A222945 -id:A222945 - cp's OEIS frontend

A226357 Number of ordered triples (i,j,k) with |i|,|j|,|k|,|ijk| <= n and gcd(i,j,k) <= 1.

1, 27, 75, 147, 243, 363, 483, 651, 819, 1011, 1179, 1443, 1683, 1995, 2211, 2475, 2763, 3171, 3459, 3915, 4251, 4611, 4923, 5475, 5883, 6411, 6771, 7275, 7707, 8403, 8811, 9555, 10059, 10611, 11067, 11715, 12291, 13179, 13683, 14331, 14931, 15915, 16419

Offset: 0

Robert Price, Jun 04 2013

nonn

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

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

|i| + |j| + |k| <= n instead of |i*j*k| <= n: A100450.
This sequence (A226357) without the GCD qualifier: A226359.
Distinct sums i+j+k with the GCD qualifier: A222947.
Distinct sums i+j+k without the GCD qualifier: A222945.
Distinct products i*j*k with or without the GCD qualifier is 2n+1: A005408.
With the further restriction i,j,k >= 0 ...
  Distinct sums i+j+k <= n with the GCD qualifier: A223133.
  Distinct sums i+j+k <= n without the GCD qualifier: A223134.
  Distinct products i*j*k with or without the GCD qualifier is n+1: A000217(n+1).
  Distinct sums i+j+k with i*j*k = n with the GCD qualifier: A223135.
  Distinct sums i+j+k with i*j*k = n without the GCD qualifier: A226378.
  Distinct products i*j*k with i*j*k = n with or without the GCD qualifier is trivial and always 1: A000012.
  Ordered triples with the product <= n with the GCD qualifier: A226001.
  Ordered triples with the product <= n without the GCD qualifier: A226600.
  Ordered triples with the product = n with the GCD qualifier: A226602.
  Ordered triples with the product = n without the GCD qualifier: A007425.

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}]

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

A270741 a(n) = (Fibonacci(n+2)-1) mod Fibonacci(floor(n/2)).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 3, 2, 0, 1, 11, 10, 0, 1, 32, 31, 0, 1, 87, 86, 0, 1, 231, 230, 0, 1, 608, 607, 0, 1, 1595, 1594, 0, 1, 4179, 4178, 0, 1, 10944, 10943, 0, 1, 28655, 28654, 0, 1, 75023, 75022, 0, 1, 196416, 196415, 0, 1, 514227, 514226, 0, 1, 1346267, 1346266, 0, 1, 3524576, 3524575, 0, 1

Offset: 2

Altug Alkan, Mar 22 2016

nonn

It appears that a(n)=1 gives same terms as A222945. - Michel Marcus, Mar 23 2016

			a(10) = 3 since (1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55) mod 5 = 143 mod 5 = 3.
a(11) = 2 since (1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 + 89) mod 5 = 2.
a(12) = 0 since (1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 + 89 + 144) mod 8 = 0.

Cf. A000045, A000071, A270708.

Table[Mod[Fibonacci[n + 2] - 1, Fibonacci[Floor[n/2]]], {n, 2, 69}] (* Michael De Vlieger, Mar 23 2016 *)

for(n=2, 1e2, print1((fibonacci(n+2)-1) % fibonacci(n\2), ", "));

Empirical g.f.: x^7*(1-x+2*x^2+x^3-3*x^4+3*x^5-6*x^6+5*x^7+x^8-x^9+2*x^10-2*x^11) / ((1-x)*(1+x^2)*(1+x^2-x^4)*(1-x^2-x^4)). - Colin Barker, Mar 23 2016

cp's OEIS Frontend

A226357 Number of ordered triples (i,j,k) with |i|,|j|,|k|,|i*j*k| <= n and gcd(i,j,k) <= 1.

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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

A270741 a(n) = (Fibonacci(n+2)-1) mod Fibonacci(floor(n/2)).

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A226357 Number of ordered triples (i,j,k) with |i|,|j|,|k|,|ijk| <= n and gcd(i,j,k) <= 1.

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.