A124056 -id:A124056 - cp's OEIS frontend

A088167 Number of earlier occurring divisors of n; a(1)=1.

1, 1, 2, 3, 2, 5, 2, 5, 3, 7, 2, 8, 2, 8, 6, 9, 2, 12, 2, 11, 5, 10, 2, 16, 5, 10, 5, 11, 2, 21, 2, 15, 6, 12, 8, 19, 2, 14, 4, 24, 2, 21, 2, 18, 11, 15, 2, 28, 3, 23, 5, 17, 2, 24, 11, 24, 6, 17, 2, 37, 2, 19, 9, 24, 8, 29, 2, 23, 7, 31, 2, 41, 2, 23, 13, 25, 8, 29, 2, 38, 7, 24, 2, 40

Offset: 1

Reinhard Zumkeller, Sep 21 2003

nonn

a(n) = #{k: 1<=k

a(n)=2 iff n is an odd prime;

a(A088168(n))=n and a(k)<>n for k < A088168(n).

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A124056 (counting divisors of a(n) instead of those of n).

A[1]:= 1:
for n from 2 to 1000 do
  A[n]:= numboccur(0, [seq(n mod A[j],j=1..n-1)])
od:
seq(A[n],n=1..1000); # Robert Israel, Sep 20 2015

Fold[Append[#1, Count[#1, x_ /; Divisible[#2, x]]] &, {1}, Range[2, 84]] (* Ivan Neretin, Sep 20 2015 *)

a(1) = 1; a(n) = [x^n] Sum_{k=1..n-1} x^a(k)/(1 - x^a(k)). - Ilya Gutkovskiy, Dec 11 2017

A354606 a(1) = 1; for n > 1, a(n) is number of terms in the first n-1 terms of the sequence that have the same number of divisors as a(n-1).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 3, 4, 1, 4, 2, 5, 6, 1, 5, 7, 8, 2, 9, 3, 10, 3, 11, 12, 1, 6, 4, 4, 5, 13, 14, 5, 15, 6, 7, 16, 1, 7, 17, 18, 2, 19, 20, 3, 21, 8, 9, 6, 10, 11, 22, 12, 4, 7, 23, 24, 1, 8, 13, 25, 8, 14, 15, 16, 2, 26, 17, 27, 18, 5, 28, 6, 19, 29, 30, 2, 31, 32, 7, 33, 20, 8, 21, 22, 23, 34

Offset: 1

Scott R. Shannon, Jul 08 2022

After 250000 terms the most common number of divisors of all terms are 4, 8, 2, 12, 16 divisors. These correspond to the uppermost five lines of the attached image. It is unknown if these stay the most common or are passed by numbers with more divisors as n gets arbitrarily large.

See A355606 for the indices where a(n) = 1.

			a(6) = 2 as a(5) = 3 which has two divisors, and the total number of terms in the first five terms with two divisors is two, namely a(3) = 2 and a(5) = 3.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 250000 terms. The green line is y = n.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Cf. A355606, A000005, A124056, A342585.

nn = 120; c[] := 0; a[1] = j = 1; Do[k = ++c[DivisorSigma[0, j]]; Set[{a[n], j}, {k, k}], {n, 2, nn}]; Array[a, nn] (* _Michael De Vlieger, Dec 12 2024 *)

from sympy import divisor_count
from collections import Counter
def f(n): return divisor_count(n)
def aupton(nn):
    an, fan, alst, inventory = 1, 1, [1], Counter([1])
    for n in range(2, nn+1):
        an = inventory[fan]
        fan = f(an)
        alst.append(an)
        inventory.update([fan])
    return alst
print(aupton(86)) # Michael S. Branicky, Jul 09 2022

A362031 a(1) = 1; for n > 1, a(n) is number of terms in the first n-1 terms of the sequence that have the same number of prime factors, counted with multiplicity, as a(n-1).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 3, 4, 1, 4, 2, 5, 6, 3, 7, 8, 1, 5, 9, 4, 5, 10, 6, 7, 11, 12, 2, 13, 14, 8, 3, 15, 9, 10, 11, 16, 1, 6, 12, 4, 13, 17, 18, 5, 19, 20, 6, 14, 15, 16, 2, 21, 17, 22, 18, 7, 23, 24, 3, 25, 19, 26, 20, 8, 9, 21, 22, 23, 27, 10, 24, 4, 25, 26, 27, 11, 28, 12, 13, 29, 30, 14, 28

Offset: 1

Scott R. Shannon, Apr 06 2023

nonn

After 1 million terms the most common numbers for the number of prime factors of the terms are 3, 2, 4, and 5. These correspond to the uppermost four lines of the attached image. It is unknown if these stay the most common or are passed by numbers with more prime factor as n gets arbitrarily large.

