A371149 -id:A371149 - cp's OEIS frontend

A371151 Numbers k >= 2 such that A362333(k)-A371148(k)/A371149(k) sets a new maximum.

2, 80, 224, 5632, 26624, 1114112, 2490368, 24117248

Offset: 1

Pontus von Brömssen, Mar 13 2024

The corresponding maxima are: 0, 2/3, 3/4, 7/8, 9/10, 14/15, 15/16, 18/19, ... .
All terms after a(1) = 2 are in A371150.
Apparently, a(n) = 2^(A080085(n+1)+1)*prime(n+1) for n >= 2, with corresponding maxima 1 - 1/A080085(n+1).

Cf. A080085, A362333, A371148, A371149, A371150.

A371150 Numbers k such that A371149(k) > 1.

Original entry on oeis.org

80, 160, 189, 224, 240, 378, 448, 480, 640, 672, 756, 896, 945, 1120, 1280, 1344, 1375, 1512, 1625, 1701, 1890, 1920, 2016, 2240, 2673, 2688, 2750, 3024, 3200, 3250, 3360, 3402, 3584, 3780, 3840, 4032, 4125, 4480, 4875, 5120, 5346, 5500, 5632, 5760, 5831, 6048

Offset: 1

Pontus von Brömssen, Mar 13 2024

nonn

Cf. A371148, A371149.

A362333 Least nonnegative integer k such that (gpf(n)!)^k is divisible by n, where gpf(n) is the greatest prime factor of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 4, 2, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 1

Offset: 1

Pontus von Brömssen, Apr 16 2023

nonn

First differs from A088388 at n = 40.

			For n = 12, gpf(n)! = 3! = 6 is not divisible by 12, but (3!)^2 = 36 is divisible by 12, so a(12) = 2.

Cf. A006530, A051903, A057109, A088388, A371148, A371149, A371151, A371152.

from sympy import factorint
def A362333(n):
    f = factorint(n)
    gpf = max(f,default=None)
    a = 0
    for p in f:
        m = gpf
        v = 0
        while m >= p:
            m //= p
            v += m
        a = max(a,-(-f[p]//v))
    return a

a(n) > 1 if and only if n is in A057109.
a(n) <= A051903(n).
a(n) = ceiling(A371148(n)/A371149(n)). - Pontus von Brömssen, Mar 16 2024

A371148 Let n = Product_{j=1..k} p_j^e_j and gpf(n)! = Product_{j=1..k} p_j^f_j, where p_j = A000040(j) is the j-th prime and p_k = gpf(n) = A006530(n) is the greatest prime factor of n. a(n) is the numerator of the maximum of e_j/f_j.

Original entry on oeis.org

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

Offset: 2

Pontus von Brömssen, Mar 13 2024

			For n = 80 = 2^4 * 3^0 * 5^1, gpf(80)! = 5! = 2^3 * 3^1 * 5^1. The ratios of the prime exponents are 4/3, 0/1, and 1/1, the greatest of which is 4/3, so a(80) = 4.

Pontus von Brömssen, Table of n, a(n) for n = 2..10000

Cf. A000040, A006530, A362333, A371149 (denominators), A371150, A371151.

from sympy import factorint,Rational
def A371148(n):
    f = factorint(n)
    gpf = max(f,default=None)
    a = 0
    for p in f:
        m = gpf
        v = 0
        while m >= p:
            m //= p
            v += m
        a = max(a,Rational(f[p],v))
    return a.p

A362333(n) = ceiling(a(n)/A371149(n)).

cp's OEIS Frontend

A371151 Numbers k >= 2 such that A362333(k)-A371148(k)/A371149(k) sets a new maximum.

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A371150 Numbers k such that A371149(k) > 1.

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A362333 Least nonnegative integer k such that (gpf(n)!)^k is divisible by n, where gpf(n) is the greatest prime factor of n.

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A371148 Let n = Product_{j=1..k} p_j^e_j and gpf(n)! = Product_{j=1..k} p_j^f_j, where p_j = A000040(j) is the j-th prime and p_k = gpf(n) = A006530(n) is the greatest prime factor of n. a(n) is the numerator of the maximum of e_j/f_j.

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