A381621 -id:A381621 - cp's OEIS frontend

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

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]}];

A380887 a(n) is the smallest positive integer s that can be partitioned into n positive integers whose product is s * 100^(n-1).

Original entry on oeis.org

1, 400, 525, 644, 759, 864, 972, 1089, 1188, 1296, 1403, 1508, 1612, 1722, 1827, 1932, 2040, 2145, 2250, 2354, 2457, 2565, 2668, 2772, 2880, 2988, 3087, 3192, 3294, 3399, 3498, 3604, 3705, 3810, 3915, 4018, 4116, 4221, 4323, 4425, 4536, 4635, 4732, 4836, 4940

Offset: 1

Markus Sigg, Feb 07 2025

nonn

The AM-GM inequality shows a(n) >= 100 * n^(n/(n-1)). This bound gives a(2) >= 400, a(3) >= 520, a(4) >= 635, a(5) >= 748, a(6) >= 859.
a(n) is 100 times the smallest price r such that n prices exist whose sum and product both are equal to r. For example (see Guardian article link) 7.11 = 1.20 + 1.25 + 1.50 + 3.16 = 1.20 * 1.25 * 1.50 * 3.16.
Upper bounds for the next terms a(31)-a(40) are 3498, 3604, 3705, 3810, 3915, 4018, 4116, 4221, 4323, 4425. - Karl-Heinz Hofmann, Mar 26 2025
a(n) is well-defined: For n > 1, the sum of the numbers 1, ..., 1, k+1, k*(k+n-1), where the first n-2 numbers are 1 and k = 100^(n-1), is an example (possibly the largest one) of a positive integer s with the required properties. - Markus Sigg, Mar 30 2025
Better upper bounds can be given for specific values of n: Let s > 1 be an integer number and n = 2^s-s. Then the n-tuple of n-s times the number 100 and s times the number 200 has the required properties, hence a(2^s-s) <= 100*(n-s) + 200*s = 100 * 2^s. For s = 6, together with the lower bound from above, this gives 6229 <= a(58) <= 6400. - Markus Sigg, Apr 22 2025
For every n, the n-tuple (100, ..., 100, 10100, n + 99) has the required properties, hence a(n) <= 101*n + 9999. - Markus Sigg, May 30 2025

			a(2) = 400 because 200 + 200 = 400 and 200 * 200 = 400 * 100^1 and no positive integer smaller than 400 exists with the requested properties.
For a(3) the sum is 525 = 150 + 175 + 200.
For a(4) it is 644 = 125 + 160 + 175 + 184.
For a(5) it is 759 = 125 + 125 + 160 + 165 + 184.

David A. Corneth, Table of n, a(n) for n = 1..300
Alex Bellos, Can you solve it? Sexy maths, 2. The 7-Eleven, The Guardian, 3 Feb 2025.
David A. Corneth, PARI program
Markus Sigg, PARI program suitable for calculating a(n) for up to about n = 30.

Cf. A381187, A381619, A381621, A382547.

PARI
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\\ See Sigg link
```
PARI
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\\ See Corneth link
```

a(8)-a(9) from Hugo Pfoertner, Feb 13 2025
a(10)-a(12) from Hugo Pfoertner, Feb 16 2025
a(13) from Karl-Heinz Hofmann, Mar 02 2025
a(14)-a(30) from Markus Sigg, Mar 27 2025
a(31)-a(32) from Markus Sigg, Apr 23 2025
a(33)-a(45) from Jinyuan Wang, May 01 2025

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.

Original entry on oeis.org

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.

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) *)

A384795 Sorted list of sums of 5 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.

Original entry on oeis.org

759, 760, 762, 765, 770, 770, 774, 777, 779, 780, 780, 780, 783, 783, 784, 785, 786, 791, 791, 792, 792, 792, 795, 798, 798, 798, 798, 798, 799, 799, 800, 804, 804, 805, 805, 805, 806, 808, 810, 810, 810, 810, 810, 810, 810, 810, 812, 812, 813, 816, 816, 816, 817, 817, 817

Offset: 1

Hugo Pfoertner, Jun 15 2025

The sequence is finite with largest term 1000000050000000400000000, corresponding to the quintuple {1, 1, 1, 100000001, 10000000400000000}. The growth of A382510 indicates that the number of terms might be in the order of 500000.

s occurs k times in the list if there exist k multisets {x_1,...,x_5} of natural numbers with s = Sum_{j=1..5} x_j = (1/100^4)*Product_{j=1..5} x_j.

			a(1) = 759 = 125 + 125 + 160 + 165 + 184; 1.25^2*1.6*1.65*1.84 = 7.59.
a(5) = a(6) = 770 = 125 + 125 + 140 + 160 + 220 = 110 + 125 + 160 + 175 + 200; 1.25^2*1.4*1.6*2.2 = 1.1*1.25*1.6*1.75*2.0 = 7.70.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000
Hugo Pfoertner, Solution quintuples sorted by sum, up to sum 5000. (2025)

Cf. A380887, A381619, A381621, A382510.

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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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A380887 a(n) is the smallest positive integer s that can be partitioned into n positive integers whose product is s * 100^(n-1).

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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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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

A384795 Sorted list of sums of 5 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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