A226357 -id:A226357 - cp's OEIS frontend

A226359 Number of ordered triples (i,j,k) with |i|, |j|, |k|, |ijk| <= n.

1, 27, 93, 183, 321, 459, 669, 855, 1121, 1379, 1685, 1967, 2393, 2723, 3125, 3551, 4049, 4475, 5045, 5519, 6137, 6707, 7301, 7871, 8681, 9323, 10013, 10735, 11545, 12259, 13189, 13951, 14881, 15739, 16621, 17527, 18673, 19579, 20557, 21559, 22753, 23755

Offset: 0

Robert Price, Jun 04 2013

nonn

Robert Price and Charles R Greathouse IV, Table of n, a(n) for n = 0..10000 (terms through 100 from Robert Price)

Cf. A100450, A226357.

f[n_]:=Length[Complement[Union[Flatten[Table[If[Abs[i*j*k] <=  n, {i, j, k}], {i, -n, n}, {j, -n, n}, {k, -n, n}], 2]], {Null}]]; Table[f[n], {n, 0, 100}]

a(n)=12*n^2+6*n+1+8*sum(i=1, n, n\i*numdiv(i)) \\ Charles R Greathouse IV, Jun 04 2013

A226602 Number of ordered triples (i,j,k) with ijk = n, i,j,k >= 0 and gcd(i,j,k) <= 1.

Original entry on oeis.org

1, 1, 3, 3, 6, 3, 9, 3, 9, 6, 9, 3, 18, 3, 9, 9, 12, 3, 18, 3, 18, 9, 9, 3, 27, 6, 9, 9, 18, 3, 27, 3, 15, 9, 9, 9, 36, 3, 9, 9, 27, 3, 27, 3, 18, 18, 9, 3, 36, 6, 18, 9, 18, 3, 27, 9, 27, 9, 9, 3, 54, 3, 9, 18, 18, 9, 27, 3, 18, 9, 27, 3, 54, 3, 9, 18, 18

Offset: 0

Robert Price, Jun 13 2013

Note that gcd(0,m) = m for any m.

a(n) is the number of cubefree divisors summed over the divisors of n. In other words, a(n) = Sum_{d|n} A073184(d). - Geoffrey Critzer, Mar 20 2015

Alois P. Heinz, Table of n, a(n) for n = 0..20000 (first 101 terms from Robert Price)

Cf. A000005, A007427, A008836, A034444, A056624, A073184, A092520, A100450, A212793, A226357, A226359.

with(numtheory):
b:= proc(n, t, g) option remember; `if`(t=0,
      `if`(igcd(n, g)=1, 1, 0), add(b(n/d, t-1,
      igcd(g, d)), d=divisors(n)))
    end:
a:= n-> `if`(n=0, 1, b(n, 2, 0)):
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 20 2015

f[n_] := Length[Complement[Union[Flatten[Table[If[i*j*k == n && GCD[i, j, k] <= 1, {i, j, k}], {i, 0,n}, {j, 0, n}, {k, 0, n}], 2]], {Null}]]; Table[f[n], {n, 0, 100}]
a[0] = a[1] = 1; a[n_] := Times @@ (3 * Last[#] & /@ FactorInteger[n]); Array[a, 100, 0] (* Amiram Eldar, Sep 14 2020 *)

from math import prod
from sympy import factorint
def A226602(n): return prod(3*e for e in factorint(n).values()) if n else 1 # Chai Wah Wu, Dec 26 2022

From Geoffrey Critzer, Mar 20 2015: (Start)
If n = p_1^e_1*p_2^e_2*...*p_r^e_r then a(n) = Product_{i=1..r} 3*e_i.
Dirichlet g.f.: zeta(s)^3/zeta(3*s). (End)
From Werner Schulte, May 13 2018: (Start)
Multiplicative with a(p^e) = 3*e, p prime and e>0.
Dirichlet inverse b(n), n>0, is multiplicative with b(1) = 1, and for p prime and e>0: b(p^e)=0 if e mod 3 = 0 otherwise b(p^e)=3*(-1)^(e mod 3).
Dirichlet convolution with A007427(n) yields A212793(n).
Dirichlet convolution with A008836(n) yields A092520(n).
Equals Dirichlet convolution of A034444(n) and A056624(n).
Equals Dirichlet convolution of A000005(n) and A212793(n). (End)
Sum_{k=1..n} a(k) ~ n/(2*Zeta(3)) * (log(n)^2 + 2*log(n) * (-1 + 3*gamma - 3*Zeta'(3)/Zeta(3)) + 2 + 6*gamma^2 - 6*sg1 + 6*Zeta'(3)/Zeta(3) + 18*Zeta'(3)^2/Zeta(3)^2 - 6*gamma*(1 + 3*Zeta'(3)/Zeta(3)) - 9*Zeta''(3)/Zeta(3)), where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Feb 07 2019
a(n) = A005361(n) * A074816(n). - Vaclav Kotesovec, Feb 27 2023

