This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A380887 #81 Jun 04 2025 00:18:41 %S A380887 1,400,525,644,759,864,972,1089,1188,1296,1403,1508,1612,1722,1827, %T A380887 1932,2040,2145,2250,2354,2457,2565,2668,2772,2880,2988,3087,3192, %U A380887 3294,3399,3498,3604,3705,3810,3915,4018,4116,4221,4323,4425,4536,4635,4732,4836,4940 %N A380887 a(n) is the smallest positive integer s that can be partitioned into n positive integers whose product is s * 100^(n-1). %C A380887 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. %C A380887 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. %C A380887 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 %C A380887 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 %C A380887 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 %C A380887 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 %H A380887 David A. Corneth, <a href="/A380887/b380887.txt">Table of n, a(n) for n = 1..300</a> %H A380887 Alex Bellos, <a href="https://www.theguardian.com/science/2025/feb/03/can-you-solve-it-sexy-maths">Can you solve it? Sexy maths</a>, 2. The 7-Eleven, The Guardian, 3 Feb 2025. %H A380887 David A. Corneth, <a href="/A380887/a380887_1.gp.txt">PARI program</a> %H A380887 Markus Sigg, <a href="/A380887/a380887.gp.txt">PARI program suitable for calculating a(n) for up to about n = 30</a>. %e A380887 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. %e A380887 For a(3) the sum is 525 = 150 + 175 + 200. %e A380887 For a(4) it is 644 = 125 + 160 + 175 + 184. %e A380887 For a(5) it is 759 = 125 + 125 + 160 + 165 + 184. %o A380887 (PARI) \\ See Sigg link %o A380887 (PARI) \\ See Corneth link %Y A380887 Cf. A381187, A381619, A381621, A382547. %K A380887 nonn %O A380887 1,2 %A A380887 _Markus Sigg_, Feb 07 2025 %E A380887 a(8)-a(9) from _Hugo Pfoertner_, Feb 13 2025 %E A380887 a(10)-a(12) from _Hugo Pfoertner_, Feb 16 2025 %E A380887 a(13) from _Karl-Heinz Hofmann_, Mar 02 2025 %E A380887 a(14)-a(30) from _Markus Sigg_, Mar 27 2025 %E A380887 a(31)-a(32) from _Markus Sigg_, Apr 23 2025 %E A380887 a(33)-a(45) from _Jinyuan Wang_, May 01 2025