A354606 -id:A354606 - cp's OEIS frontend

A355606 The indices where A354606(n) = 1.

1, 2, 4, 9, 14, 25, 37, 57, 99, 133, 182, 191, 404, 469, 595, 640, 780, 1195, 1884, 2407, 2808, 3010, 3217, 3444, 4245, 4383, 5773, 8703, 10069, 10731, 12640, 14470, 17998, 18535, 22648, 23341, 24286, 27431, 33702, 37019, 45593, 53759, 56598, 57578, 76640, 96729, 99557, 106881, 125900, 144162

Offset: 1

Scott R. Shannon, Jul 09 2022

nonn

See A354606 for further details.

Cf. A354606, A000005.

c[] := 0; j = 1; nn = 2^20; {1}~Join~Reap[Monitor[Do[k = ++c[DivisorSigma[0, j]]; If[k == 1, Sow[n]]; j = k, {n, 2, nn}], n] ][[-1, 1]] (* _Michael De Vlieger, Dec 12 2024 *)

from sympy import divisor_count
from collections import Counter
from itertools import count, islice
def f(n): return divisor_count(n)
def agen():
    n, an, fan, inventory = 1, 1, 1, Counter([1])
    yield n
    for n in count(2):
        an = inventory[fan]
        fan = f(an)
        inventory.update([fan])
        if an == 1: yield n
print(list(islice(agen(), 50))) # 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

A355621 a(1) = 1; for n > 1, a(n) is the number of terms in the first n-1 terms of the sequence that share a 1-bit with a(n-1) in their binary expansions.

Original entry on oeis.org

1, 1, 2, 1, 3, 5, 5, 6, 5, 8, 1, 8, 2, 4, 5, 11, 15, 17, 12, 11, 19, 17, 15, 23, 22, 19, 22, 21, 24, 16, 10, 18, 20, 21, 29, 33, 22, 30, 33, 23, 38, 31, 42, 28, 35, 37, 38, 37, 40, 22, 41, 40, 24, 33, 35, 46, 49, 49, 50, 47, 59, 60, 55, 61, 62, 61, 64, 1, 39, 63, 69, 58, 60, 64, 3, 60, 65, 46, 67

Offset: 1

Scott R. Shannon, Jul 10 2022

The indices where a(n) = 1 in the first 500000 terms are 1, 2, 4, 11, 68, 131, 2051, 4099. It is unknown if more exist. Many terms of the sequence are close to the line a(n) = n although only the first term is a possible fixed point. In the first 500000 terms the lowest values not to appear are 7, 9, 14, 25, 26. It is likely these and other numbers never appear although this is unknown.

			a(7) = 5 as a(6) = 5 and the total number of terms in the first six terms that share a 1-bit with 5 in their binary expansions is five, namely 1, 1, 1, 3, 5.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 500000 terms. The green line is y = n.

Cf. A355625, A030190, A129760, A353989, A352763, A354606.

from itertools import count, islice
def agen():
    an, alst = 1, [1]
    for n in count(2):
        yield an
        an = sum(1 for k in alst if k&an)
        alst.append(an)
print(list(islice(agen(), 79))) # Michael S. Branicky, Jul 10 2022

A355625 a(1) = 1; for n > 1, a(n) is the number of terms in the first n-1 terms of the sequence that share a 1-bit with n in their binary expansions.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 4, 0, 3, 2, 6, 2, 6, 7, 11, 0, 6, 9, 13, 6, 13, 13, 18, 6, 11, 17, 21, 16, 21, 22, 26, 0, 14, 16, 26, 14, 23, 25, 31, 12, 22, 27, 34, 27, 33, 34, 39, 19, 31, 35, 43, 36, 44, 44, 49, 36, 42, 48, 52, 47, 52, 53, 57, 0, 29, 32, 48, 30, 48, 48, 57, 25, 41, 46, 56, 47, 57, 58, 65, 34

Offset: 1

Scott R. Shannon, Jul 10 2022

The indices where a(n) = 1 in the first 500000 terms are 1, 3, 6. It is likely no more exist although this is unknown. Many terms of the sequence are close to the line a(n) = n although only the first term is a possible fixed point. In the first 500000 terms the lowest values not to appear are 5, 8, 10, 15, 20, 24, 28. It is likely these and other numbers never appear although this is unknown. All terms for n > 1 where n is a power of 2 equal 0.

			a(7) = 4 as the total number of terms in the first six terms that share a 1-bit with 7 in their binary expansions is four, namely 1, 1, 2, 1.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 500000 terms. The green line is y = n.

Cf. A355621, A030190, A129760, A353989, A352763, A354606.

from itertools import count, islice
def agen():
    an, alst = 1, [1]
    for n in count(2):
        yield an
        an = sum(1 for k in alst if k&n)
        alst.append(an)
print(list(islice(agen(), 80))) # Michael S. Branicky, Jul 10 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

cp's OEIS Frontend

A355606 The indices where A354606(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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A355621 a(1) = 1; for n > 1, a(n) is the number of terms in the first n-1 terms of the sequence that share a 1-bit with a(n-1) in their binary expansions.

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A355625 a(1) = 1; for n > 1, a(n) is the number of terms in the first n-1 terms of the sequence that share a 1-bit with n in their binary expansions.

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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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