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A237350 a(n) = the smallest number k such that Sum_{d|k} 1/tau(d) >= n. 5
1, 6, 24, 60, 180, 420, 840, 2520, 4620, 9240, 13860, 27720, 55440, 55440, 110880, 166320, 180180, 360360, 360360, 720720, 720720, 1441440, 1801800, 2162160, 3063060, 4084080, 6126120, 6126120, 6126120, 12252240, 12252240, 18378360, 24504480, 24504480, 30630600, 36756720 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Are there numbers n > 1 such that Sum_{d|n} 1/tau(d) is an integer?

Values of function F = Sum_{d|n} 1/tau(d) for some numbers according to their prime signature: F{} = 1; F{1} = 3/2; F{2} = 11/6; F{1, 1} = 9/4; F{3} = 25/12; F{2, 1} = 11/4; F{4} = 137/60; F{3, 1} = 25/8, ...

All terms are of the form Product_{j=1..k} prime(j)^e(j) where e(j+1)<= e(j), and thus products of (not necessarily distinct) primorials. - Robert Israel, Dec 21 2015

From David A. Corneth, Nov 05 2019: (Start)

Instead of checking all divisors of A025487(n), one could use A318277 to see how often each prime signature occurs as a divisor.

Knowing the lcm of the terms below some m drastically improves the possibility of finding terms. In hindsight, knowing the lcm of the terms below 10^25 yields having to consider 1056 terms of A025487 instead of 222124. Is there some way to accurately predict the lcm to improve computation? (End)

LINKS

David A. Corneth, Table of n, a(n) for n = 1..3338 (first 131 terms from Robert Israel, terms <= 10^25)

David A. Corneth, m, Sum_{d|a(m)} 1/tau(d) and the prime signature of a(m)

EXAMPLE

For n = 2; a(2) = 6 because 6 is the smallest number with Sum_{d|6} 1/tau(d) = 1/1 + 1/2 + 1/2 + 1/4 = 9/4 >= 2.

MAPLE

N:= 10^9: # to get all entries <= N

Primorials:= NULL:

p:= 2: P:= p:

while P <= N do

  Primorials:= Primorials, P;

  p:= nextprime(p);

  P:= P*p;

od:

Primorials:= [Primorials]:

S:= {1}:

for i from 1 to nops(Primorials) do

  S:= {seq(seq(s*Primorials[i]^j,

       j = 0 .. floor(log[Primorials[i]](N/s))), s=S)}

od:

A:= NULL:

S:= sort(convert(S, list)):

xmax:= 0:

for s in S do

  x:= floor(add(1/numtheory:-tau(d), d=numtheory:-divisors(s)));

  if x > xmax then

     A:= A, s$(x-xmax);

     xmax:= x

  fi

od:

A; # Robert Israel, Dec 21 2015

MATHEMATICA

s[1] = 1; s[n_] := DivisorSum[n, 1/DivisorSigma[0, #] &]; n = 1; k = 1; seq = {}; Do[While[s[k] < n, k++]; AppendTo[seq, k]; n++, {j, 1, 20}]; seq (* Amiram Eldar, Jan 30 2019 *)

PROG

(MAGMA) a:=1; S:=[a]; for n in [2..14] do k:=0; flag:= true; while flag do k+:=1; if &+[1/NumberOfDivisors(d): d in Divisors(k)] gt n then Append(~S, k); a:=k; flag:=false; end if; end while; end for; S;

(PARI) a(n) = {my(k=1); while(sumdiv(k, d, 1/numdiv(d)) < n, k++); k; } \\ Michel Marcus, Dec 20 2015

CROSSREFS

Cf. A000005, A002110, A025487, A253139, A265390, A265391, A265392, A318277.

Cf. A265393 (a(n) = the smallest number k such that floor(Sum_{d|k} 1/tau(d)) = n).

Sequence in context: A258351 A130669 A214308 * A265393 A292908 A293017

Adjacent sequences:  A237347 A237348 A237349 * A237351 A237352 A237353

KEYWORD

nonn

AUTHOR

Jaroslav Krizek, Dec 13 2015

EXTENSIONS

a(24)-a(30) from Michel Marcus, Dec 20 2015

a(31)-a(35) from Robert Israel, Dec 21 2015

Missing a(31) = 12252240 inserted in data section by Georg Fischer, Nov 05 2019

STATUS

approved

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Last modified July 29 17:41 EDT 2021. Contains 346346 sequences. (Running on oeis4.)