A382547 a(n) is the smallest positive integer s that can be partitioned into n distinct positive integers whose product is s * 100^(n-1), or 0 if no such s exists.

1, 405, 525, 644, 762, 882, 1038, 1155, 1302, 1428, 1638, 1863, 2079, 2187, 2457, 2673, 3078, 3213, 3402, 3861, 4374, 5103, 5103, 6174

Offset: 1

Author

Markus Sigg, Mar 31 2025

Keywords

• nonn
• fini
• hard
• more

Comments

a(n) >= A380887(n) in case of a(n) > 0.

There are only finitely many positive a(n): If x_1 < ... < x_n are positive integers with the required properties, then x_k >= k, and (n-1)! * x_n <= x_1 * ... * x_n = 100^(n-1) * (x_1 + ... + x_n) <= 100^(n-1) * n * x_n gives (n-1)! <= 100^(n-1) * n, hence n <= 274. In fact, n <= 273 must hold, see Mathematics StackExchange link. A more elaborate argumentation in the same discussions shows n <= 269.

By restricting the search space, solution tuples have been found for 25 <= n <= 42. These tuples are not guaranteed to have the smallest possible sum and thus only give upper bounds for a(n). For example, the tuple (1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 20, 25, 30, 32, 40, 50, 60, 64, 75, 80, 100, 120, 125, 128, 150, 160, 200, 225, 250, 400, 625, 800, 1000, 1250, 2500, 3125, 5000, 6250, 12500, 78125, 1953125) shows a(42) <= 2066715.

Programs used for A380887 can be adapted for this sequence.

Examples

			a(2) = 405 because 180 + 225 = 405 and 180 * 225 = 405 * 100^1 and no positive integer smaller than 405 exists with the requested properties.
		

Links

• Mathematics StackExchange user Servaes, What is the maximum number of distinct prices whose product equals their sum?, answer dated May 29 2025.

Crossrefs

Cf. A380887, A381187, A383026.

Programs

PARI

dfs(rs, rp, i, r, tp) = if(r==1, return(rs==rp&&setsearch(d, rs)>i)); if((rs/r)^r<=rp, return(0)); for(j=i+1, oo, if(tp>rp, return(0)); if(rp%d[j]==0, if(dfs(rs-d[j], rp/d[j], j, r-1, tp/d[j]), return(1))); tp*=d[j+r]/d[j]);
a(n) = if(n>1, my(p); for(s=100*n, oo, d=divisors(p=s*100^(n-1)); if(dfs(s, p, 0, n, prod(i=1, n, d[i])), return(s))), 1); \\ Jinyuan Wang, May 14 2025

Extensions

a(17)-a(21) from Markus Sigg, Apr 21 2025

a(22)-a(24) from Jinyuan Wang, May 14 2025