A381620 -id:A381620 - cp's OEIS frontend

A381619 Sorted list of sums of 3 prices in minor currency units for a currency that has a 2-decimal minor unit, such that the riddle "sum of prices equals product of prices" has a solution, with prices expressed as floating point numbers with 2 decimals.

525, 540, 546, 549, 555, 561, 567, 570, 585, 588, 600, 612, 630, 642, 660, 660, 663, 675, 726, 735, 744, 750, 759, 765, 783, 792, 798, 810, 819, 825, 840, 840, 891, 897, 900, 930, 945, 957, 966, 966, 975, 981, 996, 1050, 1050, 1071, 1080, 1092, 1125, 1134, 1155, 1155, 1170

Offset: 1

Hugo Pfoertner and Markus Sigg, Mar 02 2025

The sequence has 622 terms. See linked files for all solutions.

A natural number s occurs k times in the list if there exist k multisets {x,y,z} of natural numbers with s = x + y + z and 10000*s = x*y*z.

			a(1) = 525 because 1.50 + 1.75 + 2.00 = 1.50*1.75*2.00 = 5.25 is the solution with minimum sum;
a(15) = a(16) = 660 because there are 2 solutions:
  0.80 + 2.50 + 3.30 = 0.80*2.50*3.30 = 6.60 and
  1.10 + 1.50 + 4.00 = 1.10*1.50*4.00 = 6.60;
a(31) = a(32) = 840:
  0.60 + 2.80 + 5.00 = 0.60*2.80*5.00 = 8.40 and
  1.00 + 1.40 + 6.00 = 1.00*1.40*6.00 = 8.40;
a(622) = 100030002 is the largest term:
  0.01 + 100.01 + 1000200.00 = 0.01*100.01*1000200.00 = 1000300.02.

Hugo Pfoertner, Table of n, a(n) for n = 1..622
Hugo Pfoertner, Solution triples sorted by sum. (2025)
Hugo Pfoertner, Solution triples sorted by smallest price. (2025)
Eric Snyder and others, Finding solutions of sum a_i = product a_i = n, where the a_i are "price rationals", question in Mathematics StackExchange, Jun 29, 2022.

Cf. A380887, A381187, A381620, A381621, A382510.

A382508 a(n) is the number of solutions to the problem described in A381621 with smallest price equal to n.

Original entry on oeis.org

4728, 2314, 1165, 2169, 1429, 703, 304, 1006, 283, 1532, 129, 351, 135, 241, 595, 668, 58, 175, 72, 511, 60, 136, 52, 166, 994, 51, 36, 110, 35, 331, 15, 123, 12, 49, 109, 69, 20, 39, 12, 301, 18, 36, 20, 37, 57, 31, 19, 74, 6, 315, 11, 29, 8, 10, 38, 24, 10, 25, 6, 95

Offset: 1

Hugo Pfoertner, Mar 30 2025

			a(71) = 0 because no 4-tuple with smallest element = 71 exists.
a(91) = 1 because the only 4-tuple with smallest element 91 is [91, 100, 110, 301000].

Hugo Pfoertner, Table of n, a(n) for n = 1..125

Cf. A381619, A381620, A381621.

(* Uses a tested heuristic upper bound for the second element b in the 4-tuple; running times > 10 minutes for small n, depending on the computer speed *)
a382508[n_] := Sum[Length[Solve[10^6*(n+b+c+d) == n*b*c*d && c>=b && d>=c,{c,d}, Integers]], {b, n, 111+Floor[1600/n^0.55-n/2]}];

A382510 a(n) is the number of solutions to the "sum equals product" riddle with n prices v_j, i.e., find positive integers v_j, v_{j+1}>=v_j such that 100^(n-1)*Sum_{k=1..n} v_k = Product_{k=1..n} v_k.

Original entry on oeis.org

1, 13, 622, 22640

Offset: 1

Hugo Pfoertner, Apr 01 2025

See A380887 and A381619 for more information.

			a(1) = 1: [100] is the only solution.
a(2) = 13: the 13 solutions are [101, 10100], [102, 5100], [104, 2600], [105, 2100], [108, 1350], [110, 1100], [116, 725], [120, 600], [125, 500], [140, 350], [150, 300], [180, 225], [200, 200].
a(3) = 622 is the number of terms of A381619.
a(4) = 22640 is the number of terms of A381621.

Cf. A380887, A381619, A381620, A381621, A382508.

Length[Solve[100*(a + b) == a*b && a > 0 && b >= a, {a, b}, Integers]] (* Computes a(2) *)

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A381619 Sorted list of sums of 3 prices in minor currency units for a currency that has a 2-decimal minor unit, such that the riddle "sum of prices equals product of prices" has a solution, with prices expressed as floating point numbers with 2 decimals.

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A382508 a(n) is the number of solutions to the problem described in A381621 with smallest price equal to n.

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A382510 a(n) is the number of solutions to the "sum equals product" riddle with n prices v_j, i.e., find positive integers v_j, v_{j+1}>=v_j such that 100^(n-1)*Sum_{k=1..n} v_k = Product_{k=1..n} v_k.

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Programs

Mathematica