cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

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

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Crossrefs

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

Programs

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

Formula

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

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Crossrefs

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

Programs

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

Formula

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

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Examples

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

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Crossrefs

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

Programs

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

Formula

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

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Crossrefs

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

Programs

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

Formula

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

Views

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

Crossrefs

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

Programs

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

Formula

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

A347778 Number of compositions (ordered partitions) of n into at most 3 squarefree parts.

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 11, 14, 18, 23, 25, 26, 28, 37, 42, 44, 46, 57, 66, 70, 68, 79, 88, 96, 92, 106, 115, 124, 118, 134, 143, 149, 142, 161, 176, 187, 178, 210, 221, 235, 214, 251, 266, 280, 262, 300, 328, 335, 308, 350, 379, 385, 342, 396, 425, 447, 392, 442, 475
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 13 2021

Keywords

Crossrefs

Cf. A005117, A280194, A280210, A347777, A347779, A347780.

Programs

  • Mathematica
    Table[Length@Flatten[Permutations/@IntegerPartitions[n,3,Select[Range@n,SquareFreeQ]],1],{n,0,58}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)

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

Views

Author

Ilya Gutkovskiy, Apr 30 2019

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)

Formula

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

A282585 Number of ways to write n as an ordered sum of 3 squarefree palindromes (A071251).

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 7, 9, 12, 19, 21, 21, 18, 24, 27, 28, 18, 18, 19, 24, 15, 10, 6, 12, 12, 12, 9, 9, 12, 15, 18, 12, 9, 7, 15, 15, 15, 9, 12, 15, 18, 18, 12, 9, 9, 18, 15, 12, 0, 9, 9, 9, 0, 0, 0, 6, 6, 9, 12, 9, 12, 15, 18, 18, 12, 9, 13, 18, 18, 18, 9, 15, 18, 21, 18, 12, 9, 15, 21, 21, 21, 9, 18, 21, 24, 18
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 19 2017

Keywords

Comments

Every number can be written as the sum of 3 palindromes (see A261132 and A261422).
Conjecture: a(n) > 0 for any sufficiently large n.
Additional conjecture: every number > 3 can be written as the sum of 4 squarefree palindromes.

Examples

			a(22) = 6 because we have [11, 6, 5], [11, 5, 6] [6, 11, 5], [6, 5, 11], [5, 11, 6] and [5, 6, 11].

Links

Crossrefs

Cf. A002113, A005117, A035137, A071251, A091580, A091581, A260254, A261131, A261132, A261422, A280210, A282584.

Programs

  • Mathematica
    nmax = 85; CoefficientList[Series[Sum[Boole[SquareFreeQ[k] && PalindromeQ[k]] x^k, {k, 1, nmax}]^3, {x, 0, nmax}], x]

Formula

G.f.: (Sum_{k>=1} x^A071251(k))^3.
Showing 1-10 of 10 results.

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