A307908 -id:A307908 - cp's OEIS frontend

A088381 Numbers greater than the cube of their smallest prime factor.

10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 87, 88, 90, 92, 93, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112

Offset: 1

Reinhard Zumkeller, Sep 28 2003

nonn

a(n) > A020639(a(n))^3 = A088378(a(n)); complement of A088380;

a(n) > A088380(k) for n <= 28, a(n) < A088380(k) for n > 28.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A080257, A088378, A088380, A088383.
Cf. A020639, A138511 (subsequence).
Positions of numbers greater than 3 in A307908.

a088381 n = a088381_list !! (n-1)
a088381_list = filter f [1..] where
                      f x = p ^ 2 < div x p  where p = a020639 x
-- Reinhard Zumkeller, Jan 08 2015

filter:= n -> n > min(numtheory:-factorset(n))^3:
select(filter, [$2..200]); # Robert Israel, Aug 11 2020

isok(n) = n > factor(n)[1,1]^3; \\ Michel Marcus, Jan 08 2015

A088382 Numbers not exceeding the 4th power of their smallest prime factor.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 125, 127, 131, 133

Offset: 1

Reinhard Zumkeller, Sep 28 2003

nonn

a(n) <= A020639(a(n))^4 = A088379(a(n)); complement of A088383;

a(n) < A088383(k) for n <= 67, a(n) > A088383(k) for n > 67.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A088380, A000430, A020639, A088379, A088383.

Positions of numbers less than 5 in A307908.

a088382 n = a088382_list !! (n-1)
a088382_list = [x | x <- [1..], x <= a020639 x ^ 4]
-- Reinhard Zumkeller, Feb 06 2015

Select[Range[200],#<=FactorInteger[#][[1,1]]^4&] (* Harvey P. Dale, Jan 25 2015 *)

A088380 Numbers not exceeding the cube of their smallest prime factor.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 125, 127, 131, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 169

Offset: 1

Reinhard Zumkeller, Sep 28 2003

nonn

a(n) <= A020639(a(n))^3 = A088378(a(n)); complement of A088381;

a(n) < A088381(k) for n <= 28, a(n) > A088381(k) for n > 28.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000430, A088382, A020639, A088378, A088381.

Positions of numbers less than 4 in A307908.

a088380 n = a088382_list !! (n-1)
a088380_list = [x | x <- [1..], x <= a020639 x ^ 3]
-- Reinhard Zumkeller, Feb 06 2015

Select[Range[200],#<=FactorInteger[#][[1,1]]^3&] (* Harvey P. Dale, Apr 28 2022 *)

A088383 Numbers greater than the 4th power of their smallest prime factor.

Original entry on oeis.org

18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 87, 88, 90, 92, 93, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 116, 117, 118, 120, 122, 123, 124, 126

Offset: 1

Reinhard Zumkeller, Sep 28 2003

nonn

a(n) > A020639(a(n))^4 = A088379(a(n)); complement of A088382.

a(n) > A088382(k) for n <= 67, a(n) < A088382(k) for n > 67.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A088381, A080257, A020639, A088379, A088382.

Positions of numbers greater than 4 in A307908.

a088383 n = a088383_list !! (n-1)
a088383_list = [x | x <- [1..], x  a020639 x ^ 4]
-- Reinhard Zumkeller, Feb 06 2015

Select[Range[200],#>(FactorInteger[#][[1,1]])^4&] (* Harvey P. Dale, Aug 15 2015 *)

A307907 a(n) is the greatest k such that p^k <= n for any prime factor p of n.

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

Offset: 2

Rémy Sigrist, May 05 2019

nonn

			For n = 12:
- the prime factors of 12 are 2 and 3,
- 2^2 < 3^2 <= 12 < 3^3,
- hence a(12) = 2.

Rémy Sigrist, Colored scatterplot of the ordinal transform of n -> a(n^5) - 5*a(n) for n = 2..100000

Cf. A006530, A048098, A064052, A090081, A307908.

Array[If[PrimeQ@ #, 1, Floor@ Log[FactorInteger[#][[-1, 1]], #]] &, 105, 2] (* Michael De Vlieger, May 08 2019 *)

a(n) = my (f=factor(n)); logint(n, f[#f~, 1])

from sympy import integer_log, primefactors
def A307907(n): return integer_log(n,max(primefactors(n)))[0] # Chai Wah Wu, Oct 12 2024

a(n) = floor(log(n)/log(A006530(n))).
a(p^k) = k for any prime number p and any k > 0.
0 <= a(n^k) - k*a(n) < k for any n > 1 and any k > 0.
a(n) = 1 iff n belongs to A064052.
a(n) > 1 iff n belongs to A048098.
a(n) > 2 iff n belongs to A090081.

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A088381 Numbers greater than the cube of their smallest prime factor.

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A088382 Numbers not exceeding the 4th power of their smallest prime factor.

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A088380 Numbers not exceeding the cube of their smallest prime factor.

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A088383 Numbers greater than the 4th power of their smallest prime factor.

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A307907 a(n) is the greatest k such that p^k <= n for any prime factor p of n.

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