See A362033 for the indices where a(n) = 1.

			a(6) = 2 as the number of prime factors of a(5) = A001222(a(5)) = A001222(3) = 1, and there are two previous terms, a(3) and a(5), that have one prime factor.
a(9) = 1 as the number of prime factors of a(8) = A001222(a(8)) = A001222(4) = 2, and there is only one term, a(8), that has two prime factors.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, with a color function representing Omega(a(n-1)), where black = 0, red = 1, orange = 2, ..., magenta = 14.
Scott R. Shannon, Image of the first 1 million terms.

Cf. A362033, A001222, A354606, A000005, A124056, A342585.

nn = 120; c[] = 0; j = a[1] = c[0] = 1; m = 0; Do[Set[k, c[m]]; (Set[{a[n], j, m}, {k, k, #}]; c[#]++) &[PrimeOmega[k]], {n, 2, nn}]; Array[a, nn] (* _Michael De Vlieger, Apr 06 2023 *)

from sympy import factorint
from itertools import islice
from collections import Counter
def A362031gen(): # generator of terms
    an, c, d = 1, Counter(), dict()
    while True:
        yield an
        pf = d[an] if an in d else sum(factorint(an).values())
        c[pf] += 1
        an = c[pf]
print(list(islice(A362031gen(), 83))) # Michael S. Branicky, Apr 06 2023

from itertools import islice
from sympy import primeomega
def A362031_gen(): # generator of terms
    a, b, c = {}, {}, 1
    while True:
        yield c
        d = b[c] = b.get(c,primeomega(c))
        c = a[d] = a.get(d,0)+1
A362031_list = list(islice(A362031_gen(),20)) # Chai Wah Wu, Apr 08 2023

A359034 a(n+1) is the sum of the number of terms in all groups of contiguous terms that add up to a(n); a(1)=1.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 4, 5, 3, 5, 4, 6, 6, 7, 7, 8, 10, 11, 4, 7, 9, 9, 10, 12, 13, 14, 13, 15, 8, 11, 7, 10, 13, 16, 19, 18, 18, 19, 19, 20, 7, 11, 8, 12, 14, 14, 15, 9, 11, 9, 12, 15, 10, 14, 16, 20, 14, 17, 17, 18, 22, 22, 23, 22, 24, 23, 23, 24, 24, 25, 28, 27, 22

Offset: 1

Neal Gersh Tolunsky, Dec 12 2022

nonn

If strongly smoothened, this sequence displays growth. This growth appears to be caused by the number of groups which is increasing by growing length of the sequence roughly proportional to n^(1/2). But the length of the groups appears to be nearly uninfluenced by this. - Thomas Scheuerle, Dec 14 2022

			a(17) is 10 because in the sequence so far (1, 1, 2, 3, 3, 4, 4, 5, 3, 5, 4, 6, 6, 7, 7, 8), these are the ways of adding contiguous terms to get a(16)=8: (2, 3, 3); (4, 4); (5, 3); (3, 5); (8). This is 10 terms in total, so a(17) is 10. Notice groups (5,3) and (3,5) overlap.

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000
Thomas Scheuerle, Scatter plot of a(n) per number of groups found. For n = 1...20000. This is the mean number of contiguous terms for each n.
Thomas Scheuerle, Scatter plot, number of groups of contiguous terms for n = 1...20000

Cf. A331614, A358919. Begins the same as A124056 (until a(13)).

function a = A359034( max_n )
    a = [1 1];
    for n = 3:max_n
        s = 1; e = 1; sm = 1; c = 0;
        while e < n-1
            while sm < a(n - 1) && e < (n - 1)
                e = e + 1; sm = sm + a(e);
            end
            if sm == a(n - 1)
                c = c + (e - s) + 1;
            end
            s = s + 1;
            e = s; sm = a(s);
        end
        a(n) = c + 1;
    end
end % Thomas Scheuerle, Dec 14 2022

A362061 a(1) = 1; for n > 1, a(n) is number of terms in the first n-1 terms of the sequence that have the same number of distinct prime factors as a(n-1).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 3, 4, 5, 6, 1, 4, 7, 8, 9, 10, 2, 11, 12, 3, 13, 14, 4, 15, 5, 16, 17, 18, 6, 7, 19, 20, 8, 21, 9, 22, 10, 11, 23, 24, 12, 13, 25, 26, 14, 15, 16, 27, 28, 17, 29, 30, 1, 5, 31, 32, 33, 18, 19, 34, 20, 21, 22, 23, 35, 24, 25, 36, 26, 27, 37, 38, 28, 29, 39, 30, 2, 40, 31, 41, 42

Offset: 1

Scott R. Shannon, Apr 06 2023

nonn

After 5 million terms the most common numbers for the number of distinct prime factors of the terms are 3, 2, 4, 1, and 5, although it is likely these change as n increases.

