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.

A261985 Sum of the smaller parts of the partitions of n into two squarefree parts.

Original entry on oeis.org

0, 1, 1, 3, 2, 4, 3, 6, 5, 8, 6, 14, 11, 11, 8, 17, 18, 16, 13, 32, 25, 27, 19, 39, 32, 39, 35, 58, 40, 47, 32, 61, 47, 65, 41, 93, 58, 67, 54, 95, 73, 80, 89, 130, 109, 94, 87, 142, 110, 106, 102, 203, 129, 130, 115, 189, 148, 151, 137, 232, 170, 165, 169
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 02 2015

Keywords

Comments

Sum of the widths of the distinct rectangles with squarefree length and width such that L + W = n, W <= L. For example a(16) = 17; the rectangles are 1 X 15, 2 X 14, 3 X 13, 5 X 11, 6 X 10 and the sum of the widths of these rectangles gives 1 + 2 + 3 + 5 + 6 = 17. - Wesley Ivan Hurt, Nov 02 2017

Examples

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

Links

Crossrefs

Cf. A008683, A071068, A261927, A262351, A303221.

Programs

  • Maple
    with(numtheory): A261985:=n->add(i*mobius(i)^2*mobius(n-i)^2, i=1..floor(n/2)): seq(A261985(n), n=1..100);
  • Mathematica
    Table[Sum[i*MoebiusMu[i]^2*MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 100}]
  • PARI
    a(n) = sum(i=1, n\2, i*moebius(i)^2*moebius(n-i)^2); \\ Altug Alkan, Jan 01 2018

Formula

a(n) = Sum_{i=1..floor(n/2)} (i * mu(i)^2 * mu(n-i)^2), where mu is the Moebius function (A008683).
a(n) = A262351(n) - A261927(n).
a(n) = A303221(n)/n. - Wesley Ivan Hurt, Jan 05 2024

A262991 Number of squarefree numbers among the parts of the partitions of n into two parts.

Original entry on oeis.org

0, 2, 2, 4, 3, 5, 5, 6, 6, 7, 7, 9, 8, 10, 10, 11, 11, 12, 12, 14, 13, 15, 15, 16, 16, 17, 17, 18, 17, 19, 19, 20, 20, 22, 22, 23, 23, 25, 25, 26, 26, 28, 28, 30, 29, 30, 30, 31, 31, 31, 31, 33, 32, 33, 33, 34, 34, 36, 36, 38, 37, 39, 39, 39, 39, 41, 41, 43
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 06 2015

Keywords

Examples

			a(5)=3; there are 2 partitions of 5 into two parts: (4,1) and (3,2). Three of the parts in the partitions are squarefree, so a(5)=3.
a(6)=5; there are 3 partitions of 6 into two parts: (5,1), (4,2) and (3,3). Five of the parts in the partitions are squarefree, so a(6)=5.

Links

Crossrefs

Cf. A008683, A071068, A261985, A262351, A262869, A262870, A262871, A262992, A294101.

Programs

  • Maple
    with(numtheory): A262991:=n->add(mobius(i)^2+mobius(n-i)^2, i=1..floor(n/2)): seq(A262991(n), n=1..100);
  • Mathematica
    Table[Sum[MoebiusMu[i]^2 + MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 100}]
Table[Count[Flatten[IntegerPartitions[n,{2}]],?SquareFreeQ],{n,70}] (* _Harvey P. Dale, Aug 18 2021 *)
  • PARI
    vector(100, n, sum(k=1, n\2, moebius(k)^2 + moebius(n-k)^2)) \\ Altug Alkan, Oct 07 2015

Formula

a(n) = Sum_{i=1..floor(n/2)} mu(i)^2 + mu(n-i)^2, where mu is the Möebius function (A008683).
a(n) = A262868(n) + A262869(n).
a(n) = A294101(n) + 2*A071068(n). - Wesley Ivan Hurt, Jul 16 2025

A262868 Number of squarefree numbers appearing among the larger parts of the partitions of n into two parts.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 3, 3, 3, 3, 4, 3, 4, 4, 5, 5, 6, 6, 7, 6, 7, 7, 8, 8, 8, 8, 8, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 14, 14, 14, 15, 15, 15, 15, 16, 15, 16, 16, 17, 17, 18, 18, 19, 18, 19, 19, 19, 19, 20, 20, 21, 20, 21, 21, 22, 22
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 03 2015

