

A075036


Smaller of two smallest consecutive numbers with 2n divisors.


3



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


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


LINKS

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


FORMULA

a(n) <= A215199(n1) for n > 1. Conjecture: if p is prime, then a(p) = A215199(p1). This conjecture is true if the conjecture in A215199 is true. The bfile of A215199 thus shows that a(p) = A215199(p1) for prime p < 1279.  Chai Wah Wu, Mar 12 2019


EXAMPLE

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


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 *)


PROG

(PARI) a(n) = my(k=1); while(numdiv(k)!=2*n  numdiv(k+1)!=2*n, k++); k \\ Felix FrÃ¶hlich, Jun 24 2016


CROSSREFS

Cf. A000005, A005237, A039832, A049103, A174456, A215197, A215199, A274357, A274358, A274359.
Sequence in context: A195960 A268684 A333052 * A212902 A091405 A085929
Adjacent sequences: A075033 A075034 A075035 * A075037 A075038 A075039


KEYWORD

nonn


AUTHOR

Amarnath Murthy, Sep 03 2002


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


STATUS

approved



