A253139 -id:A253139 - cp's OEIS frontend

A334470 a(n) = Product_{d|n} (A253139(n) / tau(d)) where A253139(n) = lcm_{d|n} tau(d).

1, 2, 2, 36, 2, 16, 2, 864, 36, 16, 2, 10368, 2, 16, 16, 6480000, 2, 10368, 2, 10368, 16, 16, 2, 11943936, 36, 16, 864, 10368, 2, 4096, 2, 64800000, 16, 16, 16, 2176782336, 2, 16, 16, 11943936, 2, 4096, 2, 10368, 10368, 16, 2, 1343692800000000, 36, 10368, 16

Offset: 1

Jaroslav Krizek, May 01 2020

nonn

			For n = 6; divisors d of 6: {1, 2, 3, 6}; tau(d): {1, 2, 2, 4}; lcm_{d|6} tau(d) = 4; a(6) = 4/1 * 4/2 * 4/2 * 4/4 = 16.

Cf. A334471 (similar sequence with sigma(d)).

Cf. A000005, A000040, A211776, A253139.

[&*[ LCM([#Divisors(d): d in Divisors(n)]) / #Divisors(d): d in Divisors(n)]: n in [1..100]]

a[n_] := (LCM @@ (s = DivisorSigma[0, Divisors[n]]))^Length[s] / Times @@ s; Array[a, 51] (* Amiram Eldar, May 02 2020 *)

a(n) = {my(d=divisors(n), lcmd = lcm(vector(#d, k, numdiv(d[k])))); vecprod(vector(#d, k, lcmd/numdiv(d[k])));} \\ Michel Marcus, May 02 2020

a(n) = ((lcm_{d|n} tau(d))^tau(n)) / Product_{d|n} tau(d).
a(n) = A253139(n)^A000005(n) / A211776(n).
a(p) = 2 for p = primes (A000040).
a(n) = 2^(k*2^(k-1)) if n is a product of k distinct primes.

A069934 a(n) = lcm_{d|n} sigma(d).

Original entry on oeis.org

1, 3, 4, 21, 6, 12, 8, 105, 52, 18, 12, 84, 14, 24, 24, 3255, 18, 156, 20, 126, 32, 36, 24, 420, 186, 42, 520, 168, 30, 72, 32, 9765, 48, 54, 48, 1092, 38, 60, 56, 630, 42, 96, 44, 252, 312, 72, 48, 13020, 456, 558, 72, 294, 54, 1560, 72, 840, 80, 90, 60, 504, 62, 96

Offset: 1

Vladeta Jovovic, Apr 25 2002

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to lcm's

Cf. A000203, A051549, A253139.

Table[LCM@@(DivisorSigma[1,#]&/@Divisors[n]),{n,70}] (* Harvey P. Dale, Feb 22 2023 *)

A069934(n) = my(d = divisors(n)); lcm(vector(#d, k, sigma(d[k]))); \\ Antti Karttunen, Sep 10 2017

Multiplicative with a(p^e) = Product_{k=2..e+1} Phi_k(p), where Phi_k(x) is k-th cyclotomic polynomial.

A265391 a(n) = numerator of Sum_{d|n} 1 / tau(d).

Original entry on oeis.org

1, 3, 3, 11, 3, 9, 3, 25, 11, 9, 3, 11, 3, 9, 9, 137, 3, 11, 3, 11, 9, 9, 3, 25, 11, 9, 25, 11, 3, 27, 3, 49, 9, 9, 9, 121, 3, 9, 9, 25, 3, 27, 3, 11, 11, 9, 3, 137, 11, 11, 9, 11, 3, 25, 9, 25, 9, 9, 3, 33, 3, 9, 11, 363, 9, 27, 3, 11, 9, 27, 3, 275, 3, 9, 11

Offset: 1

Jaroslav Krizek, Dec 08 2015

a(n) = numerator of Sum_{d|n} 1 / A000005(d).