See A362062 for the indices where a term with k distinct prime factors first appears.

			a(9) = 5 as the number of distinct prime factors of a(8) = A001221(a(8)) = A001221(4) = 1, and there are five previous terms, a(3), a(5) a(6), a(7) and a(8), that have one prime factor.
a(11) = 1 as the number of distinct prime factors of a(10) = A001221(a(10)) = A001221(6) = 2, and there is only one term, a(10), that has two prime factors.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20, with a color function representing omega(a(n-1)), where black = 0, red = 1, yellow = 2, ..., magenta = 6.
Scott R. Shannon, Image of the first 250000 terms.

Cf. A362062, A362031, A362033, A001221, A354606, A000005, A124056, A342585.

nn = 120; c[] = 0; j = a[1] = c[0] = 1; m = 0; Do[Set[k, c[m]]; (Set[{a[n], j, m}, {k, k, #}]; c[#]++) &[PrimeNu[k]], {n, 2, nn}]; Array[a, nn] (* _Michael De Vlieger, Apr 06 2023 *)

from itertools import islice
from sympy import primefactors
from collections import Counter
def A362061gen(): # generator of terms
    an, c, d = 1, Counter(), dict()
    while True:
        yield an
        dpf = d[an] if an in d else len(primefactors(an))
        c[dpf] += 1
        an = c[dpf]
print(list(islice(A362061gen(), 81))) # Michael S. Branicky, Apr 06 2023

A362077 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that is a multiple of Omega(a(n-1)).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 15, 14, 16, 20, 18, 21, 22, 24, 28, 27, 30, 33, 26, 32, 5, 7, 11, 13, 17, 19, 23, 25, 34, 36, 40, 44, 39, 38, 42, 45, 48, 35, 46, 50, 51, 52, 54, 56, 60, 64, 66, 57, 58, 62, 68, 63, 69, 70, 72, 55, 74, 76, 75, 78, 81, 80, 65, 82, 84, 88, 92, 87, 86, 90, 96, 102, 93

Offset: 1

Scott R. Shannon, Apr 08 2023

nonn

Other than the first three terms the only other primes in the first 500000 terms are the consecutive terms a(24)..a(30) = 5, 7, 11, 13, 17, 19, 23. It is unknown if more exist.

In the same range the fixed points are 1, 2, 3, 4, and 48559, although it is possible more exist.

			a(4) = 4 as Omega(a(3)) = A001222(3) = 1, and 4 is the smallest unused number that is a multiple of 1.
a(10) = 15 as Omega(a(9)) = A001222(12) = 3, and 15 is the smallest unused number that is a multiple of 3.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 200000 terms. The green line is a(n) = n.

Cf. A001222, A362031, A354606, A000005, A124056, A342585.

from sympy import primeomega
from itertools import count, islice
def A362077_gen(): # generator of terms
    a, b = {1,2}, 2
    yield from (1,2)
    while True:
        for b in count(p:=primeomega(b),p):
            if b not in a:
                yield b
                a.add(b)
                break
A362077_list = list(islice(A362077_gen(),20)) # Chai Wah Wu, Apr 11 2023

A362178 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that is a multiple of omega(a(n-1)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 12, 14, 16, 11, 13, 15, 18, 20, 22, 24, 26, 28, 30, 21, 32, 17, 19, 23, 25, 27, 29, 31, 33, 34, 36, 38, 40, 42, 39, 44, 46, 48, 50, 52, 54, 56, 58, 60, 45, 62, 64, 35, 66, 51, 68, 70, 57, 72, 74, 76, 78, 63, 80, 82, 84, 69, 86, 88, 90, 75, 92, 94, 96, 98, 100

Offset: 1

Scott R. Shannon, Apr 11 2023

nonn

Unlike A362077 numerous primes appear in the sequence; in the first 500000 terms there are seventy-four in total. In the same range there are twelve fixed points, the last being 57. It is unknown whether more exist.

			a(5) = 5 as omega(a(4)) = A001221(4) = 1, and 5 is the smallest unused number that is a multiple of 1.
a(7) = 8 as omega(a(6)) = A001221(6) = 2, and 8 is the smallest unused number that is a multiple of 2.

