A067274 -id:A067274 - cp's OEIS frontend

A069274 13-almost primes (generalization of semiprimes).

8192, 12288, 18432, 20480, 27648, 28672, 30720, 41472, 43008, 45056, 46080, 51200, 53248, 62208, 64512, 67584, 69120, 69632, 71680, 76800, 77824, 79872, 93312, 94208, 96768, 100352, 101376, 103680, 104448, 107520, 112640, 115200

Offset: 1

Rick L. Shepherd, Mar 13 2002

nonn

Product of 13 not necessarily distinct primes.

Divisible by exactly 13 prime powers (not including 1).

D. W. Wilson, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Almost Prime.

Sequences listing r-almost primes, that is, the n such that A001222(n) = r: A000040 (r = 1), A001358 (r = 2), A014612 (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), this sequence (r = 13), A069275 (r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20). - Jason Kimberley, Oct 02 2011

Select[Range[30000], Plus @@ Last /@ FactorInteger[ # ] == 13 &] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2008 *)
Select[Range[116000],PrimeOmega[#]==13&] (* Harvey P. Dale, Mar 11 2019 *)

k=13; start=2^k; finish=130000; v=[]; for(n=start,finish, if(bigomega(n)==k,v=concat(v,n))); v

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A067274(n):
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x, a, b, c, m): yield from (((d, ) for d in enumerate(primerange(b, isqrt(x//c)+1), a)) if m==2 else (((a2, b2), )+d for a2, b2 in enumerate(primerange(b, integer_nthroot(x//c, m)[0]+1), a) for d in g(x, a2, b2, c*b2, m-1)))
    def f(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x, 0, 1, 1, 13)))
    return bisection(f, n, n) # Chai Wah Wu, Nov 03 2024

Product p_i^e_i with Sum e_i = 13.

A091627 Number of ordered integer pairs (b,c) with 1 <= b,c <= n such that both roots of x^2+bx+c=0 are integers.

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 7, 8, 10, 12, 14, 15, 18, 19, 21, 23, 26, 27, 30, 31, 34, 36, 38, 39, 43, 45, 47, 49, 52, 53, 57, 58, 61, 63, 65, 67, 72, 73, 75, 77, 81, 82, 86, 87, 90, 93, 95, 96, 101, 103, 106, 108, 111, 112, 116, 118, 122, 124, 126, 127, 133, 134, 136, 139, 143

Offset: 0

Eric W. Weisstein, Jan 24 2004

nonn

Also number of ordered pairs of positive integers (i, j) such that i+j <= n and i*j <= n. - Seiichi Manyama, Sep 04 2021

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Quadratic Equation

Cf. A000005, A067274, A091626, A094820.

Accumulate[ Join[{0, 0}, Table[ Ceiling[ DivisorSigma[0, n]/2], {n, 2, 64}]]]  (* Jean-François Alcover, Oct 23 2012, after Vladeta Jovovic *)

a(n) = sum(i=1, n, sum(j=i, n-i, i*j<=n)); \\ Seiichi Manyama, Sep 04 2021

N=66; x='x+O('x^N); concat([0, 0], Vec((-x+sum(k=1, sqrtint(N), x^k^2/(1-x^k)))/(1-x))) \\ Seiichi Manyama, Sep 04 2021

from math import isqrt
def A091627(n):
    m = isqrt(n)
    return 0 if n == 0 else sum(n//k for k in range(1, m+1))-m*(m-1)//2-1 # Chai Wah Wu, Oct 07 2021

a(n) = A091626(n) - n - 1. a(n) = a(n-1) + ceiling(tau(n)/2), n>1. Partial sums of A038548. - Vladeta Jovovic, Jun 12 2004

G.f.: (1/(1 - x)) * (-x + Sum_{k>=1} x^(k^2)/(1 - x^k)). - Seiichi Manyama, Sep 04 2021

A091626 Number of ordered integer pairs (b,c) with 0 <= b, c <= n such that both roots of x^2+bx+c=0 are integers.