			For n = 6; divisors d of 6: {1, 2, 3, 6}; tau(d): {1, 2, 2, 4}; Sum_{d|6} 1 / tau(d) = 1/1 + 1/2 + 1/2 + 1/4 = 9 / 4; a(n) = 9 (numerator).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000005, A253139, A265390, A265392 (denominator), A265393.

[Numerator(&+[1/NumberOfDivisors(d): d in Divisors(n)]): n in [1..100]]

Table[Numerator[Sum[1/DivisorSigma[0, d], {d, Divisors@ n}]], {n, 75}] (* Michael De Vlieger, Dec 09 2015 *)

a(n) = numerator(sumdiv(n, d, 1/numdiv(d))); \\ Michel Marcus, Dec 09 2015

a(n) = [Sum_{d|n} 1 / tau(d)] * A265392(n) = A265390(n) * A265392(n) / A253139(n).

a(1) = 1; a(p) = 3 for p = prime.

A265392 a(n) = denominator of Sum_{d|n} 1 / tau(d).

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 2, 12, 6, 4, 2, 4, 2, 4, 4, 60, 2, 4, 2, 4, 4, 4, 2, 8, 6, 4, 12, 4, 2, 8, 2, 20, 4, 4, 4, 36, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 2, 40, 6, 4, 4, 4, 2, 8, 4, 8, 4, 4, 2, 8, 2, 4, 4, 140, 4, 8, 2, 4, 4, 8, 2, 72, 2, 4, 4, 4, 4, 8, 2, 40, 60, 4, 2

Offset: 1

Jaroslav Krizek, Dec 08 2015

a(n) = denominator of Sum_{d|n} 1 / A000005(d).

			For n = 6; divisors d of 6: {1, 2, 3, 6}; tau(d): {1, 2, 2, 4}; Sum_{d|6} 1 / tau(d) = 1/1 + 1/2 + 1/2 + 1/4 = 9 / 4; a(n) = 4 (denominator).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000005, A253139, A265390, A265391 (numerator), A265392, A265393.

[Denominator(&+[1/NumberOfDivisors(d): d in Divisors(n)]): n in [1..1000]]

Table[Denominator[Sum[1/DivisorSigma[0, d], {d, Divisors@ n}]], {n, 83}] (* Michael De Vlieger, Dec 09 2015 *)

a(n) = denominator(sumdiv(n, d, 1/numdiv(d))); \\ Michel Marcus, Dec 09 2015

a(n) = A265391(n) / [Sum_{d|n} 1 / tau(d)] = A265391(n) * A253139(n) / A265390(n).

a(1) = 1; a(p) = 2 for p = prime; a(n) = n for numbers 1, 2, 36, 72, ...

A265390 a(n) = lcm_{d|n} tau(d) * Sum_{d|n} 1/tau(d), where tau(d) represents the number of divisors of d (A000005(d)).

Original entry on oeis.org

1, 3, 3, 11, 3, 9, 3, 25, 11, 9, 3, 33, 3, 9, 9, 137, 3, 33, 3, 33, 9, 9, 3, 75, 11, 9, 25, 33, 3, 27, 3, 147, 9, 9, 9, 121, 3, 9, 9, 75, 3, 27, 3, 33, 33, 9, 3, 411, 11, 33, 9, 33, 3, 75, 9, 75, 9, 9, 3, 99, 3, 9, 33, 1089, 9, 27, 3, 33, 9, 27, 3, 275, 3, 9, 33, 33, 9, 27, 3, 411, 137, 9, 3, 99, 9, 9, 9, 75, 3, 99, 9, 33

Offset: 1

Jaroslav Krizek, Dec 08 2015

			For n = 6; divisors d of 6: {1, 2, 3, 6}; tau(d): {1, 2, 2, 4}; LCM_{d|6} tau(d) = 4; a(6) = 4/1 + 4/2 + 4/2 + 4/4 = 9.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A025529, A253139, A265391, A265392, A265393.

