A213212 -id:A213212

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A100450

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

+10
17

1, 7, 19, 51, 99, 195, 291, 483, 675, 963, 1251, 1731, 2115, 2787, 3363, 4131, 4899, 6051, 6915, 8355, 9507, 11043, 12483, 14595, 16131, 18531, 20547, 23139, 25443, 28803, 31107, 34947, 38019, 41859, 45315, 49923, 53379, 58851, 63171, 68547

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

I would also like to get the sequences of the numbers of distinct sums i+j+k (also distinct products i*j*k) over all ordered triples (i,j,k) with |i| + |j| + |k| <= n; also over all ordered triples (i,j,k) with |i| + |j| + |k| <= n and gcd(i,j,k) <= 1.

Also the sequences of the numbers of distinct sums i+j+k (also distinct products i*j*k) over all ordered triples (i,j,k) with i >= 0, j >= 0, k >= 0 and i + j + k = n; also over all ordered triples (i,j,k) with i >= 0, j >= 0, k >= 0, i + j + k = n and gcd(i,j,k) <= 1.

Also the number of ordered triples (i,j,k) with i >= 0, j >= 0, k >= 0, i + j + k = n and gcd(i,j,k) <= 1.

From Robert Price, Mar 05 2013: (Start)

The sequences that address the previous comments are:

Distinct sums i+j+k with or without the GCD qualifier results in a(n)=2n+1 (A005408).

Distinct products i*j*k without the GCD qualifier is given by A213207.

Distinct products i*j*k with the GCD qualifier is given by A213208.

With the restriction i,j,k >= 0 ...

Distinct sums or products equal to n is trivial and always equals one (A000012).

Distinct sums <= n results in a(n)=n (A001477).

Distinct products <= n without the GCD qualifier is given by A213213.

Distinct products <= n with the GCD qualifier is given by A213212.

Ordered triples with sum = n without the GCD qualifier is A000217(n+1).

Ordered triples with sum = n with the GCD qualifier is A048240.

Ordered triples with sum <= n without the GCD qualifier is A000292.

Ordered triples with sum <= n with the GCD qualifier is A048241. (End)

This sequence (A100450) without the GCD qualifier results in A001845. - Robert Price, Jun 04 2013

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

FORMULA

G.f.: (3 + Sum_{k>=1} (moebius(k)*((1+x^k)/(1-x^k))^3))/(1-x). - Vladeta Jovovic, Nov 22 2004. [Sketch of proof: Let b(n) = number of ordered triples (i, j, k) with |i| + |j| + |k| = n and gcd(i, j, k) <= 1. Then a(n) = A100450(n) = partial sums of b(n) and Sum_{d divides n} b(d) = 4*n^2+2 = A005899(n) with g.f. ((1+x)/(1-x))^3.]

MAPLE

f:=proc(n) local i, j, k, t1, t2, t3; t1:=0; for i from -n to n do for j from -n to n do t2:=gcd(i, j); for k from -n to n do if abs(i) + abs(j) + abs(k) <= n then t3:=gcd(t2, k); if t3 <= 1 then t1:=t1+1; fi; fi; od: od: od: t1; end;

MATHEMATICA

f[n_] := Length[ Union[ Flatten[ Table[ If[ Abs[i] + Abs[j] + Abs[k] <= n && GCD[i, j, k] <= 1, {i, j, k}, {0, 0, 0}], {i, -n, n}, {j, -n, n}, {k, -n, n}], 2]]]; Table[ f[n], {n, 0, 40}] (* Robert G. Wilson v, Dec 14 2004 *)

CROSSREFS

Cf. A000124, A000292, A018805, A027430, A048240, A048241, A100448, A100449, A213207, A213208, A213212, A213213.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Nov 21 2004

STATUS

approved

A213207

Number of distinct products i*j*k over all triples (i,j,k) with |i| + |j| + |k| <= n.

+10
10

1, 1, 1, 3, 5, 9, 13, 19, 25, 35, 43, 55, 65, 79, 91, 111, 127, 149, 167, 193, 217, 249, 273, 311, 339, 383, 419, 463, 501, 551, 591, 643, 693, 751, 799, 869, 925, 995, 1057, 1133, 1199, 1281, 1347, 1439, 1515, 1615, 1697, 1801, 1883, 2001, 2101, 2219, 2313

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

This sequence is in reply to an extension request made in A100450.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 0..100 from Robert Price)

MAPLE

h:= proc() true end:

b:= proc(n) local c, i, j, p;

c:=0;

for i to iquo(n, 3) do

for j from i to iquo(n-i, 2) do p:= i*j*(n-i-j);

if h(p) then h(p):= false; c:=c+1 fi

od; c

end:

a:= proc(n) a(n):= `if`(n=0, 1, a(n-1) +2*b(n)) end:

seq(a(n), n=0..60); # Alois P. Heinz, Mar 02 2013

MATHEMATICA

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

CROSSREFS

Cf. A018805, A027430, A100448, A100449, A100450, A213208, A213212, A213213.

KEYWORD

nonn,easy

AUTHOR

Robert Price, Mar 01 2013

STATUS

approved

A213208

Number of distinct products i*j*k over all triples (i,j,k) with |i| + |j| + |k| <= n and gcd(i,j,k) <= 1.