Original entry on oeis.org

1, 2, 4, 6, 9, 11, 14, 16, 19, 22, 25, 27, 31, 33, 36, 39, 43, 45, 49, 51, 55, 58, 61, 63, 68, 71, 74, 77, 81, 83, 88, 90, 94, 97, 100, 103, 109, 111, 114, 117, 122, 124, 129, 131, 135, 139, 142, 144, 150, 153, 157, 160, 164, 166, 171, 174, 179, 182, 185, 187

Offset: 0

Eric W. Weisstein, Jan 24 2004

nonn

Also number of ordered pairs of nonnegative integers (i, j) such that i+j <= n and i*j <= n. - Seiichi Manyama, Sep 04 2021

			The six quadratics for a(3)=6 and their roots are as follows:
x^2 + 0*x + 0; x=0.
x^2 + 1*x + 0; x=0, x=-1.
x^2 + 2*x + 0; x=0, x=-2.
x^2 + 2*x + 1; x=-1.
x^2 + 3*x + 0; x=0, x=-3.
x^2 + 3*x + 2; x=-1, x=-2.

Griffin N. Macris, Table of n, a(n) for n = 0..9999
Eric Weisstein's World of Mathematics, Quadratic Equation

Cf. A000005, A006218, A038548, A067274, A091627.

a[n_] := a[n] = a[n-1] + Ceiling[ DivisorSigma[0, n]/2] + 1; a[0]=1; a[1]=2; Table[a[n], {n, 0, 59}] (* Jean-François Alcover, Nov 08 2012, after Vladeta Jovovic *)

a(n) = sum(i=0, n, sum(j=i, n-i, i*j<=n)); \\ Seiichi Manyama, Sep 04 2021

from math import isqrt
def A091626(n):
    m = isqrt(n)
    return 1 if n == 0 else n+sum(n//k for k in range(1, m+1))-m*(m-1)//2 # Chai Wah Wu, Oct 07 2021

a(n) = a(n-1) + ceiling(tau(n)/2) + 1, n>1. - Vladeta Jovovic, Jun 12 2004

a(n) = n + floor(sqrt(n))/2 + A006218(n)/2, n>0. - Griffin N. Macris, Jun 14 2016

A379597 a(n) is the number of distinct solution sets to the quadratic equations ux^2 + vx + w = 0 with integer coefficients u, v, w, abs(u) + abs(v) + abs(w) <= n having a nonnegative discriminant.

Original entry on oeis.org

1, 4, 12, 24, 50, 80, 134, 192, 276, 366, 510, 632, 834, 1018, 1262, 1502, 1858, 2136, 2584, 2956, 3448, 3910, 4576, 5076, 5834, 6488, 7320, 8066, 9136, 9892, 11118, 12114, 13358, 14482, 15978, 17108, 18862, 20272, 22024, 23532, 25700, 27216, 29600, 31486, 33746

Offset: 1

Felix Huber, Feb 18 2025

nonn

Quadratic equations u*x^2 + v*x + w = 0 with real coefficients u, v, w and nonnegative discriminant v^2 - 4*u*w have two real solutions.

a(n) is even for n >= 2.

			a(3) = 12 because there are 12 equations with abs(u) + abs(v) + abs(w) <= 3 and distinct solution set having a nonnegative discriminant: (u, v, w) = (1, 0, 0), (1, -1, 0), (1, 1, 0), (1, 0, -1), (1, -1, -1), (1, 1, -1), (1, -2, 0), (1, 2, 0), (1, 0, -2), (2, -1, 0), (2, 1, 0), and (2, 0, -1). Multiplied equations like 2*(1, 0, 0) = (2, 0, 0) or (-1)*(1, -1, 0) = (-1, 1, 0) do not have a distinct solution set.

Felix Huber, Table of n, a(n) for n = 1..3800
Eric Weisstein's World of Mathematics, Quadratic Equation

Cf. A067274, A091626, A091627, A364384, A364385, A365876, A365877, A381710, A381711.

A379597:=proc(n)
   option remember;
   local a,u,v,w;
   if n=1 then
      1
   else
      a:=0;	
      for u to n-1 do
         for v from 0 to n-u do
            w:=n-u-v;
            if igcd(u,v,w)=1 then
               if v=0 then
                  a:=a+1
               elif w=0 or w>=v^2/(4*u) then
                  a:=a+2
               else
                  a:=a+4
               fi
            fi
         od
      od;
      a+procname(n-1)
   fi;
end proc;
seq(A379597(n),n=1..45);

A381710 a(n) is the number of distinct solution sets to the quadratic equations ux^2 + vx + w = 0 with integer coefficients u, v, w, abs(u) + abs(v) + abs(w) <= n having a negative discriminant.