[&+[LCM([NumberOfDivisors(d): d in Divisors(n)]) / NumberOfDivisors(d): d in Divisors(n) ]: n in [1..100]]

Table[LCM @@ DivisorSigma[0, Divisors@ n] Sum[1/DivisorSigma[0, d], {d, Divisors@ n}], {n, 74}] (* Michael De Vlieger, Dec 09 2015 *)

A253139(n) = my(d = divisors(n)); lcm(vector(#d, k, numdiv(d[k]))); \\ This function from Michel Marcus, Jan 23 2015
A265390(n) = (A253139(n) * sumdiv(n,d,(1/numdiv(d)))); \\ Antti Karttunen, Nov 24 2017

a(n) = A253139(n) * Sum_{d|n} 1/A000005(d) = A265391(n) * A253139(n) / A265392(n).

Multiplicative with a(p^e) = A025529(e+1) = (1/1 + 1/2 + 1/3 + ... + 1/(e+1)) * lcm{1, 2, 3, ..., e+1}.

More terms from Antti Karttunen, Nov 24 2017

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

Original entry on oeis.org

1, 6, 24, 60, 180, 420, 840, 2520, 4620, 9240, 13860, 27720, 60060, 55440, 110880, 166320, 180180, 480480, 360360, 900900, 720720, 1441440, 1801800, 2162160, 3063060, 4084080, 7207200, 12612600, 6126120, 27027000, 12252240, 18378360, 43243200, 24504480

Offset: 1

Jaroslav Krizek, Dec 08 2015

nonn

Further known terms: a(29) = 6126120, a(31) = 12252240.
Are there numbers n > 1 such that Sum_{d|n} 1/tau(d) is an integer?
Sequences of numbers n such that floor(Sum_{d|n} 1/tau(d)) = k for k = 1..6:
k=1: 1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, ... (A166684);
k=2: 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 26, 27, 28, 32, 33, 34, 35, ...;
k=3: 24, 30, 36, 40, 42, 48, 54, 56, 66, 70, 72, 78, 80, 88, 96, 100, ...;
k=4: 60, 84, 90, 120, 126, 132, 140, 144, 150, 156, 168, 198, 204, 216, ...;
k=5: 180, 210, 240, 252, 300, 330, 336, 360, 390, 396, 450, 462, 468, ...;
k=6: 420, 630, 660, 720, 780, 900, 924, 990, 1008, 1020, 1050, 1080, ....
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, ...

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

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..5000

Cf. A000005, A253139, A265390, A265391, A265392.

Cf. A237350 (a(n) = the smallest number k such that Sum_{d|k} 1/tau(d) >= n).

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

Table[k = 1; While[Floor@ Sum[1/DivisorSigma[0, d], {d, Divisors@ k}] != n, k++]; k, {n, 17}] (* Michael De Vlieger, Dec 09 2015 *)

a(n) = {k=1; while(k, if(floor(sumdiv(k, d, 1/numdiv(d))) == n, return(k)); k++)} \\ Altug Alkan, Dec 09 2015

More terms from Michel Marcus, Dec 23 2015

a(33)-a(34) from Hiroaki Yamanouchi, Dec 31 2015

A237350 a(n) = the smallest number k such that Sum_{d|k} 1/tau(d) >= n.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Dec 13 2015

nonn

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)

			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.

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)

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).

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;

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

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 *)

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

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

A250270 Products of terms of A003418.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 192, 216, 240, 256, 288, 360, 384, 420, 432, 480, 512, 576, 720, 768, 840, 864, 960, 1024, 1152, 1296, 1440, 1536, 1680, 1728, 1920, 2048, 2160, 2304, 2520, 2592, 2880, 3072, 3360, 3456

Offset: 1

Matthew Vandermast, Dec 16 2014

nonn

Includes all factorials and Jordan-Polya numbers, since n! = Product_{i = 1..n} A003418(floor(n/i)) for positive n.

			720 = 2*6*60 = 12*60. Since 2, 6, 12 and 60 are all terms of A003418, 720 is a term of this sequence.

Michel Marcus, Table of n, a(n) for n = 1..5000

Range of values of A253139. Subsequences include A000142, A001013, A001813, A025527, A064350, A166338, A250569.

