A382093 -id:A382093 - cp's OEIS frontend

A381522 Sequence where k is appended after every k^2 occurrences of 1, with multiple values following a 1 listed in order.

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 2, 4, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 5, 1, 1, 3, 1, 2, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 2, 3, 6, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 4, 1, 7, 1, 5, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 4, 8, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 6

Offset: 1

Jwalin Bhatt, Feb 26 2025

nonn

The frequencies of the terms follow the zeta distribution with parameter value 2.
The geometric mean approaches exp(-zeta'(2)/zeta(2)) A381456 in the limit. In general, if the sequence was formed by every k^s occurrences, it would approach e^(-zeta'(s)/zeta(s)).
Considered as an irregular triangle, the n-th row lists the divisors of the square root of the largest square dividing n.

			After every 4 ones we see a 2, after every 9 ones we see a 3 and so on.

Jwalin Bhatt, Table of n, a(n) for n = 1..10000
Wikipedia, Zeta distribution

Cf. A000188, A084580, A381456, A381900, A382093.

lista(n)={my(L=List()); for(n=1, n, fordiv(sqrtint(n/core(n)), d, listput(L,d))); Vec(L[1..n])} \\ Andrew Howroyd, Feb 26 2025

from itertools import islice
def zeta_distribution_generator():
    num_ones, num_reached = 0, 1
    while num_ones := num_ones+1:
        yield 1
        for num in range(2, num_reached+2):
            if num_ones % (num*num) == 0:
                yield num
                num_reached += num == num_reached+1
A381522 = list(islice(zeta_distribution_generator(), 120))

A383238 A sequence constructed by greedily sampling the Poisson distribution for parameter value 1, 1/(e*(i-1)!) to minimize discrepancy selecting the smallest value in case of ties.

Original entry on oeis.org

1, 2, 3, 1, 2, 4, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 5, 1, 2, 3, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 4, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 4, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 4, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 5, 1, 2, 3, 1, 2, 4, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 4, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 6, 1, 2, 3, 1, 2

Offset: 1

Jwalin Bhatt, Apr 20 2025

nonn

The geometric mean approaches A382095 = exp(Sum_{k>=2} log(k)/(k-1)!) in the limit.

			Let p(k) denote the probability of k and c(k) denote the count of occurrences of k so far, then the expected occurrences of k at n-th step are given by n*p(k).
We subtract the actual occurrences c(k) from the expected occurrences and pick the one with the highest value.
| n | n*p(1) - c(1) | n*p(2) - c(2) | n*p(3) - c(3) | choice |
|---|---------------|---------------|---------------|--------|
| 1 |     0.367     |     0.367     |     0.183     |   1    |
| 2 |    -0.264     |     0.735     |     0.367     |   2    |
| 3 |     0.103     |     0.103     |     0.551     |   3    |
| 4 |     0.471     |     0.471     |    -0.264     |   1    |
| 5 |    -0.160     |     0.839     |    -0.080     |   2    |

Jwalin Bhatt, Table of n, a(n) for n = 1..10000
Wikipedia, Poisson distribution

Cf. A381617, A382093, A382095, A382961.

probCountDiff[j_,k_,count_]:=N[k/(E*Factorial[j-1])]-Lookup[count,j,0]
samplePDF[n_]:=Module[{coeffs,unreachedVal,counts,k,probCountDiffs,mostProbable},
  coeffs=ConstantArray[0,n];unreachedVal=1;counts=<||>;
  Do[probCountDiffs=Table[probCountDiff[i,k,counts],{i,1,unreachedVal}];
    mostProbable=First@FirstPosition[probCountDiffs,Max[probCountDiffs]];
    If[mostProbable==unreachedVal,unreachedVal++];coeffs[[k]]=mostProbable;
    counts[mostProbable]=Lookup[counts,mostProbable,0]+1;,{k,1,n}];coeffs]
A383238=samplePDF[120]

A382095 Decimal expansion of exp((Sum_{k>=2} log(k)/(k-1)!)/e).

Original entry on oeis.org

1, 7, 7, 4, 2, 9, 4, 3, 7, 5, 7, 8, 8, 8, 1, 3, 0, 6, 3, 4, 0, 6, 2, 8, 6, 5, 7, 3, 1, 9, 7, 1, 0, 8, 9, 4, 2, 9, 2, 4, 2, 2, 2, 9, 1, 4, 2, 9, 7, 5, 4, 2, 1, 8, 0, 1, 4, 8, 0, 8, 5, 1, 7, 2, 5, 1, 0, 0, 4, 1, 3, 1, 8, 2, 1, 1, 5, 7, 6, 3, 9, 1, 0, 6, 3, 8, 7, 2, 7, 4, 9, 6, 0, 8, 5, 1, 4, 2, 6, 7, 7, 5, 3, 8, 9, 4, 3, 3, 0, 3, 6, 2, 7, 5, 3, 0, 0, 6, 8, 2

Offset: 1

Jwalin Bhatt, Mar 25 2025

The geometric mean of the Poisson distribution with parameter value 1 (A382093) approaches this constant.

			1.774294375788813063406286573197109...

Cf. A193424, A381456, A381898, A382093.

N[Exp [Sum[Log[i]/Factorial[i-1], {i, 2, Infinity}] / E ], 120]

Equals exp((Sum_{k>=2} log(k)/(k-1)!)/e) = exp(A193424/e).

Equals (Product_{k>=2} k^(1/(k-1)!)) ^ (1/e).

A385685 Sequence where k is appended after every k! occurrences of 1, with multiple values following a 1 listed in order.

Original entry on oeis.org

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

Offset: 0

Jwalin Bhatt, Jul 06 2025

nonn

The frequencies of the terms follow the Poisson distribution with parameter value 1.

The geometric mean approaches A385686 in the limit. In general, for parameter value p it approaches Product_{k>=2} k^(((p^(k-1))*((e-1)^(-p)))/k!).

			Every 1 is followed by a 1 because 1! = 1,
after every (2!=2) ones we see a 2,
after every (3!=6) ones we see a 3 and so on.

Jwalin Bhatt, Table of n, a(n) for n = 0..9999
Wikipedia, Poisson distribution

Cf. A000142 (n!), A382093, A385686.

A385685[n_] := Module[{N1 = 0,NR= 1, result = {},i=1}, While[Length[result] < n,N1++;AppendTo[result, 1]; Do[If[Mod[N1, Factorial[k]] == 0,    AppendTo[result, k];f[k == NR + 1, NR++]],{k, 2, NR + 1}];If[Length[result] > n, result = Take[result, n]]];result];A385685[92] (* James C. McMahon, Jul 11 2025 *)

from itertools import islice
from math import factorial
def poisson_distribution_generator():
    num_ones, num_reached = 0, 1
    while num_ones := num_ones+1:
        yield 1
        for num in range(2, num_reached+2):
            if num_ones % factorial(num) == 0:
                yield num
                num_reached += num == num_reached+1
A385685 = list(islice(poisson_distribution_generator(), 120))

cp's OEIS Frontend

A381522 Sequence where k is appended after every k^2 occurrences of 1, with multiple values following a 1 listed in order.

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A383238 A sequence constructed by greedily sampling the Poisson distribution for parameter value 1, 1/(e*(i-1)!) to minimize discrepancy selecting the smallest value in case of ties.

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A382095 Decimal expansion of exp((Sum_{k>=2} log(k)/(k-1)!)/e).

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A385685 Sequence where k is appended after every k! occurrences of 1, with multiple values following a 1 listed in order.

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Examples

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Mathematica

Python