A222945 Number of distinct sums i+j+k with |i|, |j|, |k|, |ijk| <= n.

Original entry on oeis.org

1, 7, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97, 101, 105, 109, 113, 117, 121, 125, 129, 133, 137, 141, 145, 149, 153, 157, 161, 165, 169, 173, 177, 181, 185, 189, 193, 197, 201, 205, 209, 213, 217, 221, 225

Offset: 0

Robert Price, Jun 12 2013

nonn

Apparently a(n) = A004766(n) for n>=2. - R. J. Mathar, May 26 2024

Robert Price, Table of n, a(n) for n = 0..100

Cf. A004766, A226357, A226359, A222947.

f[n_] := Length[Complement[Union[Flatten[Table[If[Abs[i*j*k] ≤ n, {i + j + k}], {i, -n, n}, {j, -n, n}, {k, -n, n}], 2]], {Null}]]; Table[f[n], {n, 0, 100}]

A222947 Number of distinct sums i+j+k with |i|, |j|, |k|, |ijk| <= n and gcd(i,j,k) <= 1.

Original entry on oeis.org

1, 7, 9, 11, 15, 19, 21, 27, 29, 35, 37, 43, 45, 51, 53, 59, 61, 67, 69, 75, 77, 83, 85, 91, 93, 99, 101, 107, 109, 115, 117, 123, 125, 131, 133, 139, 141, 147, 149, 155, 157, 163, 165, 171, 173, 179, 181, 187, 189, 195, 197, 203, 205, 211, 213, 219, 221

Offset: 0

Robert Price, Jun 12 2013

nonn

Note that gcd(0,m) = m for any m.

Robert Price, Table of n, a(n) for n = 0..100

Cf. A226357, A226359, A222945.

f[n_] := Length[Complement[Union[Flatten[Table[If[Abs[i*j*k] ≤ n && GCD[i, j, k] <= 1, {i + j +  k}], {i, -n, n}, {j, -n, n}, {k, -n, n}], 2]], {Null}]]; Table[f[n], {n, 0, 100}]

A223133 Number of distinct sums i+j+k with i,j,k >= 0, ijk <= n and gcd(i,j,k) <= 1.

Original entry on oeis.org

1, 4, 5, 6, 8, 10, 11, 14, 15, 18, 19, 22, 23, 26, 27, 30, 31, 34, 35, 38, 39, 42, 43, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63, 66, 67, 70, 71, 74, 75, 78, 79, 82, 83, 86, 87, 90, 91, 94, 95, 98, 99, 102, 103, 106, 107, 110, 111, 114, 115, 118, 119, 122, 123

Offset: 0

Robert Price, Jun 12 2013

nonn

Note that gcd(0,m) = m for any m.

Robert Price, Table of n, a(n) for n = 0..100

Cf. A226357, A226359, A222945, A222947.

f[n_] := Length[Complement[Union[Flatten[Table[If[i*j*k <= n && GCD[i, j, k] <= 1, {i + j + k}], {i, 0, n}, {j, 0, n}, {k, 0, n}], 2]], {Null}]]; Table[f[n], {n, 0, 100}]

A223134 Number of distinct sums i+j+k with i,j,k >= 0, ijk <= n.

Original entry on oeis.org

1, 4, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125

Offset: 0

Robert Price, Jun 12 2013

nonn

Appears to be essentially the same as A176271, A140139, A130773, A062545. - R. J. Mathar, Aug 23 2024

Robert Price, Table of n, a(n) for n = 0..100

Cf. A226357, A226359, A222945, A222947, A223133.

f[n_] := Length[Complement[Union[Flatten[Table[If[i*j*k <= n, {i + j + k}], {i, 0, n}, {j, 0, n}, {k, 0, n}], 2]], {Null}]]; Table[f[n], {n, 0, 100}]

A226378 Number of distinct sums i+j+k with i,j,k >= 0, ijk = n.