Subsequence of A025487.

f(n) = lcm(vector(n, i, i)); \\ A003418
mul(x,y) = x*y;
lista(nn) = {my(v = vector(nn, k, f(k)), lim = f(nn+1), ok = 0, nv); while (!ok,  nv = select(x->(xMichel Marcus, May 09 2021

A253141 If n is a prime power, then a(n) = lambda(tau(n)) = A014963(A000005(n)); otherwise, a(n) = 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 1, 2, 2, 3, 1, 2, 1, 2, 1, 1, 5, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 7, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 5, 1, 2, 1, 1, 1, 1

Offset: 1

Matthew Vandermast, Dec 27 2014

For any integer sequence a, the sequence b such that b(n) = Product_{d|n} a(d) is a divisibility sequence. Since A253139(n) = Product_{d|n} a(d), A253139 is a divisibility sequence.

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

			2 is a prime number, i.e., a prime power with 2 divisors; a(2) = A014963(2) = 2.
6 = 2*3 is not a prime power; a(6) = 1.
8 = 2^3 is a prime power with 4 divisors; a(8) = A014963(4) = 2.
32 = 2^5 is a prime power with 6 divisors; a(32) = A014963(6) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Morgan Ward, A note on divisibility sequences, Bull. Amer. Math. Soc., 45 (1939), 334-336.
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A014963, A253139.

Table[If[PrimePowerQ[n], Exp[MangoldtLambda[DivisorSigma[0, n]]], 1], {n, 1, 100}] (* Indranil Ghosh, Jul 19 2017 *)

A014963(n) = ispower(n, , &n); if(isprime(n), n, 1); \\ This function from Charles R Greathouse IV, Jun 10 2011
A253141(n) = if(1==omega(n), A014963(numdiv(n)), 1); \\ Antti Karttunen, Jul 19 2017

A266226 a(n) = floor(Sum_{d|n} 1 / tau(d)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3

Offset: 1

Jaroslav Krizek, Dec 24 2015

a(n) = floor(Sum_{d|n} 1 / A000005(d)).
Sequences of numbers n such that floor(Sum_{d|n} 1/tau(d)) = k for k = 1..6:
k=1: 1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, ... (A166684);
k=2: 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 26, 27, 28, 32, 33, 34, 35, ...;
k=3: 24, 30, 36, 40, 42, 48, 54, 56, 66, 70, 72, 78, 80, 88, 96, 100, ...;
k=4: 60, 84, 90, 120, 126, 132, 140, 144, 150, 156, 168, 198, 204, 216, ...;
k=5: 180, 210, 240, 252, 300, 330, 336, 360, 390, 396, 450, 462, 468, ...;
k=6: 420, 630, 660, 720, 780, 900, 924, 990, 1008, 1020, 1050, 1080, ....
See A265393 - the smallest number n such that a(n) = k for k>= 1.

			For n = 6; a(6) = floor(Sum_{d|6} 1/tau(d)) = floor(1/1 + 1/2 + 1/2 + 1/4) = floor(9/4) = 2.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000005, A237350, A253139, A265390, A265391, A265392, A265393

[Floor(&+[1/NumberOfDivisors(d): d in Divisors(n)]): n in [1..100]];

Table[Floor[Sum[1/DivisorSigma[0, d], {d, Divisors[ n]}]], {n, 1, 100}] (* G. C. Greubel, Dec 24 2015 *)

cp's OEIS Frontend

A334470 a(n) = Product_{d|n} (A253139(n) / tau(d)) where A253139(n) = lcm_{d|n} tau(d).

Views

Author

Keywords

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A069934 a(n) = lcm_{d|n} sigma(d).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A265391 a(n) = numerator of Sum_{d|n} 1 / tau(d).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A265392 a(n) = denominator of Sum_{d|n} 1 / tau(d).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A265390 a(n) = lcm_{d|n} tau(d) * Sum_{d|n} 1/tau(d), where tau(d) represents the number of divisors of d (A000005(d)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A237350 a(n) = the smallest number k such that Sum_{d|k} 1/tau(d) >= n.

Views

Author

Keywords

Comments

Examples