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.

3, 78, 10728, 28920, 53752896, 4157280, 278628512256, 90323520, 1658908800, 21499810560, 7487812494923563008, 13005699840, 155267279705546496147456, 111451576596480, 8599694054400, 468208581120, 4172630516011611848266349543424, 5202323481600, 21630916595004029113587563614961664, 67421367982080

Offset: 1

Author

Hartmut F. W. Hoft, Nov 27 2024

Keywords

• nonn

Comments

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.

This sequence is column 2 in the array of A367377 and a(2) = A370206(1).

Examples

			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.
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.
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.
		

Crossrefs

Cf. A003056, A082662, A162247, A235791, A237270, A237591, A237593, A239929, A250071, A341969, A370206.

Programs

Mathematica

(* function f[ ] by T. D. Noe in A162247 *)
sF[n_] := Min[Map[Apply[Times, Prime[Range[2, Length[#]+1]]^#]&, Map[Reverse[#-1]&, f[n]]]]
f2U[n_] := Module[{s=sF[n], k, p}, k=Floor[Log[2, s]]; p=NextPrime[2^(k+1) s]; 2^k s p]
a378470[n_] := Map[f2U, Range[n]]
a378470[20]

Formula

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 ).