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

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

Author

Markus Sigg, Feb 07 2025

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A381187, A381619, A381621, A382547.

Programs

PARI

\\ See Sigg link

PARI

\\ See Corneth link

Extensions

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