11, 18, 12, 13, 11, 21, 13, 14, 13, 20, 14, 13, 14, 20, 16, 13, 13, 22, 15

Offset: 1

Benoit Cloitre, May 26 2002

The natural density of the squarefree numbers is 6/Pi^2, so An < a(n) < Bn for all large enough n with A < 6/Pi^2 - 1/2 and B > 3/Pi^2. The Schnirelmann density of the squarefree numbers is 53/88 > 1/2, and so a(n) > 0 for all n > 1 (in fact, a(n+1) >= 9n/88). It follows from Theoreme 3 bis. in Cohen, Dress, & El Marraki along with finite checking up to 16089908 that 0.10792n < a(n) < 0.303967n for n > 36. (The lower bound holds for n > 1.) - Charles R Greathouse IV, Feb 02 2016

			12=1+11=2+10=5+7=6+6 hence a(12)=4.

T. D. Noe, Table of n, a(n) for n = 1..10000
Henri Cohen, Francois Dress, and Mahomed El Marraki, Explicit estimates for summatory functions linked to the Möbius μ-function, Funct. Approx. Comment. Math. 37:1 (2007), pp. 51-63.

Cf. A005117, A008683 (mu), A008966, A098235, A262991, A294101.

Table[Sum[Abs[MoebiusMu[i] MoebiusMu[n - i]],{i, 1, Floor[n/2]}],{n, 1, 85}] (* Indranil Ghosh, Mar 10 2017 *)
Table[Count[IntegerPartitions[n,{2}],?(AllTrue[#,SquareFreeQ]&)],{n,90}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Aug 13 2020 *)

a(n)=sum(i=1,n\2,issquarefree(i)&&issquarefree(n-i)) \\ Charles R Greathouse IV, May 21 2013

list(lim)=my(n=lim\1); concat(0, ceil(Vec((Polrev(vector(n, k, issquarefree(k-1))) + O('x^(n+1)))^2)/2)) \\ Charles R Greathouse IV, May 21 2013

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

a(n) = Sum_{k=1..floor(n/2)} mu(k)^2 * mu(n-k)^2. - Wesley Ivan Hurt, May 20 2013

a(n) = (A262991(n) - A294101(n))/2. - Wesley Ivan Hurt, Jul 16 2025

A098236 Number of ways to write n as the sum of two positive distinct squarefree numbers.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 3, 2, 1, 2, 3, 3, 2, 3, 5, 4, 4, 3, 5, 4, 4, 4, 7, 5, 4, 5, 6, 5, 4, 5, 8, 6, 6, 6, 11, 7, 6, 7, 11, 8, 7, 9, 12, 10, 7, 8, 13, 10, 8, 7, 13, 10, 10, 7, 13, 10, 10, 9, 14, 11, 10, 11, 15, 11, 10, 11, 17, 12, 12, 11, 21, 13, 13, 13, 19, 14, 12, 14, 20, 16, 12, 13, 21

Offset: 1

Ralf Stephan, Aug 31 2004

Number of distinct rectangles with squarefree length and width such that L + W = n, W < L. - Wesley Ivan Hurt, Oct 29 2017

Cf. A005117, A098235.

with(numtheory): A098236:=n->add(mobius(i)^2*mobius(n-i)^2, i=1..floor(n/2)-((n+1) mod 2)): seq(A098236(n), n=1..150); # Wesley Ivan Hurt, Oct 29 2017

Table[Sum[MoebiusMu[i]^2*MoebiusMu[n - i]^2, {i, Floor[(n-1)/2]}], {n, 80}] (* Wesley Ivan Hurt, Oct 26 2017 *)

a(n) = sum(i=1, n\2-(n+1)%2, moebius(i)^2*moebius(n-i)^2); \\ Michel Marcus, Oct 27 2017

a(n) = Sum_{k=1..floor((n-1)/2)} mu(k)^2 * mu(n-k)^2. - Wesley Ivan Hurt, Oct 26 2017

A280210 Expansion of (Sum_{k>=1} mu(k)^2*x^k)^3, where mu(k) is the Moebius function (A008683).

