A379442 -id:A379442 - cp's OEIS frontend

A379558 Index where prime(n) appears as a term in A379442.

2, 6, 17, 25, 77, 83, 89, 259, 319, 329, 539, 545, 1010, 1016, 1026, 1032, 2128, 2134, 2140, 2146, 2152, 2158, 3196, 3202, 3222, 3228, 3234, 5465, 5471, 5487, 5493, 6300, 6308, 6314, 6320, 8252, 8258, 8264, 8270, 8276, 8282, 8288, 13775, 13949, 13957, 13965, 13971, 13977, 13983, 13989, 13995, 14001, 27677, 27683, 27689, 27695, 27701, 27707, 27713, 27719

Offset: 1

Scott R. Shannon, Dec 25 2024

nonn

See A379442 for further details.

Michael S. Branicky, Table of n, a(n) for n = 1..300 (terms 1..122 Scott R. Shannon)

Cf. A379442, A379557, A379559, A379290.

A379559 Index where n appears as a term in A379442.

Original entry on oeis.org

1, 2, 6, 3, 17, 4, 25, 9, 5, 15, 77, 8, 83, 13, 19, 10, 89, 7, 259, 12, 23, 28, 319, 11, 16, 34, 21, 27, 329, 30, 539, 40, 56, 36, 73, 31, 545, 46, 58, 38, 1010, 32, 1016, 14, 22, 48, 1026, 39, 24, 18, 60, 29, 1032, 64, 79, 44, 62, 52, 2128, 35, 2134, 50, 20, 41, 81, 54, 2140, 33, 97, 71, 2146, 69, 2152, 103, 72, 37, 75, 101, 2158, 43, 65, 107, 3196, 47

Offset: 1

Scott R. Shannon, Dec 25 2024

nonn

See A379442 for further details.

Michael S. Branicky, Table of n, a(n) for n = 1..1992 (terms 1..676 from Scott R. Shannon)

Cf. A379442, A379557, A379558, A379293.

A379557 Number k such that A379442(k) = k.

Original entry on oeis.org

1, 2, 30, 285, 750, 822, 826, 952, 2824, 3016, 6112

Offset: 1

Scott R. Shannon, Dec 25 2024

These are the known fixed points of A379442. See that sequence for further details.

Cf. A379442, A379558, A379559, A379292.

A379440 a(1) = 1, a(2) = 2, for a(n) > 2, a(n) is the smallest unused positive number that shares a factor with a(n-1) such that the exponents of each distinct prime factor of a(n) differ by one from those of the same prime factors of a(n-1).

Original entry on oeis.org

1, 2, 4, 6, 9, 3, 18, 12, 8, 16, 24, 20, 14, 44, 10, 25, 5, 50, 15, 63, 27, 45, 21, 49, 7, 98, 28, 22, 52, 30, 36, 26, 60, 34, 76, 40, 48, 32, 64, 96, 80, 56, 68, 38, 84, 46, 116, 42, 92, 58, 124, 66, 117, 33, 90, 39, 99, 51, 126, 57, 153, 54, 81, 135, 162, 108, 62, 132, 70, 75, 35, 147, 77, 121, 11, 242, 55, 150, 65, 169, 13, 338, 91, 245, 119, 289, 17

Offset: 1

Scott R. Shannon, Dec 23 2024

Like A379442, for the terms studied, prime terms p are preceded by p^2 and followed by 2*p^2, can be divisors of terms before they appear as a term themselves, and are distributed in groups of primes, with many primes within the groups differing by six terms. Unlike A379442 not all primes appear in their natural order, although the occurrence of such primes is rare - only four primes are out of order in the first 250000 terms, namely a(6787) = 179, a(18355) = 353, a(43516) = 593, a(201498) = 1579. In the same range the fixed points are 1, 2, 30, 34, 46, 130, 352, 456, although more may exist. The sequence is conjectured to be a permutation of the positive integers.

			a(14) = 44 as 44 = 2^2*11^1, and a(13) = 14 = 2*7 which contains 2^1 as a factor, whose power differs by one from 2^2, while not containing any power of 11. This is the smallest unused number satisfying these criteria. Note that 36 = 2^2*3^2 cannot be chosen as a(13) contains no power of 3 - this is the first term to differ from A379441.

Scott R. Shannon, Table of n, a(n) for n = 1..20000
Scott R. Shannon, Image of the first 250000 terms. The green line is a(n) = n.