+10
10

1, 1, 1, 3, 5, 9, 11, 19, 23, 33, 39, 51, 57, 75, 87, 103, 117, 143, 155, 187, 207, 235, 259, 297, 319, 363, 395, 441, 473, 525, 555, 615, 659, 721, 765, 831, 875, 959, 1017, 1091, 1147, 1239, 1291, 1397, 1467, 1553, 1631, 1743, 1813, 1937, 2023, 2141, 2233, 2379, 2465

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

This sequence is in reply to an extension request made in A100450.

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 0..100 from Robert Price)

MAPLE

h:= proc() true end:

b:= proc(n) local c, i, j, p;

c:=0;

for i to iquo(n, 3) do

for j from i to iquo(n-i, 2) do

if igcd(i, j, n-i-j)=1 then p:= i*j*(n-i-j);

if h(p) then h(p):= false; c:=c+1 fi

od; c

end:

a:= proc(n) a(n):= `if`(n=0, 1, a(n-1) +2*b(n)) end:

seq(a(n), n=0..60); # Alois P. Heinz, Mar 01 2013

MATHEMATICA

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

CROSSREFS

Cf. A018805, A027430, A100448, A100449, A100450, A213207, A213212, A213213.

KEYWORD

nonn,easy

AUTHOR

Robert Price, Mar 01 2013

STATUS

approved

A213213

Number of distinct products i*j*k over all triples (i,j,k) with i,j,k>=0 and i+j+k <= n.

+10
7

1, 1, 1, 2, 3, 5, 7, 10, 13, 18, 22, 28, 33, 40, 46, 56, 64, 75, 84, 97, 109, 125, 137, 156, 170, 192, 210, 232, 251, 276, 296, 322, 347, 376, 400, 435, 463, 498, 529, 567, 600, 641, 674, 720, 758, 808, 849, 901, 942, 1001, 1051, 1110, 1157, 1225, 1275

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

This sequence is in reply to an extension request made in A100450.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 0..200 from Robert Price)

Cristina Ballantine, George Beck, Mircea Merca, and Bruce Sagan, Elementary symmetric partitions, arXiv:2409.11268 [math.CO], 2024. See p. 20.

FORMULA

a(n) = (A213207(n)+1)/2.

MAPLE

h:= proc() true end:

b:= proc(n) local c, i, j, p;

c:=0;

for i to iquo(n, 3) do

for j from i to iquo(n-i, 2) do p:= i*j*(n-i-j);

if h(p) then h(p):= false; c:=c+1 fi

od; c

end:

a:= proc(n) a(n):= `if`(n=0, 1, a(n-1) +b(n)) end:

seq(a(n), n=0..60); # Alois P. Heinz, Mar 02 2013

MATHEMATICA

f[n_] := Length[ Union[ Flatten[ Table[ If[ i+j+k <= n, i*j*k, 0], {i, 0, n}, {j, 0, n}, {k, 0, n}], 2]]]; Table[ f[n], {n, 0, 200}]

CROSSREFS

Cf. A018805, A027430, A100448, A100449, A100450, A213207, A213208, A213212.

KEYWORD

nonn,easy

AUTHOR

Robert Price, Mar 02 2013

STATUS

approved

A225531

Number of ordered pairs (i, j) with i, j >= 0, i + j <= n and gcd(i, j) <= 1.

+10
4

1, 3, 4, 6, 8, 12, 14, 20, 24, 30, 34, 44, 48, 60, 66, 74, 82, 98, 104, 122, 130, 142, 152, 174, 182, 202, 214, 232, 244, 272, 280, 310, 326, 346, 362, 386, 398, 434, 452, 476, 492, 532, 544, 586, 606, 630, 652, 698, 714, 756, 776

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

This sequence is in reply to an extension request made in A100449.

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

LINKS

Robert Price, Table of n, a(n) for n = 0..400

MATHEMATICA

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

PROG

(PARI) alist(N) = my(c=2); vector(N, i, if(1==i, 1, c+=eulerphi(i-1))); \\ Ruud H.G. van Tol, Jul 09 2024

CROSSREFS

Cf. A018805, A027430, A100448, A100449, A100450, A213207, A213208, A213212, A213213.

KEYWORD

nonn,easy

AUTHOR

Robert Price, May 09 2013

STATUS

approved

A225530

Number of ordered pairs (i,j) with i,j >= 0, i + j = n and gcd(i,j) <= 1.

+10
3

1, 2, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 10, 22, 8, 20, 12, 18, 12, 28, 8, 30, 16, 20, 16, 24, 12, 36, 18, 24, 16, 40, 12, 42, 20, 24, 22, 46, 16, 42, 20, 32, 24, 52, 18, 40, 24, 36, 28, 58, 16, 60, 30, 36, 32, 48, 20, 66

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

This sequence is in reply to an extension request made in A100449.

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

Apparently a(n) = A000010(n), n >= 2. - R. J. Mathar, May 11 2013

LINKS

Robert Price, Table of n, a(n) for n = 0..400

MATHEMATICA

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

CROSSREFS

Cf. A018805, A027430, A100448, A100449, A100450, A213207, A213208, A213212, A213213.

KEYWORD

nonn,easy

AUTHOR

Robert Price, May 09 2013

STATUS

approved

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