Keywords

Comments

Number of distinct rectangles with squarefree length and integer width such that L + W = n, W <= L. For example, a(14) = 4; the rectangles are 1 X 13, 3 X 11, 4 X 10 and 7 X 7. - Wesley Ivan Hurt, Nov 02 2017
a(10) = 3, a(100) = 30, a(10^3) = 302, a(10^4) = 3041, a(10^5) = 30393, a(10^6) = 303968, a(10^7) = 3039658, a(10^8) = 30396350, a(10^9) = 303963598, a(10^10) = 3039635373, a(10^11) = 30396355273, a(10^12) = 303963551068, a(10^13) = 3039635509338, a(10^14) = 30396355094469, a(10^15) = 303963550926043, a(10^16) = 3039635509271763, a(10^17) = 30396355092700721, and a(10^18) = 303963550927014110. The limit of a(n)/n is 3/Pi^2. - Charles R Greathouse IV, Nov 04 2017

Examples

			a(4)=2; there are two partitions of 4 into two parts: (3,1) and (2,2). Both of the larger parts are squarefree, thus a(4)=2.
a(5)=1; there are two partitions of 5 into two parts: (4,1) and (3,2). Among the larger parts, only 3 is squarefree, thus a(5)=1.

Links

Crossrefs

Cf. A008683, A071068, A261985, A262351, A262869, A262870, A262871, A262991, A262992, A013928.

Programs

  • Maple
    with(numtheory): A262868:=n->add(mobius(n-i)^2, i=1..floor(n/2)): seq(A262868(n), n=1..100);
  • Mathematica
    Table[Sum[MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 100}]
Table[Count[IntegerPartitions[n,{2}][[All,1]],?SquareFreeQ],{n,80}] (* _Harvey P. Dale, Jan 03 2022 *)
  • PARI
    a(n) = sum(i=1, n\2, moebius(n-i)^2); \\ Michel Marcus, Oct 04 2015
  • PARI
    f(n)=my(s); forfactored(k=1,sqrtint(n),s+=n\k[1]^2*moebius(k)); s
a(n)=n--; f(n) - f(n\2) \\ Charles R Greathouse IV, Nov 04 2017

Formula

a(n) = Sum_{i=1..floor(n/2)} mu(n-i)^2, where mu is the Möbius function A008683.
a(n) = A262991(n) - A262869(n).
a(n) ~ 3*n/Pi^2. - Charles R Greathouse IV, Nov 04 2017

A262869 Number of squarefree numbers appearing among the smaller parts of the partitions of n into two parts.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 18, 18, 19, 19, 20, 20, 20, 20, 21, 21, 22, 22, 23, 23
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 03 2015

Keywords

Comments

Number of distinct rectangles with integer length and squarefree width such that L + W = n, W <= L. For example, a(14) = 6; the rectangles are 13 X 1, 12 X 2, 11 X 3, 9 X 5, 8 X 6, 7 X 7. - Wesley Ivan Hurt, Nov 04 2017

Examples

			a(5)=2; there are two partitions of 5 into two parts: (4,1) and (3,2). Both of the smaller parts are squarefree, thus a(5)=2.
a(6)=3; there are three partitions of 6 into two parts: (5,1), (4,2) and (3,3). Among the three smaller parts, all are squarefree, thus a(6)=3.

Links

Crossrefs

Cf. A008683, A013928, A071068, A261985, A262351, A262868, A262870, A262871, A262991, A262992.

Programs

  • Maple
    with(numtheory): A262869:=n->add(mobius(i)^2, i=1..floor(n/2)): seq(A262869(n), n=1..100);
  • Mathematica
    Table[Sum[MoebiusMu[i]^2, {i, Floor[n/2]}], {n, 100}]
Table[Count[IntegerPartitions[n,{2}][[All,2]],?SquareFreeQ],{n,80}] (* _Harvey P. Dale, Oct 17 2021 *)
  • PARI
    a(n) = sum(i=1, n\2, moebius(i)^2); \\ Michel Marcus, Oct 04 2015
  • PARI
    a(n)=my(s); n\=2; forsquarefree(k=1, sqrtint(n), s += n\k[1]^2*moebius(k)); s \\ Charles R Greathouse IV, Jan 08 2018

Formula

a(n) = Sum_{i=1..floor(n/2)} mu(i)^2, where mu is the Möebius function (A008683).
a(n) = A262991(n) - A262868(n).
a(n) = A013928(floor(n/2)+1). - Georg Fischer, Nov 29 2022

A262870 Sum of the squarefree numbers appearing among the larger parts of the partitions of n into two parts.