Original entry on oeis.org

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

Offset: 0

Robert Price, Jun 12 2013

nonn

			From _Antti Karttunen_, Aug 30 2017: (Start)
For n = 4 = 1*1*4 = 1*2*2, 1+1+4 = 6 and 1+2+2 = 5, so there are two distinct sums, and a(4) = 2.
For n = 6 = 1*1*6 = 1*2*3, 1+1+6 = 8 and 1+2+3 = 6, so there are two distinct sums, and a(6) = 2.
For n = 36, of its A034836(36) = 8 factorizations as x*y*z with 1 <= x <= y <= z: 1*1*36 = 1*2*18 = 1*3*12 = 1*4*9 = 1*6*6 = 2*2*9 = 2*3*6 = 3*3*4, sums 1+6+6 and 2+2+9 are both 13, while other triples yield unique sums, thus a(36) = 8-1 = 7. (End)

Robert Price and Michael De Vlieger, Table of n, a(n) for n = 0..10000 (first 100 terms from Robert Price).
Michael De Vlieger, Records and first positions of records in A226378(n), with 0 <= n <= 10^6.

Cf. A226357, A226359, A222945, A222947, A223133, A223134, A223135.
Cf. A008578 (gives the positions of 1's after a(0)=1).
Differs from A034836 for the first time at n=36.

f[n_] := Length[Complement[Union[Flatten[Table[If[i*j*k == n, {i + j + k}], {i, 0, n}, {j, 0, n}, {k, 0, n}], 2]], {Null}]]; Table[f[n], {n, 0, 100}]
(* Second program, more efficient: *)
{1}~Join~Table[With[{D = Divisors@ n}, Length@ Union@ Reap[Map[Function[a, Map[Function[b, Map[Function[c, If[a b c == n, Sow[a + b + c]]], Select[D, # <= n/a b &]]], Select[D, # <= n/a &]]], D]][[-1, 1]] ], {n, 100}] (* Michael De Vlieger, Aug 24 2017 *)

A226378(n) = { my(sums=Set()); if(!n,1,fordiv(n, i, for(j=i, (n/i), if(!(n%j),for(k=j, n/(i*j), if(i*j*k==n, sums = Set(concat(sums, (i+j+k)))))))); length(sums)); }; \\ Antti Karttunen, Aug 30 2017

For n >= 1, a(n) <= A034836(n). - Antti Karttunen, Aug 30 2017

A223135 Number of distinct sums i + j + k with i, j, k >= 0, ijk = n and gcd(i,j,k) <= 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 5, 2, 2, 2, 4, 1, 5, 1, 3, 2, 2, 2, 7, 1, 2, 2, 5, 1, 5, 1, 4, 4, 2, 1, 7, 2, 4, 2, 4, 1, 5, 2, 5, 2, 2, 1, 10, 1, 2, 4, 4, 2, 5, 1, 4, 2, 5, 1, 10, 1, 2, 4, 4, 2, 5, 1, 7, 3, 2, 1, 10, 2, 2, 2, 5, 1, 8, 2, 4, 2, 2, 2, 7, 1, 4, 4, 8, 1, 5, 1, 5, 5

Offset: 0

Robert Price, Jun 12 2013

nonn

Note that gcd(0,m) = m for any m.

Antti Karttunen, Table of n, a(n) for n = 0..10001 (terms n = 0..100 from Robert Price)

Cf. A226357, A226359, A222945, A222947, A223133, A223134, A226378.

f[n_] := Length[Complement[Union[Flatten[Table[If[i*j*k == n && GCD[i, j, k] ≤ 1, {i + j + k}], {i, 0, n}, {j, 0, n}, {k, 0, n}], 2]], {Null}]]; Table[f[n], {n, 0, 100}]

A223135(n) = { my(sums=Set()); if(!n,1,fordiv(n, i, for(j=i, (n/i), if(!(n%j),for(k=j, n/(i*j), if((i*j*k==n)&&(gcd(i,gcd(j,k))<=1), sums = Set(concat(sums, (i+j+k)))))))); length(sums)); }; \\ Antti Karttunen, Oct 21 2017

More terms from Antti Karttunen, Oct 21 2017

A226600 Number of ordered triples (i,j,k) with ijk <= n and i,j,k >= 0.

