A081303 -id:A081303 - cp's OEIS frontend

A081306 Numbers n with prime factors less than 2*spf(n), where spf(m) is the smallest prime factor of m (A020639).

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 23, 24, 25, 27, 29, 31, 32, 35, 36, 37, 41, 43, 45, 47, 48, 49, 53, 54, 59, 61, 64, 67, 71, 72, 73, 75, 77, 79, 81, 83, 89, 91, 96, 97, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 135, 137, 139, 143, 144

Offset: 1

Reinhard Zumkeller, Mar 17 2003

nonn

A081303(a(n)) < 0, A006530(a(n)) < A020639(a(n))*2.

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

Cf. A006530, A020639, A081303.
Complement of A069900.
Union of {1} and A069899. [R. J. Mathar, Sep 18 2008]

filter:= proc(n) local F;
  F:= numtheory:-factorset(n);
  max(F) < 2*min(F);
end proc:
select(filter, [$1..200]); # Robert Israel, Mar 28 2018

Select[Range[200], Max[F = FactorInteger[#][[All, 1]]] < 2 Min[F]&] (* Jean-François Alcover, Mar 04 2019 *)

A069900 Numbers k such that the integer quotient of largest and smallest prime factors of k is greater than one.

Original entry on oeis.org

10, 14, 20, 21, 22, 26, 28, 30, 33, 34, 38, 39, 40, 42, 44, 46, 50, 51, 52, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 74, 76, 78, 80, 82, 84, 85, 86, 87, 88, 90, 92, 93, 94, 95, 98, 99, 100, 102, 104, 105, 106, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120

Offset: 1

Labos Elemer, Apr 10 2002

nonn

Numbers k such that A069897(k) = floor(P(k)/p(k)) > 1, where P(k) and p(k) are largest and least prime factor of k, respectively.

Also numbers having at least one prime factor greater than twice the smallest prime factor: complement of A081306. - Reinhard Zumkeller, Mar 17 2003

			Composites with at least two and sufficiently deviating prime factors are here, like 2q, where q = prime >= 5: {10, ..., 254}.
Numbers with such divisors like 30 are also included.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A006530, A020639, A069897, A069899, A081303, A081306.

Select[Range@ 120, #[[-1]] > 2 #[[1]] &@ FactorInteger[#][[All, 1]] &] (* Michael De Vlieger, Dec 08 2018 *)

is(k) = if(k == 1, 0, my(p = factor(k)[,1]); p[#p] > 2*p[1]); \\ Amiram Eldar, Feb 10 2025

A081303(a(n)) > 0. - Reinhard Zumkeller, Mar 17 2003

More terms from Reinhard Zumkeller, Mar 17 2003

A081305 Number of numbers m <= n with at least one prime factor greater than 2*spf(m), where spf(m) is the smallest prime factor of m (A020639).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 5, 5, 5, 5, 6, 6, 7, 7, 8, 8, 8, 9, 10, 10, 10, 10, 11, 12, 13, 13, 14, 14, 15, 15, 16, 16, 16, 16, 17, 18, 19, 19, 19, 20, 21, 22, 23, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 31, 31, 31, 31, 32, 32, 33, 33, 34, 34, 35

Offset: 1

Reinhard Zumkeller, Mar 17 2003

nonn

a(n)+A081304(n)=n; a(114)=A081304(114)=57;

a(n)<=n/2 for n<=114, a(n)>n/2 for n>114.

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

Cf. A069900, A081303.

f:= proc(n) local R; R:= numtheory:-factorset(n); if max(R) > 2*min(R) then 1 else 0 fi end proc:
ListTools:-PartialSums(map(f, [$1..100])); # Robert Israel, Jul 27 2020

pfg[n_]:=Module[{f=Transpose[FactorInteger[n]][[1]]},If[Last[f]> 2*First[ f], 1,0]]; Accumulate[Array[pfg,80]] (* Harvey P. Dale, Apr 28 2014 *)

A081304 Number of numbers m <= n with prime factors less than 2*spf(m), where spf(m) is the smallest prime factor of m (A020639).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 16, 17, 17, 17, 17, 18, 19, 20, 20, 21, 21, 22, 22, 23, 24, 24, 24, 25, 26, 27, 27, 27, 27, 28, 28, 29, 29, 30, 30, 31, 32, 33, 33, 33, 33, 34, 35, 35, 35, 35, 35, 36, 36, 37, 37, 37, 38, 38, 38, 39, 39, 39, 39, 40, 41, 42

Offset: 1

Reinhard Zumkeller, Mar 17 2003

nonn

a(n)+A081305(n)=n; a(114)=A081305(114)=57;

a(n)>=n/2 for n<=114, a(n)114.

Cf. A081306, A081303.

cp's OEIS Frontend

A081306 Numbers n with prime factors less than 2*spf(n), where spf(m) is the smallest prime factor of m (A020639).

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A069900 Numbers k such that the integer quotient of largest and smallest prime factors of k is greater than one.

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A081305 Number of numbers m <= n with at least one prime factor greater than 2*spf(m), where spf(m) is the smallest prime factor of m (A020639).

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A081304 Number of numbers m <= n with prime factors less than 2*spf(m), where spf(m) is the smallest prime factor of m (A020639).

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