Original entry on oeis.org

0, 1, 2, 5, 3, 8, 11, 18, 18, 18, 23, 34, 28, 41, 48, 63, 63, 80, 80, 99, 89, 110, 121, 144, 144, 144, 157, 157, 143, 172, 187, 218, 218, 251, 268, 303, 303, 340, 359, 398, 398, 439, 460, 503, 481, 481, 504, 551, 551, 551, 551, 602, 576, 629, 629, 684, 684
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 03 2015

Keywords

Examples

			a(4)=5; there are two partitions of 4 into two parts: (3,1) and (2,2). The sum of the larger squarefree parts is 3+2=5, thus a(4)=5.
a(5)=3; there are two partitions of 5 into two parts: (4,1) and (3,2). Of the larger parts, 3 is the only squarefree part, so a(5)=3.

Links

Crossrefs

Cf. A008683, A071068, A261985, A262351, A262868, A262869, A262871, A262991, A262992.

Programs

  • Maple
    with(numtheory): A262870:=n->add((n-i)*mobius(n-i)^2, i=1..floor(n/2)): seq(A262870(n), n=1..100);
  • Mathematica
    Table[Sum[(n - i) MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 70}]
  • PARI
    a(n) = sum(i=1, n\2, (n-i) * moebius(n-i)^2); \\ Michel Marcus, Oct 04 2015
  • PARI
    a(n)=my(s); forsquarefree(k=(n+1)\2,n-1, s += k[1]); s \\ Charles R Greathouse IV, Jan 08 2018

Formula

a(n) = Sum_{i=1..floor(n/2)} (n-i) * mu(n-i)^2, where mu is the Möebius function (A008683).
a(n) = A262992(n) - A262871(n).

A262871 Sum of the squarefree numbers appearing among the smaller parts of the partitions of n into two parts.

Original entry on oeis.org

0, 1, 1, 3, 3, 6, 6, 6, 6, 11, 11, 17, 17, 24, 24, 24, 24, 24, 24, 34, 34, 45, 45, 45, 45, 58, 58, 72, 72, 87, 87, 87, 87, 104, 104, 104, 104, 123, 123, 123, 123, 144, 144, 166, 166, 189, 189, 189, 189, 189, 189, 215, 215, 215, 215, 215, 215, 244, 244, 274
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 03 2015

Keywords

Examples

			a(5)=3; there are two partitions of 5 into two parts: (4,1) and (3,2). The sum of the smaller squarefree parts is 1+2=3. Thus a(5)=3.
a(6)=6; there are three partitions of 6 into two parts: (5,1), (4,2) and (3,3). All of the smaller parts are squarefree, so a(6) = 1+2+3 = 6.

Links

Crossrefs

Cf. A008683, A071068, A261985, A262351, A262868, A262869, A262870, A262991, A262992.

Programs

  • Maple
    with(numtheory): A262871:=n->add(i*mobius(i)^2, i=1..floor(n/2)): seq(A262871(n), n=1..100);
  • Mathematica
    Table[Sum[i*MoebiusMu[i]^2, {i, Floor[n/2]}], {n, 70}]
  • PARI
    a(n) = sum(i=1, n\2, i * moebius(i)^2); \\ Michel Marcus, Oct 04 2015
  • PARI
    a(n)=my(s); forsquarefree(k=1,n\2, s += k[1]); s \\ Charles R Greathouse IV, Jan 08 2018

Formula

a(n) = Sum_{i=1..floor(n/2)} i * mu(i)^2, where mu is the Möebius function (A008683).
a(n) = A262992(n) - A262870(n).

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

Original entry on oeis.org

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

Views

Author

Wesley Ivan Hurt, Oct 06 2015

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • Maple
    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);
  • Mathematica
    Table[Sum[i*MoebiusMu[i]^2 + (n - i)*MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 100}]
  • PARI
    vector(100, n, sum(k=1, n\2, k*moebius(k)^2 + (n-k)*moebius(n-k)^2)) \\ Altug Alkan, Oct 07 2015
  • PARI
    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

Formula

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).