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 7, 9, 12, 19, 21, 21, 21, 30, 36, 37, 36, 48, 58, 63, 57, 70, 78, 87, 78, 96, 105, 114, 105, 123, 133, 138, 126, 148, 162, 174, 156, 195, 207, 220, 192, 234, 250, 261, 237, 280, 312, 318, 282, 330, 363, 370, 315, 375, 405, 432, 366, 421, 453, 483, 417, 468, 507, 532, 474, 537, 568, 591, 519, 601, 630, 666, 570

Offset: 0

Ilya Gutkovskiy, Dec 29 2016

nonn

Number of ordered ways of writing n as sum of three squarefree numbers (A005117).

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

Eric Weisstein's World of Mathematics, Squarefree

Cf. A005117, A008683, A098235.

Cf. A098238, A121550.

nmax = 72; CoefficientList[Series[(Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}])^3, {x, 0, nmax}], x]

G.f.: (Sum_{k>=1} mu(k)^2*x^k)^3.

A341064 Number of ways to write n as an ordered sum of 4 squarefree numbers.

Original entry on oeis.org

1, 4, 10, 16, 23, 32, 50, 68, 83, 92, 116, 148, 178, 192, 224, 276, 335, 360, 400, 460, 547, 580, 634, 704, 821, 868, 938, 1024, 1162, 1212, 1288, 1392, 1572, 1628, 1742, 1876, 2123, 2172, 2308, 2460, 2761, 2820, 2964, 3176, 3550, 3628, 3778, 4028, 4481, 4528, 4686, 4932, 5513, 5564

Offset: 4

Ilya Gutkovskiy, Feb 04 2021

nonn

Cf. A005117, A008683, A008966, A098235, A280210, A308767, A341065, A341066, A341067, A341068, A341069, A341070.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(numtheory[issqrfree](j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 4):
seq(a(n), n=4..57);  # Alois P. Heinz, Feb 04 2021

nmax = 57; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}]^4, {x, 0, nmax}], x] // Drop[#, 4] &

G.f.: (Sum_{k>=1} mu(k)^2 * x^k)^4.

A341065 Number of ways to write n as an ordered sum of 5 squarefree numbers.

Original entry on oeis.org

1, 5, 15, 30, 50, 76, 120, 180, 250, 315, 401, 520, 670, 805, 955, 1160, 1445, 1715, 1980, 2290, 2741, 3180, 3605, 4040, 4690, 5341, 5985, 6600, 7490, 8380, 9251, 10060, 11240, 12415, 13595, 14670, 16295, 17850, 19425, 20780, 22905, 24905, 26895, 28600, 31335, 33966, 36485, 38620

Offset: 5

Ilya Gutkovskiy, Feb 04 2021

nonn

Cf. A005117, A008683, A008966, A098235, A280210, A308840, A341064, A341066, A341067, A341068, A341069, A341070.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(numtheory[issqrfree](j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..52);  # Alois P. Heinz, Feb 04 2021

nmax = 52; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}]^5, {x, 0, nmax}], x] // Drop[#, 5] &

G.f.: (Sum_{k>=1} mu(k)^2 * x^k)^5.

A341066 Number of ways to write n as an ordered sum of 6 squarefree numbers.

Original entry on oeis.org

1, 6, 21, 50, 96, 162, 267, 426, 645, 902, 1218, 1632, 2187, 2826, 3543, 4402, 5547, 6906, 8397, 10032, 12108, 14578, 17298, 20112, 23517, 27534, 32034, 36592, 41892, 48018, 54886, 61758, 69549, 78408, 88365, 98274, 109478, 122058, 136230, 150114, 165759, 183114, 202630, 221484

Offset: 6

Ilya Gutkovskiy, Feb 04 2021

nonn

			G.f.: (Sum_{k>=1} mu(k)^2 * x^k)^6.

Cf. A005117, A008683, A008966, A098235, A280210, A308902, A341064, A341065, A341067, A341068, A341069, A341070.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(numtheory[issqrfree](j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 6):
seq(a(n), n=6..49);  # Alois P. Heinz, Feb 04 2021

nmax = 49; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}]^6, {x, 0, nmax}], x] // Drop[#, 6] &

A341067 Number of ways to write n as an ordered sum of 7 squarefree numbers.

