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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 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

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..48

Ray Chandler, Known terms and upper limits.

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

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 Fröhlich, 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

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Last modified April 20 12:07 EDT 2021. Contains 343135 sequences. (Running on oeis4.)