Cf. A379441, A379442, A379248, A124010, A027746, A051903, A064413, A348086.

from sympy import factorint
from itertools import islice
from collections import Counter
fcache = dict()
def myfactors(n):
    global fcache
    if n in fcache: return fcache[n]
    ans = Counter({p:e for p, e in factorint(n).items()})
    fcache[n] = ans
    return ans
def agen(): # generator of terms
    yield 1
    an, a, m = 2, {1, 2}, 3
    while True:
        yield an
        k, fan = m-1, myfactors(an)
        sfan = set(fan)
        while True:
            k += 1
            if k in a: continue
            fk = myfactors(k)
            sfk = set(fk)
            if sfk & sfan and all(abs(fk[p]-fan[p])==1 for p in sfk):
                an = k
                break
        a.add(an)
print(list(islice(agen(), 87))) # Michael S. Branicky, May 25 2025

A379441 a(1) = 1, a(2) = 2, for a(n) > 2, a(n) is the smallest unused positive number that shares a factor with a(n-1) such that the exponents of each distinct prime factor of a(n-1) differ by one from those of the same prime factors of a(n).

Original entry on oeis.org

1, 2, 4, 6, 9, 3, 18, 12, 8, 16, 24, 20, 14, 36, 30, 25, 5, 50, 15, 63, 27, 45, 21, 49, 7, 98, 28, 10, 44, 26, 60, 22, 52, 34, 76, 40, 48, 32, 64, 96, 80, 56, 68, 38, 84, 46, 100, 70, 75, 35, 147, 77, 121, 11, 242, 33, 72, 108, 90, 39, 99, 42, 92, 54, 81, 135, 117, 51, 126, 57, 144, 120, 112, 88, 116, 62, 132, 58, 124, 66, 140, 74, 156, 82, 148, 78, 153, 69

Offset: 1

Scott R. Shannon, Dec 23 2024

nonn

Like A379442, for the terms studied, prime terms p are preceded by p^2 and followed by 2*p^2, can be divisors of terms before they appear as a term themselves, and are distributed in groups of primes, with many primes within the groups differing by six terms. Unlike A379442 not all primes appear in their natural order, although the occurrence of such primes is rare - only three primes are out of order in the first 250000 terms, namely a(13350) = 149, a(18410) = 179, a(21382) = 191. The sequence contains numerous fixed points, these being 1, 2, 34, 46, 218, 370, 410, 462, 474, 1954, 5592, 19186,... . The sequence is conjectured to be a permutation of the positive integers.

			a(14) = 36 as 36 = 2^2*3^2 while a(13) = 14 = 2*7 which contains 2^1 as a factor, whose power differs by one from 2^2, and 7^1 as a factor, and 36 contains no power of 7. This is the smallest unused number satisfying these criteria. This is the first term to differ from A379440.

Scott R. Shannon, Table of n, a(n) for n = 1..20000
Scott R. Shannon, Image of the first 250000 terms. The green line is a(n) = n.

Cf. A379440, A379442, A379248, A124010, A027746, A051903, A064413, A348086.

from sympy import factorint
from itertools import islice
from collections import Counter
fcache = dict()
def myfactors(n):
    global fcache
    if n in fcache: return fcache[n]
    ans = Counter({p:e for p, e in factorint(n).items()})
    fcache[n] = ans
    return ans
def agen(): # generator of terms
    yield 1
    an, a, m = 2, {1, 2}, 3
    while True:
        yield an
        k, fan = m-1, myfactors(an)
        sfan = set(fan)
        while True:
            k += 1
            if k in a: continue
            fk = myfactors(k)
            sfk = set(fk)
            if sfk & sfan and all(abs(fk[p]-fan[p])==1 for p in sfan):
                an = k
                break
        a.add(an)
print(list(islice(agen(), 88))) # Michael S. Branicky, May 25 2025

cp's OEIS Frontend

A379558 Index where prime(n) appears as a term in A379442.

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A379559 Index where n appears as a term in A379442.

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A379557 Number k such that A379442(k) = k.

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A379440 a(1) = 1, a(2) = 2, for a(n) > 2, a(n) is the smallest unused positive number that shares a factor with a(n-1) such that the exponents of each distinct prime factor of a(n) differ by one from those of the same prime factors of a(n-1).

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A379441 a(1) = 1, a(2) = 2, for a(n) > 2, a(n) is the smallest unused positive number that shares a factor with a(n-1) such that the exponents of each distinct prime factor of a(n-1) differ by one from those of the same prime factors of a(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python