A088380 -id:A088380 - 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 *)

A087719 Least number m such that the number of numbers k <= m with k > spf(k)^n exceeds the number of numbers with k <= spf(k)^n.

Original entry on oeis.org

15, 27, 57, 135, 345, 927, 2577, 7335, 21225, 62127, 183297, 543735, 1618905, 4832127, 14447217, 43243335, 129533385, 388206927, 1163834337, 3489930135, 10466644665, 31393642527, 94168344657, 282479868135

Offset: 1

Reinhard Zumkeller, Sep 29 2003

nonn

mspf(k)^n & 1<=k<=m} <= m/2;

m>=a(n): #{k: k>spf(k)^n & 1<=k<=m} > m/2.

Cf. A088382, A088382, A088380, A088381, A000430, A080257.

Numbers so far satisfy a(n) = 3^n + 3*2^n + 6. - Ralf Stephan, May 10 2004

Empirical G.f.: 3*x*(5-21*x+20*x^2)/(1-x)/(1-2*x)/(1-3*x). - Colin Barker, Feb 22 2012

a(14)-a(24) from Giovanni Resta, May 23 2013

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

PARI

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

A087719 Least number m such that the number of numbers k <= m with k > spf(k)^n exceeds the number of numbers with k <= spf(k)^n.

Views

Author

Keywords

Comments

Crossrefs

Formula

Extensions