A075036 Smaller of two smallest consecutive numbers with 2n divisors.

2, 14, 44, 104, 2511, 735, 29888, 2295, 6075, 5264, 2200933376, 5984, 689278976, 156735, 180224, 21735, 2035980763136, 223244, 9399153082499072, 458864, 41680575, 701443071, 2503092614937444351, 201824, 2707370000, 29785673727, 46977524, 5475519, 1737797404898095794225152

Offset: 1

Author

Amarnath Murthy, Sep 03 2002

Keywords

• nonn

Comments

There cannot be two consecutive numbers with the same odd number of divisors as both cannot be squares.

These numbers have the property that a(n) * (a(n) + 1) has 4*n^2 divisors. - David A. Corneth, Jun 24 2016

Conjecture: if a term k is even, the highest p-adic order of k (the maximum may be attained by several p's) occurs at p=2 and the highest p-adic order of k+1 occurs at p=3. If a term k is odd, the highest p-adic order of k occurs at p=3 and the highest p-adic order of k+1 occurs at p=2. - Chai Wah Wu, Mar 12 2019

a(49) = 378401464109375, a(58) = 79921490583489592950783. - Jon E. Schoenfield, May 07 2022

a(51) = 34210814718574592, a(55) = 2481402804069375, a(57) = 394311388855795712. - Jon E. Schoenfield, Nov 06 2023 - Nov 08 2023

Examples

			a(4) = 104 as tau(104) = tau(105) = 8.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..48
• Ray Chandler, Known terms and upper limits.

Crossrefs

Cf. A000005, A005237, A039832, A049103, A174456, A215197, A215199, A274357, A274358, A274359.

Programs

Mathematica

a[n_] := (For[k=1, ! (DivisorSigma[0, k] == 2*n && DivisorSigma[0, k+1] == 2*n), k++]; k); Array[a, 10] (* Giovanni Resta, Jun 24 2016 *)

PARI

a(n) = my(k=1); while(numdiv(k)!=2*n || numdiv(k+1)!=2*n, k++); k \\ Felix Fröhlich, Jun 24 2016

Formula

a(n) <= A215199(n-1) for n > 1. Conjecture: if p is prime, then a(p) = A215199(p-1). This conjecture is true if the conjecture in A215199 is true. The b-file of A215199 thus shows that a(p) = A215199(p-1) for prime p < 1279. - Chai Wah Wu, Mar 12 2019

Extensions

a(5)-a(24) from Max Alekseyev, Mar 12 2009

a(25)-a(28) from Giovanni Resta, Jun 24 2016

a(29) from Chai Wah Wu, Mar 12 2019