A381619 -id:A381619 - cp's OEIS frontend

A381620 a(n) is the number of solutions to the problem described in A381619 with smallest price equal to n.

50, 50, 12, 30, 25, 26, 8, 30, 9, 25, 5, 15, 2, 4, 13, 5, 2, 8, 3, 30, 3, 3, 1, 8, 25, 4, 1, 4, 2, 12, 0, 10, 2, 1, 5, 5, 0, 1, 0, 15, 2, 4, 1, 3, 8, 2, 1, 2, 0, 15, 1, 2, 0, 1, 2, 2, 0, 1, 1, 15, 1, 0, 2, 0, 3, 3, 1, 2, 2, 5, 1, 6, 1, 2, 9, 3, 1, 0, 0, 5, 1, 4

Offset: 1

Hugo Pfoertner, Mar 12 2025

			a(13) = 2 because there are 2 triples {x,y,z} satisfying 100^2*(x+y+z)=x*y*z with x=13:
  {13, 770, 783000} and {13, 800, 20325};
a(23) = 1: {23, 435, 416000} is the only triple with smallest term 23: 10000*(23+435+916000) = 23*435*916000 = 9164580000 = 10000*A381619(578).

Hugo Pfoertner, Table of n, a(n) for n = 1..150
Hugo Pfoertner, Solution triples sorted by smallest price. (2025)

Cf. A380887, A381187, A381619.

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

A381621 Sorted list of sums of 4 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

644, 651, 660, 663, 665, 672, 675, 675, 678, 680, 684, 684, 686, 689, 693, 693, 702, 705, 707, 707, 708, 711, 713, 714, 714, 720, 720, 720, 725, 726, 728, 728, 729, 735, 735, 735, 737, 747, 750, 752, 756, 756, 756, 756, 762, 765, 765, 765, 765, 767, 770, 770, 774, 774, 774, 777

Offset: 1

Hugo Pfoertner, Mar 04 2025

The sequence is finite with 22640 terms.

A natural number s occurs k times in the list if there exist k multisets {a,b,c,d} of natural numbers with s = a + b + c + d and 100^3*s = a*b*c*d.

			a(1) = 644 = A380887(4) because 1.25 + 1.60 + 1.75 + 1.84 = 1.25*1.60*1.75*1.84 = 6.44 is the solution with minimum sum;
a(7) = a(8) = 675 because there are 2 solutions:
  1.00 + 1.50 + 2.00 + 2.25 = 1.00*1.50*2.00*2.25 = 6.75 and
  1.20 + 1.25 + 1.80 + 2.50 = 1.20*1.25*1.80*2.50 = 6.75;
a(22) = 711 corresponds to the solution of the puzzle "The 7-Eleven" quoted from "The Guardian" in A380887.
a(22640) = 1000004000003 is the largest term, corresponding to the quadruple of prices [0.01, 0.01, 10000.01, 10000030000.00].

Hugo Pfoertner, Table of n, a(n) for n = 1..10000
Hugo Pfoertner, Solution quadruples sorted by sum, up to sum 10^5. (2025)
Hugo Pfoertner, Complete list of 22640 solution quadruples. (2025)

Cf. A380887, A381187, A381619, A382508, 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) *)

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.

cp's OEIS Frontend

A381620 a(n) is the number of solutions to the problem described in A381619 with smallest price equal to n.

Views

Author

Keywords

Examples

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Extensions

A381621 Sorted list of sums of 4 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs