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 A378470 #7 Dec 12 2024 23:25:36
%S A378470 3,78,10728,28920,53752896,4157280,278628512256,90323520,1658908800,
%T A378470 21499810560,7487812494923563008,13005699840,155267279705546496147456,
%U A378470 111451576596480,8599694054400,468208581120,4172630516011611848266349543424,5202323481600,21630916595004029113587563614961664,67421367982080
%N A378470 a(n) is the smallest number k for which the width pattern of the symmetric representation of sigma(k), SRS(k), consists of two unimodal parts of maximum width n.
%C A378470 Let the nonincreasing multiset cL = { c_1, ... , c_s } be a factorization of n, let dL = { d_1, ... , d_s } be any set of s distinct odd primes, let q = dL^(cL - 1) = d_1^(c_1 - 1) * ... * d_s^(c_s - 1), and let k satisfy 2^k < q < 2^(k+1). Then SRS(2^k * q) is unimodal of maximum height n, 2^k * q has 2n odd divisors and its width pattern has 2n-1 entries. The smallest possible choice for 2^k * q is with the increasing sequence of odd primes d_i = p_(i+1), 1 <= i <= s. The overall smallest 2^k * q is the minimum among all factorizations of n. The smallest number m for which SRS(m) has two unimodal parts of maximum width n requires the additional prime factor r > 2^(k+1) * q which yields m = 2^k * q * r.
%C A378470 This sequence is column 2 in the array of A367377 and a(2) = A370206(1).
%F A378470 a(p) = 2^k * 3^(p-1) * r, for odd primes p, with 2^k < 3^(p-1) < 2^(k+1) and r > 2^(k+1) * 3^(p-1) least prime, i.e., k = floor( (p-1)*(log_2 (3)) ) and r = prime( primepi(2^(k+1) * 3^(p-1)) + 1 ).
%e A378470 a(2) = 78 is in the sequence since SRS(78) consists of two parts with width pattern 1 2 1 0 1 2 1 and 78 is the smallest number with those properties.
%e A378470 a(3) = 10728 = 2^3 * 3^2 * 149 is in the sequence since SRS(10728) consists of two parts with width pattern 1 2 3 2 1 0 1 2 3 2 1 and 10728 is the smallest number with those properties.
%e A378470 a(6) = 4157280 = 2^5 * 3^2 * 5 * 2887 is in the sequence. The two factorizations of 6 are {6} and {3, 2} so that with 3^5 = 243 and 3^2 * 5^1 = 45 the inequality 2^5 < 45 < 2^6 determines the single unimodular SRS(32 * 45) of maximum width 6, A250071(6) = 1440. Since 2887 is the smallest prime exceeding 2^6 * 3^2 * 5, 4157280 is the smallest number with SRS(4157280) consisting of two unimodular parts of maximum width 6.
%t A378470 (* function f[ ] by _T. D. Noe_ in A162247 *)
%t A378470 sF[n_] := Min[Map[Apply[Times, Prime[Range[2, Length[#]+1]]^#]&, Map[Reverse[#-1]&, f[n]]]]
%t A378470 f2U[n_] := Module[{s=sF[n], k, p}, k=Floor[Log[2, s]]; p=NextPrime[2^(k+1) s]; 2^k s p]
%t A378470 a378470[n_] := Map[f2U, Range[n]]
%t A378470 a378470[20]
%Y A378470 Cf. A003056, A082662, A162247, A235791, A237270, A237591, A237593, A239929, A250071, A341969, A370206.
%K A378470 nonn
%O A378470 1,1
%A A378470 _Hartmut F. W. Hoft_, Nov 27 2024