A162538 -id:A162538 - cp's OEIS frontend

A324990 a(n) = the smallest number k such that floor(sigma(k)/tau(k)) = n, or 0 if no such number k exists.

1, 3, 5, 7, 0, 11, 13, 21, 17, 19, 40, 23, 34, 39, 29, 31, 63, 46, 37, 57, 41, 43, 76, 47, 0, 99, 53, 74, 0, 59, 61, 93, 86, 67, 116, 71, 73, 111, 125, 79, 175, 83, 171, 121, 89, 122, 0, 141, 97, 0, 101, 103, 0, 107, 109, 188, 113, 250, 0, 158, 169, 183, 166

Offset: 1

Jaroslav Krizek, Mar 23 2019

Floor(sigma(n)/tau(n)) = floor(A000203(n)/A000005(n)) = A057022(n) for n >= 1.
Odd primes are terms.
a(n) = 0 for numbers n =  5, 25, 29, 47, 50, 53, 59, 83, 89, ...

			For n = 4; number 7 is the smallest number k with floor(sigma(k)/tau(k)) = 4; floor(sigma(7)/tau(7)) = floor(8/2) = 4.

Robert Israel, Table of n, a(n) for n = 1..2000

Cf. A000005, A000203, A057022, A162538, A324991.

Magma
```
Ax:=func; [Ax(n): n in[1..80]]
```

N:= 100: # for a(1)..a(N)
V:= Vector(N):
for k from 1 to N^2 do
  v:= floor(numtheory:-sigma(k)/numtheory:-tau(k));
  if v <= N and V[v]=0 then V[v]:= k fi
od:
convert(V,list); # Robert Israel, Sep 13 2020

A324991 a(n) = the largest number k such that floor(sigma(k)/tau(k)) = n, or 0 if no such number k exists.

Original entry on oeis.org

2, 4, 8, 12, 0, 18, 24, 21, 30, 36, 40, 48, 45, 60, 56, 72, 63, 84, 90, 75, 96, 120, 108, 112, 0, 144, 110, 140, 0, 180, 160, 156, 136, 67, 116, 210, 240, 200, 198, 252, 175, 224, 208, 225, 288, 228, 0, 360, 336, 0, 172, 315, 0, 330, 272, 420, 294, 306, 0, 396

Offset: 1

Jaroslav Krizek, Mar 23 2019

nonn

Floor(sigma(n)/tau(n)) = floor(A000203(n)/A000005(n)) = A057022(n) for n >= 1.

a(n) = 0 for numbers n = 5, 25, 29, 47, 50, 53, 59, 83, 89, ...

			For n = 4; number 12 is the largest number k with floor(sigma(k)/tau(k)) = 4; floor(sigma(12)/tau(12)) = floor(28/6) = 4.

Cf. A000005, A000203, A057022, A162538, A324990.

[Max([n: n in[1..10^5] | Floor(SumOfDivisors(n)/ NumberOfDivisors(n)) eq k]): k in [1..4]] cat [0] cat [Max([n: n in[1..10^5] | Floor(SumOfDivisors(n)/ NumberOfDivisors(n)) eq k]): k in [6..24]];

A327054 a(n) is the smallest number m such that the antiharmonic mean of the divisors is n, or 0 if no such m exists.

Original entry on oeis.org

1, 0, 4, 0, 0, 0, 9, 0, 0, 0, 16, 0, 20, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 50, 0, 0, 0, 0, 0, 0, 0, 49, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 81, 0, 100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 144, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jaroslav Krizek, Oct 06 2019

nonn

a(n) = the smallest number m such that sigma_2(n) / sigma_1(n) = A001157(m) / A000203(m) = n, or 0 if no such m exists.
Zeros occur if n is not in A176799.
See A000290, A091911 and A162538 for like sequences for geometric, arithmetic and harmonic means of the divisors.

			a(3) = 4 because 4 is the smallest number m with sigma_2(m) / sigma_1(m) = 3; sigma_2(4) / sigma_1(4) = 21 / 7 = 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A020487, A176797, A176799, A176800.

Cf. A000290, A004394, A091911, A162538.

A327054:=func; [A327054(n): n in[1..100]];

# This uses the b-file for A004394
# See comment at A176799
K:= 100: # to get terms <= K
M:= 36 * K^2/Pi^4:
for i from 1 while A004394[i] < M do od:
r:= numtheory:-sigma(A004394[i])/A004394[i]:
V:= Vector(K):
for m from 1 to r*K do
  F:= numtheory:-divisors(m);
  v:= add(d^2, d=F)/add(d, d=F);
  if v::integer and v <= K and V[v] = 0 then V[v]:= m fi;
od:
convert(V,list); # Robert Israel, Sep 05 2024

A355848 Irregular triangle read by rows in which row n lists the numbers whose divisors have arithmetic mean n, or 0 if no such number exists.

Original entry on oeis.org

1, 3, 5, 6, 7, 0, 11, 14, 15, 13, 20, 21, 17, 22, 30, 19, 27, 0, 23, 33, 35, 42, 45, 39, 44, 60, 29, 38, 54, 56, 31, 0, 46, 51, 55, 66, 70, 37, 49, 57, 41, 65, 68, 78, 96, 43, 0, 47, 62, 69, 77, 105, 0, 99, 126, 53, 85, 102, 110, 91, 92, 132, 140, 0, 59, 87, 95, 114, 135, 168

Offset: 1

Mohammed Yaseen, Jul 20 2022

			Triangle begins:
  n=1: 1;
  n=2: 3;
  n=3: 5, 6;
  n=4: 7;
  n=5: 0;
  n=6: 11, 14, 15;
  n=7: 13, 20;
  n=8: 21;
  n=9: 17, 22, 30;
  ...

Cf. A162538 (left border).

Cf. A000005, A000203, A003601, A157847, A324990, A324991.

nmax=30; a={}; For[n=1, n<=nmax, n++, nok=0; For[k=1, k<=n(n+1)/2, k++, If[DivisorSum[k,#&]==n*DivisorSigma[0,k], AppendTo[a,k]; nok=1]]; If[nok==0, AppendTo[a,0]]]; a (* Stefano Spezia, Jul 20 2022 *)

cp's OEIS Frontend

A324990 a(n) = the smallest number k such that floor(sigma(k)/tau(k)) = n, or 0 if no such number k exists.

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A324991 a(n) = the largest number k such that floor(sigma(k)/tau(k)) = n, or 0 if no such number k exists.

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Magma

A327054 a(n) is the smallest number m such that the antiharmonic mean of the divisors is n, or 0 if no such m exists.

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Author

Keywords

Comments

Examples

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Programs

Magma

Maple

A355848 Irregular triangle read by rows in which row n lists the numbers whose divisors have arithmetic mean n, or 0 if no such number exists.

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Mathematica