A380887 a(n) is the smallest positive integer s such that n positive integers exist whose sum is s and whose product is s * 100^(n-1), or 0 if no such s exists.

1, 400, 525, 644, 759, 864, 972, 1089, 1188, 1296, 1403, 1508, 1612

Offset

1,2

Comments

In the case of a(n) > 0, the AGM 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(14)-a(18) are 1722, 1827, 1932, 2040, 2145. - Karl-Heinz Hofmann, Mar 02 2025

Links

Table of n, a(n) for n=1..13.

Alex Bellos, Can you solve it? Sexy maths, 2. The 7-Eleven, The Guardian, 3 Feb 2025.

Example

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.

Prog

(PARI) hasTuple(n, divs, currentTuple, currentSum, currentPro, targetSum, targetPro, i0) = {
  my(newSum, newPro);
  if(#currentTuple == n, return(currentSum == targetSum && currentPro == targetPro));
  for(i = i0, #divs,
    newSum = currentSum + divs[i];
    newPro = currentPro * divs[i];
    if(newSum <= targetSum && targetPro % newPro == 0,
      if(hasTuple(n, divs, concat(currentTuple, [divs[i]]), newSum, newPro, targetSum, targetPro, i), return(1));
    );
  );
  return(0);
}
hasSolution(n, targetSum) = {
  my(targetPro = targetSum * 100^(n-1));
  hasTuple(n, divisors(targetPro), [], 0, 1, targetSum, targetPro, 1);
};
a(n) = {
  if(n == 1, return(1));
  for(s = ceil(100 * n^(n/(n-1))), +oo, if(hasSolution(n, s), return(s)));
};
for (n = 1, 5, print(a(n)));

Crossrefs

Cf. A381187, A381619, A381621.

Sequence in context: A223284 A061042 A223481 * A037991 A200837 A250847

Adjacent sequences: A380884 A380885 A380886 * A380888 A380889 A380890

Keyword

nonn,more,changed

Author

Markus Sigg, Feb 07 2025

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

Status

approved