A139270 -id:A139270 - cp's OEIS frontend

A176100 Even numbers that are not semiprimes, or, twice the nonprimes.

0, 2, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 48, 50, 52, 54, 56, 60, 64, 66, 68, 70, 72, 76, 78, 80, 84, 88, 90, 92, 96, 98, 100, 102, 104, 108, 110, 112, 114, 116, 120, 124, 126, 128, 130, 132, 136, 138, 140, 144, 148, 150, 152, 154, 156, 160, 162, 164

Offset: 1

Juri-Stepan Gerasimov, Apr 08 2010, Apr 11 2010

Cf. A001358, A005843, A100484, A141468.

2 Prepend[Select[Range@ 82, ! PrimeQ@ # &], 0] (* Michael De Vlieger, Feb 20 2017 *)

isok(n) = !(n%2) && !isprime(n/2); \\ Michel Marcus, Feb 20 2017

a(n)= 2*A141468(n).
A005843 \ A100484.
a(n) = A139270(n-1). [R. J. Mathar, May 03 2010]

Entries checked by R. J. Mathar, Apr 16 2010

A218243 Triangle numbers: m = abc such that the integers a,b,c are the sides of a triangle with integer area.

Original entry on oeis.org

60, 150, 200, 480, 780, 1200, 1530, 1600, 1620, 1690, 1950, 2040, 2100, 2730, 2860, 3570, 3840, 4050, 4056, 4200, 4350, 4624, 5100, 5400, 5460, 6240, 7500, 8120, 8250, 8670, 8750, 9600, 10812, 11050, 11900, 12180, 12240, 12800, 12960, 13260, 13520, 13650

Offset: 1

Michel Lagneau, Oct 24 2012

nonn

A triangle number m is an integer with at least one decomposition m = a*b*c such that the area of the triangle of sides (a,b,c) is an integer. Because this property is not always unique, we introduce the notion of "triangle order" for each triangle number m, denoted by TO(m). For example, TO(60) = 1 because the decomposition 60 = 3*4*5 is unique with the triangle (3,4,5) whose area A is given by Heron's formula: A = sqrt(s*(s-a)*(s-b)*(s-c)), where s = (a+b+c)/2 => A = sqrt(6*(6-3)*(6-4)*(6-5)) = 6, but TO(780) = 2 because 780 = 4*13*15 = 5*12*13 and the area of the triangle (4,13,15) is sqrt(16*(16-4)*(16-13)*(16-15)) = 24 and the area of the triangle (5,12,13) is sqrt(15*(15-5)*(15-12)*(15-13)) = 30.
Given an area A of A188158, there exists either a unique triangle number (for example for A = 6 => m = 60 = 3*4*5), or several triangle numbers (for example for A=60 => m1 = 4350 = 6*25*29, m2 = 2040 = 8*15*17, m3 = 1690 = 13*13*10).
The number of ways to write m = a*b*c with 1<=a<=b<=c<=m is given by A034836, thus: TO(m) <= A034836(m).
If n is in this sequence, so is nk^3 for any k > 0. Thus this sequence is infinite. - Charles R Greathouse IV, Oct 24 2012
In view of the preceding comment, one might call "primitive" the elements of the sequence for which there is no k>1 such that n/k^3 is again a term of the sequence. These elements 60, 150, 200, 780, 1530, 1690, 1950,... are listed in A218392. - M. F. Hasler, Oct 27 2012

			60 is in the sequence because 60 = 3*4*5 and the corresponding area is sqrt(6*(6-3)*(6-4)*(6-5)) = 6 = A188158(1).

Charles R Greathouse IV, Table of n, a(n) for n = 1..2000
Eric Weisstein's World of Mathematics, Triangle

Subsequence of A139270.

Cf. A034836, A188158, A083875.

nn = 500; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[0 < area2 <= nn^2 && IntegerQ[Sqrt[area2]], AppendTo[lst, a*b*c]]], {a, nn}, {b, a}, {c, b}]; Union[lst] (* Program from T. D. Noe, adapted for this sequence - see A188158 *)

Heron(a,b,c)=a*=a;b*=b;c*=c;((a+b+c)^2-2*(a^2+b^2+c^2))
is(n)=fordiv(n,a, if(a^3<=n, next); fordiv(n/a,b, my(c=n/a/b,h); if(a>=b && b>=c && aCharles R Greathouse IV, Oct 24 2012

A256420 a(n) = 2n unless n is prime, in which case a(n) = first term not yet present in the sequence.

Original entry on oeis.org

2, 1, 3, 8, 4, 12, 5, 16, 18, 20, 6, 24, 7, 28, 30, 32, 9, 36, 10, 40, 42, 44, 11, 48, 50, 52, 54, 56, 13, 60, 14, 64, 66, 68, 70, 72, 15, 76, 78, 80, 17, 84, 19, 88, 90, 92, 21, 96, 98, 100, 102, 104, 22, 108, 110, 112, 114, 116, 23, 120

Offset: 1

N. J. A. Sloane, Apr 07 2015

nonn

This is a permutation of the positive integers: twice nonprimes (A139270), interspersed with (odd numbers and twice primes, A256421).

John Mason, Email message, Apr 07 2015

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

Cf. A139270, A256421.

N:= 100:
S:= {$1..N}:
R:= NULL;
for n from 1 do
  if isprime(n) then if S = {} then break else t:= min(S) fi else t:= 2*n fi;
  R:= R, t;
  S:= S minus {t}
od:
R;

from sympy import primepi, isprime
def A256420(n):
    r = int(primepi(n))
    def iterfun(f,n=0):
        m, k = n, f(n)
        while m != k: m, k = k, f(k)
        return m
    def f(x): return int(r+(m:=x>>1)-primepi(m))
    return iterfun(f,r) if isprime(n) else n<<1 # Chai Wah Wu, Oct 15 2024

A282671 Twice composite numbers.

Original entry on oeis.org

8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 48, 50, 52, 54, 56, 60, 64, 66, 68, 70, 72, 76, 78, 80, 84, 88, 90, 92, 96, 98, 100, 102, 104, 108, 110, 112, 114, 116, 120, 124, 126, 128, 130, 132, 136, 138, 140, 144, 148, 150, 152, 154, 156, 160, 162, 164, 168, 170, 172, 174

Offset: 1

Alessandro Polcini, Feb 20 2017

Even numbers greater than 2 that do not appear in A001747.

Cf. A001747, A139270, A176100.

is(n)=n%2==0 && n>6 && !isprime(n/2) \\ Charles R Greathouse IV, Feb 21 2017

a(n) = 2*A002808(n). - R. J. Mathar, Feb 23 2017

a(n) = A139270(n+1). - R. J. Mathar, Feb 25 2017

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A176100 Even numbers that are not semiprimes, or, twice the nonprimes.

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A218243 Triangle numbers: m = a*b*c such that the integers a,b,c are the sides of a triangle with integer area.

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A256420 a(n) = 2n unless n is prime, in which case a(n) = first term not yet present in the sequence.

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A282671 Twice composite numbers.

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A218243 Triangle numbers: m = abc such that the integers a,b,c are the sides of a triangle with integer area.