A263207 -id:A263207 - cp's OEIS frontend

A382672 Number of integer solutions to Product_{k=1..n} (3 + c(k)) = 3 * Product_{k=1..n} c(k) with 0 < c(k) <= c(k+1).

0, 2, 17, 450, 35472, 12127741

Offset: 1

Zhining Yang, Apr 03 2025

			For n=3, a(3) = 17 because 3*x*y*z = (x + 3)*(y + 3)*(z + 3), 0 < x <= y <= z has 17 positive integer solutions: {{2,16,285}, {2,17,150}, {2,18,105}, {2,20,69}, {2,21,60}, {2,24,45}, {2,25,42}, {2,30,33}, {3,7,60}, {3,8,33}, {3,9,24}, {3,12,15}, {4,5,42}, {4,6,21}, {4,7,15}, {5,6,12}, {6,6,9}}.

Cf. A263207, A375787, A380749, A381644.

a[n_]:=(p=c/@Range[n];Length@Solve[3 Times@@p==Times@@(3+p)&&LessEqual@@Flatten[{0,p}],p,Integers]);Array[a,5]

A383223 Number of integer solutions to Product_{k=1..n} (4 + c(k)) = 4 * Product_{k=1..n} c(k) with 0 < c(k) <= c(k+1).

Original entry on oeis.org

0, 2, 15, 375, 28901, 5185573

Offset: 1

Zhining Yang, Apr 19 2025

			For n=3, a(3) = 15 because 4*x*y*z = (x + 4)*(y + 4)*(z + 4), 0 < x <= y <= z has 15 positive integer solutions: {{2,13,204}, {2,14,108}, {2,15,76}, {2,16,60}, {2,18,44}, {2,20,36}, {2,24,28}, {3,6,140}, {3,7,44}, {3,8,28}, {3,12,14}, {4,5,36}, {4,6,20}, {4,8,12}, {5,6,12}}.

Cf. A263207, A375787, A380749, A381644, A382672.

a[n_]:=(p=c/@Range[n]; Length@Solve[4 Times@@p==Times@@(4+p)&&LessEqual@@Flatten[{0, p}], p, Integers]); Array[a, 5]

cp's OEIS Frontend

A382672 Number of integer solutions to Product_{k=1..n} (3 + c(k)) = 3 * Product_{k=1..n} c(k) with 0 < c(k) <= c(k+1).

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A383223 Number of integer solutions to Product_{k=1..n} (4 + c(k)) = 4 * Product_{k=1..n} c(k) with 0 < c(k) <= c(k+1).

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Mathematica