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 A111869 #30 Aug 16 2025 10:05:15
%S A111869 1,3,5,15,13,5,25,63,41,13,61,15,85,25,14,255,145,41,181,23,25,61,265,
%T A111869 95,313,85,365,27,421,14,481,1023,61,145,39,53,685,181,86,63,841,25,
%U A111869 925,61,44,265,1105,383,1201,313,145,85,1405,365,63,95,181,421,1741,23,1861
%N A111869 Least number k such that C(2k,k) is divisible by n^2.
%C A111869 From _David A. Corneth_, Aug 15 2025: (Start)
%C A111869 Conjecture 1: a(n) = (n^2 - 1)/2 + 1 for odd prime n.
%C A111869 Conjecture 2: Let q be the largest prime factor of n. Let e be the multiplicity of q in the factorization of n. Then a(n) >= (q^(2*e) - 1)/2 + 1. for n != 2.
%C A111869 These conjectures hold for n = 1..4002.
%C A111869 Conjecture 3: a(2^k) = 4^k - 1 for k >= 1.
%C A111869 This conjecture holds for k = 1..11. (End)
%H A111869 David A. Corneth, <a href="/A111869/b111869.txt">Table of n, a(n) for n = 1..4002</a> (first 500 terms from Seiichi Manyama, terms n = 501..840 from Chai Wah Wu)
%H A111869 David A. Corneth, <a href="/A111869/a111869_1.gp.txt">PARI program</a>
%H A111869 David A. Corneth, <a href="/A111869/a111869.gp.txt">Upper bounds on a(n) for n = 1..10000</a>
%F A111869 a(n) = A073078(n^2).
%e A111869 From _David A. Corneth_, Aug 15 2025: (Start)
%e A111869 a(4) = 15 as 4^2 = 16 | binomial(2*15, 15) = binomial(30, 15) and for any k < 15 we have 16 does not divide binomial(2*k, k). We don't really need to compute binomial(30, 15) and not the previous binomial(2*k, k) but just find how many factors 2 they have. binomial(30, 15) = 30! / (15!)^2.
%e A111869 We find the 2-adic valuation of 30! as follows: let b(0) = 30 and let b(n + 1) = floor(b(n) / 2). Then the 2-adic valuation of 30! is Sum{k = 1..floor(log(30)/log(2))} b(k) = b(1) + b(2) + b(3) + b(4) = 15 + 7 + 3 + 1 = 26.
%e A111869 Similar for 15! it is 7 + 3 + 1 = 11. 26 - 2*11 = 4 >= 4 so a(4) <= 15 and checking the others gives a(4) = 15. (End)
%t A111869 f[n_] := Block[{k = 1, m = n^2}, While[ Mod[ Binomial[2k, k], m] != 0, k++ ]; k]; Array[f, 61]
%o A111869 (PARI) See Corneth link
%Y A111869 Cf. A000984, A059097, A073078, A370980.
%K A111869 nonn
%O A111869 1,2
%A A111869 _Robert G. Wilson v_, Nov 23 2005