Original entry on oeis.org

1, 7, 28, 77, 168, 315, 553, 932, 1505, 2282, 3297, 4634, 6447, 8771, 11607, 15029, 19390, 24885, 31500, 39137, 48335, 59584, 73003, 88109, 105525, 126112, 150472, 177632, 208160, 243194, 284102, 329357, 379379, 435477, 500108, 571124, 648998, 735112, 833483, 940765, 1057679

Offset: 7

Ilya Gutkovskiy, Feb 04 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..10000

Cf. A005117, A008683, A008966, A098235, A280210, A308952, A341064, A341065, A341066, A341068, A341069, A341070.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(numtheory[issqrfree](j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 7):
seq(a(n), n=7..47);  # Alois P. Heinz, Feb 04 2021

nmax = 47; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}]^7, {x, 0, nmax}], x] // Drop[#, 7] &

G.f.: (Sum_{k>=1} mu(k)^2 * x^k)^7.

A341068 Number of ways to write n as an ordered sum of 8 squarefree numbers.

Original entry on oeis.org

1, 8, 36, 112, 274, 568, 1072, 1912, 3263, 5280, 8128, 12048, 17474, 24824, 34428, 46600, 62163, 82160, 107452, 138392, 176116, 222560, 279756, 348168, 428954, 524848, 639976, 775448, 932376, 1113808, 1326748, 1573656, 1855767, 2175728, 2544048, 2965280, 3441568, 3974744, 4580060

Offset: 8

Ilya Gutkovskiy, Feb 04 2021

nonn

Cf. A005117, A008683, A008966, A098235, A280210, A326443, A341064, A341065, A341066, A341067, A341069, A341070.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(numtheory[issqrfree](j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 8):
seq(a(n), n=8..46);  # Alois P. Heinz, Feb 04 2021

nmax = 46; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}]^8, {x, 0, nmax}], x] // Drop[#, 8] &

G.f.: (Sum_{k>=1} mu(k)^2 * x^k)^8.

A341069 Number of ways to write n as an ordered sum of 9 squarefree numbers.

Original entry on oeis.org

1, 9, 45, 156, 423, 963, 1959, 3708, 6669, 11410, 18594, 29052, 44046, 65196, 94284, 133104, 184248, 251406, 338995, 450936, 591885, 768657, 990567, 1265832, 1602273, 2010294, 2506572, 3107136, 3825675, 4676643, 5686347, 6882912, 8290431, 9928305, 11834289, 14052816, 16624846

Offset: 9

Ilya Gutkovskiy, Feb 04 2021

nonn

Cf. A005117, A008683, A008966, A098235, A280210, A326522, A341064, A341065, A341066, A341067, A341068, A341070.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(numtheory[issqrfree](j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 9):
seq(a(n), n=9..45);  # Alois P. Heinz, Feb 04 2021

nmax = 45; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}]^9, {x, 0, nmax}], x] // Drop[#, 9] &

G.f.: (Sum_{k>=1} mu(k)^2 * x^k)^9.

A341070 Number of ways to write n as an ordered sum of 10 squarefree numbers.

Original entry on oeis.org

1, 10, 55, 210, 625, 1552, 3400, 6840, 12960, 23330, 40028, 65740, 104230, 160670, 241640, 354772, 509620, 718980, 999645, 1370720, 1853903, 2476250, 3274110, 4289810, 5568820, 7162184, 9138045, 11579180, 14574755, 18215900, 22619016, 27929990, 34311845, 41921710, 50946945

Offset: 10

Ilya Gutkovskiy, Feb 04 2021

nonn

Cf. A005117, A008683, A008966, A098235, A280210, A326626, A341064, A341065, A341066, A341067, A341068, A341069.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(numtheory[issqrfree](j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 10):
seq(a(n), n=10..44);  # Alois P. Heinz, Feb 04 2021

nmax = 44; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}]^10, {x, 0, nmax}], x] // Drop[#, 10] &

G.f.: (Sum_{k>=1} mu(k)^2 * x^k)^10.

cp's OEIS Frontend

A071068 Number of ways to write n as a sum of two unordered squarefree numbers.

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A098236 Number of ways to write n as the sum of two positive distinct squarefree numbers.

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A280210 Expansion of (Sum_{k>=1} mu(k)^2*x^k)^3, where mu(k) is the Moebius function (A008683).

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A341064 Number of ways to write n as an ordered sum of 4 squarefree numbers.

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A341065 Number of ways to write n as an ordered sum of 5 squarefree numbers.

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A341066 Number of ways to write n as an ordered sum of 6 squarefree numbers.

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A341067 Number of ways to write n as an ordered sum of 7 squarefree numbers.

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A341068 Number of ways to write n as an ordered sum of 8 squarefree numbers.

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