Original entry on oeis.org

1, 8, 23, 44, 74, 107, 152, 197, 255, 315, 384, 453, 543, 624, 717, 816, 927, 1032, 1158, 1275, 1413, 1548, 1689, 1830, 2004, 2160, 2325, 2497, 2683, 2860, 3067, 3256, 3469, 3676, 3889, 4108, 4360, 4585, 4822, 5065, 5335, 5584, 5863, 6124, 6406, 6694, 6979

Offset: 0

Robert Price, Jun 13 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (terms n = 0..100 from Robert Price)

Cf. A100450, A226357, A226359.

f[n_] := Length[Complement[Union[Flatten[Table[If[i*j*k <= n, {i, j, k}], {i, 0, n}, {j, 0, n}, {k, 0, n}], 2]], {Null}]]; Table[f[n], {n, 0, 100}]

a(n) = A003215(n) + A061201(n). - Alois P. Heinz, Jun 13 2013

A226601 Number of ordered triples (i,j,k) with ijk <= n, i,j,k >= 0 and gcd{i,j,k} <= 1.

Original entry on oeis.org

1, 8, 17, 32, 50, 77, 98, 137, 170, 212, 245, 308, 350, 425, 470, 527, 587, 686, 740, 851, 917, 998, 1067, 1202, 1277, 1403, 1484, 1601, 1691, 1862, 1937, 2120, 2231, 2360, 2465, 2618, 2726, 2945, 3062, 3215, 3338, 3581, 3680, 3935, 4073, 4235, 4376, 4655

Offset: 0

Robert Price, Jun 13 2013

nonn

Note that gcd(0,m) = m for any m.

Robert Price, Table of n, a(n) for n = 0..100

Cf. A100450, A226357, A226359, A226600.

f[n_] := Length[Complement[Union[Flatten[Table[If[i*j*k <= n && GCD[i, j, k] ≤ 1, {i, j, k}], {i, 0, n}, {j, 0, n}, {k, 0, n}], 2]], {Null}]]; Table[f[n], {n, 0, 100}]

cp's OEIS Frontend

A226359 Number of ordered triples (i,j,k) with |i|, |j|, |k|, |i*j*k| <= n.

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A226602 Number of ordered triples (i,j,k) with i*j*k = n, i,j,k >= 0 and gcd(i,j,k) <= 1.

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A222945 Number of distinct sums i+j+k with |i|, |j|, |k|, |i*j*k| <= n.

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A222947 Number of distinct sums i+j+k with |i|, |j|, |k|, |i*j*k| <= n and gcd(i,j,k) <= 1.

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A223133 Number of distinct sums i+j+k with i,j,k >= 0, i*j*k <= n and gcd(i,j,k) <= 1.

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A223134 Number of distinct sums i+j+k with i,j,k >= 0, i*j*k <= n.

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A226378 Number of distinct sums i+j+k with i,j,k >= 0, i*j*k = n.

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A223135 Number of distinct sums i + j + k with i, j, k >= 0, i*j*k = n and gcd(i,j,k) <= 1.

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Extensions

A226359 Number of ordered triples (i,j,k) with |i|, |j|, |k|, |ijk| <= n.

A226602 Number of ordered triples (i,j,k) with ijk = n, i,j,k >= 0 and gcd(i,j,k) <= 1.

A222945 Number of distinct sums i+j+k with |i|, |j|, |k|, |ijk| <= n.

A222947 Number of distinct sums i+j+k with |i|, |j|, |k|, |ijk| <= n and gcd(i,j,k) <= 1.

A223133 Number of distinct sums i+j+k with i,j,k >= 0, ijk <= n and gcd(i,j,k) <= 1.

A223134 Number of distinct sums i+j+k with i,j,k >= 0, ijk <= n.

A226378 Number of distinct sums i+j+k with i,j,k >= 0, ijk = n.

A223135 Number of distinct sums i + j + k with i, j, k >= 0, ijk = n and gcd(i,j,k) <= 1.

A226600 Number of ordered triples (i,j,k) with ijk <= n and i,j,k >= 0.

A226601 Number of ordered triples (i,j,k) with ijk <= n, i,j,k >= 0 and gcd{i,j,k} <= 1.