A382547 -id:A382547 - cp's OEIS frontend

A383026 Triangle T(n,k) read by rows whose n-th row is the lexicographically first n-tuple of ordered distinct positive integers with sum A382547(n) and product A382547(n) * 100^(n-1), or an n-tuple of zeros when A382547(n) = 0.

1, 180, 225, 150, 175, 200, 125, 160, 175, 184, 125, 127, 150, 160, 200, 100, 125, 140, 150, 175, 192, 80, 100, 125, 150, 160, 173, 250, 80, 100, 110, 125, 140, 150, 200, 250, 50, 100, 112, 125, 150, 155, 160, 200, 250, 50, 80, 100, 125, 128, 150, 170, 175, 200, 250

Offset: 1

Markus Sigg, Apr 13 2025

Because A382547(n) > 0 for only finitely many n, the triangle has only finitely many nonzero rows.

			Triangle begins:
    1,
  180, 225,
  150, 175, 200,
  125, 160, 175, 184,
  125, 127, 150, 160, 200,
  100, 125, 140, 150, 175, 192,
   80, 100, 125, 150, 160, 173, 250,
   80, 100, 110, 125, 140, 150, 200, 250,
   50, 100, 112, 125, 150, 155, 160, 200, 250,
   50,  80, 100, 125, 128, 150, 170, 175, 200, 250,
   50,  65,  75, 100, 125, 128, 150, 175, 200, 250, 320,
   25,  50,  80, 100, 125, 128, 150, 200, 225, 230, 250, 300,
  ...
For n = 6 there are three 6-tuples with sum A382547(6) = 882 and product 100^5 * 882, namely (100, 125, 140, 150, 175, 192), (100, 125, 147, 150, 160, 200), (112, 120, 125, 150, 175, 200). The first of these is the lexicographically smallest and thus is row 6 of the triangle.

Markus Sigg, Table of n, a(n) for n = 1..231, rows 1..21, flattened.

Cf. A382547, A380887, A381187.

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

A381187 Triangle T(n,k) read by rows whose n-th row is the lexicographically first n-tuple of ordered positive integers with sum A380887(n) and product A380887(n) * 100^(n-1).

Original entry on oeis.org

1, 200, 200, 150, 175, 200, 125, 160, 175, 184, 125, 125, 160, 165, 184, 125, 125, 144, 150, 160, 160, 125, 125, 128, 144, 150, 150, 150, 110, 125, 125, 125, 128, 150, 150, 176, 125, 125, 125, 125, 128, 128, 132, 150, 150, 120, 120, 125, 125, 125, 125, 128, 128, 150, 150

Offset: 1

Markus Sigg, Feb 16 2025

			Triangle begins:
    1,
  200, 200,
  150, 175, 200,
  125, 160, 175, 184,
  125, 125, 160, 165, 184,
  125, 125, 144, 150, 160, 160,
  125, 125, 128, 144, 150, 150, 150,
  110, 125, 125, 125, 128, 150, 150, 176,
  125, 125, 125, 125, 128, 128, 132, 150, 150,
  120, 120, 125, 125, 125, 125, 128, 128, 150, 150,
  115, 122, 125, 125, 125, 125, 125, 125, 128, 128, 160,
  104, 125, 125, 125, 125, 125, 125, 125, 128, 128, 128, 145,
  ...
For n = 8 there are three 8-tuples with sum A380887(8) = 1089 and product 100^7 * 1089, namely (110, 125, 125, 125, 128, 150, 150, 176), (120, 125, 125, 125, 125, 128, 165, 176), (121, 125, 125, 125, 125, 128, 160, 180). The first of these is the lexicographically smallest and thus is row 8 of the triangle.

David A. Corneth, Table of n, a(n) for n = 1..45150 (first 528 terms from Hugo Pfoertner and Markus Sigg)

Cf. A380887, A382547, A383026.

cp's OEIS Frontend

A383026 Triangle T(n,k) read by rows whose n-th row is the lexicographically first n-tuple of ordered distinct positive integers with sum A382547(n) and product A382547(n) * 100^(n-1), or an n-tuple of zeros when A382547(n) = 0.

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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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PARI

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A381187 Triangle T(n,k) read by rows whose n-th row is the lexicographically first n-tuple of ordered positive integers with sum A380887(n) and product A380887(n) * 100^(n-1).

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