A362333 -id:A362333 - 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.

A371152 Array read by antidiagonals: row n lists numbers m >= 2 with A362333(m) = n.

Original entry on oeis.org

2, 3, 4, 5, 9, 8, 6, 12, 24, 16, 7, 18, 27, 48, 32, 10, 25, 54, 81, 96, 64, 11, 36, 72, 144, 243, 192, 128, 13, 45, 108, 162, 288, 576, 384, 256, 14, 49, 125, 324, 486, 729, 1152, 768, 512, 15, 50, 135, 405, 864, 1458, 2187, 2304, 1536, 1024

Offset: 1

Pontus von Brömssen, Mar 13 2024

			Table begins:
  n\k|    1    2    3     4     5     6      7      8      9     10
  ---+-------------------------------------------------------------
   1 |    2    3    5     6     7    10     11     13     14     15
   2 |    4    9   12    18    25    36     45     49     50     75
   3 |    8   24   27    54    72   108    125    135    216    250
   4 |   16   48   81   144   162   324    405    432    625    648
   5 |   32   96  243   288   486   864    972   1215   1944   2430
   6 |   64  192  576   729  1458  1728   2916   3645   5184   5832
   7 |  128  384 1152  2187  3456  4374   8748  10368  10935  17496
   8 |  256  768 2304  6561  6912 13122  20736  26244  32805  52488
   9 |  512 1536 4608 13824 19683 39366  41472  78732  98415 124416
  10 | 1024 3072 9216 27648 59049 82944 118098 236196 248832 295245

Cf. A048839 (first row), A057109 (terms in rows 2, 3, ...), A362333.

T(n,1) = 2^n.

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)).

A371149 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 denominator of the maximum of e_j/f_j.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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) = 3.

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

Cf. A000040, A006530, A362333, A371148 (numerators), A371150, A371151.

from sympy import factorint, Rational
def A371149(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.q

A362333(n) = ceiling(A371148(n)/a(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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A371152 Array read by antidiagonals: row n lists numbers m >= 2 with A362333(m) = 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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Formula

A371149 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 denominator of the maximum of e_j/f_j.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Formula