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

Views

Author

Wesley Ivan Hurt, Oct 02 2015

Keywords

Examples

			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.

Links

Crossrefs

Cf. A008683, A071068, A261985, A262351.

Programs

  • Maple
    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);
  • Mathematica
    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 *)

Formula

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).

A294247 Sum of the parts in the partitions of n into exactly two distinct squarefree parts.

Original entry on oeis.org

0, 0, 3, 4, 5, 6, 14, 24, 18, 10, 22, 36, 39, 28, 45, 80, 68, 72, 57, 100, 84, 88, 92, 168, 125, 104, 135, 168, 145, 120, 155, 256, 198, 204, 210, 396, 259, 228, 273, 440, 328, 294, 387, 528, 450, 322, 376, 624, 490, 400, 357, 676, 530, 540, 385, 728, 570
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 25 2017; recomputed Oct 26 2017 with thanks to Andrey Zabolotskiy

Keywords

Comments

Sum of the semiperimeters of the distinct rectangles with squarefree length and width such that L + W = n, W < L.

Examples

			For n = 4,5,6,7 the partitions are respectively 1+3 (sum a(4) = 4), 2+3 (sum 5), 1+5 (sum 6), 1+6 and 2+5 (sum 7+7=14). - _N. J. A. Sloane_, Oct 28 2017

Links

Crossrefs

Cf. A008683, A262351.

Programs

  • Mathematica
    Table[n*Sum[MoebiusMu[i]^2*MoebiusMu[n - i]^2, {i, Floor[(n-1)/2]}], {n, 80}]
  • Python
    from sympy import mobius
def a(n): return n*sum(mobius(i)**2*mobius(n - i)**2 for i in range(1, ((n - 1)//2) + 1))
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Nov 07 2017
  • R
    require(numbers)
a <- function(n) {
  if (n<3) return(0)
  S <- numeric()
  for (i in 1:floor((n-1)/2)) S <- c(S, moebius(i)^2*moebius(n-i)^2)
  return(n*sum(S))
}
sapply(1:100, a) # Indranil Ghosh, Nov 07 2017

Formula

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

A303222 Total volume of all rectangular prisms with dimensions p, q and (p + q)/2 such that p and q are squarefree, n = p + q and p <= q.

Original entry on oeis.org

0, 1, 3, 14, 15, 42, 56, 136, 144, 230, 220, 612, 611, 665, 675, 1576, 1768, 1836, 1729, 4200, 3528, 4279, 3404, 7524, 6625, 8333, 8289, 14336, 11165, 12675, 10323, 20592, 17688, 23307, 17570, 40410, 27861, 30153, 28899, 52180, 42804, 45864, 55169, 84920
Offset: 1

Views

Author

Wesley Ivan Hurt, Apr 19 2018

Keywords

Links

Crossrefs

Cf. A008683, A262351.

Programs

  • Magma
    [0] cat [&+[k*(n-k)*(MoebiusMu(k)^2*MoebiusMu(n-k)^2)*n/2: k in [1..Floor(n/2)]]: n in [2..60]]; // Vincenzo Librandi, Apr 21 2018
  • Maple
    N:= 100: # for a(1)..a(N)
A:= Vector(N):
SF:= select(numtheory:-issqrfree, [$1..N-1]):
for iq from 1 to nops(SF) do
  q:= SF[iq];
  for ip from 1 to iq do
    p:= SF[ip];
    n:= p+q;
    if n > N then break fi;
    A[n]:= A[n] + p*q*(p+q)/2
  od
od:
convert(A,list); # Robert Israel, Jun 12 2018
  • Mathematica
    Table[(n/2)*Sum[i (n - i)*MoebiusMu[i]^2 MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 80}]
  • PARI
    a(n) = n*sum(i=1, n\2, i*(n-i)*moebius(i)^2*moebius(n-i)^2)/2; \\ Michel Marcus, Apr 21 2018

Formula

a(n) = (n/2) * Sum_{i=1..floor(n/2)} i * (n-i) * mu(i)^2 * mu(n-i)^2, where mu is the Möbius function (A008683).
Showing 1-10 of 10 results.

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