Scott R. Shannon, Image of the first 500000 terms. The green line is a(n) = n.

Cf. A001221, A362077, A362031, A354606, A000005, A124056, A342585.

from itertools import count, islice
from sympy import primenu
def A362178_gen(): # generator of terms
    a, b = {1,2}, 2
    yield from (1,2)
    while True:
        for b in count(p:=primenu(b),p):
            if b not in a:
                yield b
                a.add(b)
                break
A362178_list = list(islice(A362178_gen(),20)) # Chai Wah Wu, Apr 12 2023

A364934 a(n+1) = 1 + number of previous terms that share a factor > 1 with a(n); a(1) = 2.

Original entry on oeis.org

2, 2, 3, 2, 4, 5, 2, 6, 8, 8, 9, 4, 10, 12, 14, 13, 2, 14, 15, 8, 16, 17, 2, 18, 22, 20, 23, 2, 22, 23, 3, 8, 24, 29, 2, 26, 28, 28, 29, 3, 10, 32, 31, 2, 32, 33, 13, 4, 34, 36, 42, 43, 2, 38, 39, 17, 4, 40, 43, 3, 15, 21, 21, 22, 43, 4, 43, 5, 9, 19, 3, 20, 48, 58, 48, 60, 63, 28, 52, 53, 2, 51, 28

Offset: 1

Rok Cestnik, Aug 15 2023

nonn

There are prominent lines that have more terms, their coefficients are approximately: 0.519, 0.329, 0.689, 0.188, 0.615, ... (see the frequency link). They seem to be distorted prime harmonic lines: 1/2, 1/3, 2/3, 1/5, 3/5, ... from A016035.

It appears limsup a(n)/n is approximately 0.83.

			[2,*]          1 term  shares a factor with 2, so a(2) = 1+1 = 2.
[2,2,*]        2 terms share  a factor with 2, so a(3) = 1+2 = 3.
[2,2,3,*]      1 term  shares a factor with 3, so a(4) = 1+1 = 2.
[2,2,3,2,*]    3 terms share  a factor with 2, so a(5) = 1+3 = 4.
[2,2,3,2,4,*]  4 terms share  a factor with 4, so a(6) = 1+4 = 5.

Rok Cestnik, Table of n, a(n) for n = 1..1000
Rok Cestnik, a(n)/n frequency for 3*10^5 terms.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.

Cf. A016035, A124056.

nn = 120; c[] := 0; s = {}; a[1] = j = 2; Do[c[j]++; If[c[j] == 1, AppendTo[s, j]]; k = 1 + Total@ Map[c[#] Boole[GCD[#, j] > 1] &, s]; Set[{a[n], j}, {k, k}], {n, 2, nn}]; Array[a, nn] (* _Michael De Vlieger, Aug 16 2023 *)

from sympy.ntheory import primefactors
a=[2]; p = [{2}];
for n in range(1,1000):
    a.append(sum(1 if p[-1].intersection(p[i]) else 0 for i in range(n))+1)
    p.append(set(primefactors(a[-1])))

from math import gcd, prod
from sympy import factorint
from itertools import islice
from collections import Counter
def agen(): # generator of terms
    a=[2]; f=2; c=Counter([f])
    while True:
        yield a[-1]
        a.append(1 + sum(c[fi] for fi in c if gcd(f,fi)>1))
        f=prod(factorint(a[-1]).keys())
        c[f] += 1
a=list(islice(agen(), 1000)) # Michael S. Branicky, Aug 16 2023

Showing 1-8 of 8 results.

cp's OEIS Frontend

A088167 Number of earlier occurring divisors of n; a(1)=1.

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A354606 a(1) = 1; for n > 1, a(n) is number of terms in the first n-1 terms of the sequence that have the same number of divisors as a(n-1).

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A362031 a(1) = 1; for n > 1, a(n) is number of terms in the first n-1 terms of the sequence that have the same number of prime factors, counted with multiplicity, as a(n-1).

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A359034 a(n+1) is the sum of the number of terms in all groups of contiguous terms that add up to a(n); a(1)=1.

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MATLAB

A362061 a(1) = 1; for n > 1, a(n) is number of terms in the first n-1 terms of the sequence that have the same number of distinct prime factors as a(n-1).

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A362077 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that is a multiple of Omega(a(n-1)).

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A362178 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that is a multiple of omega(a(n-1)).

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A364934 a(n+1) = 1 + number of previous terms that share a factor > 1 with a(n); a(1) = 2.

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