Original entry on oeis.org

0, 1, 5, 11, 25, 39, 69, 99, 143, 189, 265, 327, 437, 529, 653, 777, 965, 1107, 1343, 1531, 1783, 2021, 2367, 2619, 3013, 3343, 3771, 4153, 4707, 5087, 5721, 6229, 6865, 7437, 8197, 8767, 9677, 10391, 11279, 12043, 13155, 13919, 15147, 16101, 17249, 18301, 19763

Offset: 1

Felix Huber, Mar 06 2025

nonn

Quadratic equations u*x^2 + v*x + w = 0 with real coefficients u, v, w and negative discriminant v^2 - 4*u*w have two complex solutions.

a(n) is odd for n >= 2.

			a(3) = 5 because there are 5 equations with abs(u) + abs(v) + abs(w) <= 3 and distinct solution set having a negative discriminant: (u, v, w) = (1, 0, 1), (1, -1, 1), (1, 1, 1), (1, 0, 2), (2, 0, 1). Multiplied equations like (-1)*(1, -1, 1) = (-1, 1, -1) do not have a distinct solution set.

Felix Huber, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Quadratic Equation

Cf. A067274, A091626, A091627, A364384, A364385, A365876, A365877, A379597, A381711.

A381710:=proc(n)
   option remember;
   local a,u,v,w;
      if n=1 then
      0
   else
      a:=0;
      for u to n-1 do
         for v from 0 to n-u do
            w:=n-u-v;
               if igcd(u,v,w)=1 then
                  if v=0 then
                     a:=a+1
                  elif w>v^2/(4*u) then
                    a:=a+2
               fi
            fi
         od
      od;
      a+procname(n-1)
   fi;
end proc;
seq(A381710(n),n=1..47);

A381711 a(n) = A379597(n) - A381710(n).

Original entry on oeis.org

1, 3, 7, 13, 25, 41, 65, 93, 133, 177, 245, 305, 397, 489, 609, 725, 893, 1029, 1241, 1425, 1665, 1889, 2209, 2457, 2821, 3145, 3549, 3913, 4429, 4805, 5397, 5885, 6493, 7045, 7781, 8341, 9185, 9881, 10745, 11489, 12545, 13297, 14453, 15385, 16497, 17517, 18917

Offset: 1

Felix Huber, Mar 08 2025

nonn

a(n) is odd.

Felix Huber, Table of n, a(n) for n = 1..4051
Eric Weisstein's World of Mathematics, Quadratic Equation

Cf. A067274, A091626, A091627, A364384, A364385, A365876, A365877, A379597, A381710.

A381711:=proc(n)
   option remember;
   local a,u,v,w;
   if n=1 then
      1
   else
      a:=0;
      for u to n-1 do
         for v from 0 to n-u do
            w:=n-u-v;
            if igcd(u,v,w)=1 and v<>0 then
               if w=0 or w=v^2/(4*u) then
                  a:=a+2
               elif wA381711(n),n=1..47);

a(n) = A379597(n) - A381710(n).

A358398 a(n) is the number of reducible monic cubic polynomials x^3 + rx^2 + sx + t with integer coefficients bounded by naïve height n (abs(r), abs(s), abs(t) <= n).

Original entry on oeis.org

15, 53, 117, 215, 329, 493, 657, 877, 1103, 1383, 1643, 2017, 2325, 2721, 3131, 3601, 4009, 4575, 5031, 5647, 6221, 6849, 7409, 8211, 8849, 9593, 10335, 11199, 11899, 12915, 13671, 14655, 15559, 16535, 17473, 18711, 19619, 20711, 21787, 23099, 24095, 25507, 26571, 27931, 29259

Offset: 1

Lorenz H. Menke, Jr., Nov 13 2022

nonn

Artūras Dubickas, On the number of reducible polynomials of bounded naive height, manuscripta math. 144, 439-456 (2014).
Phyllis Lefton, On the Galois groups of cubics and trinomials, Bull. Amer. Math. Soc., vol. 82 (1976), pp. 754-756.
Phyllis Lefton, On the Galois groups of cubics and trinomials, Acta Arithmetica (1979) Volume: 35, Issue: 3, page 239-246.

Cf. A067274.

{ a(n) =
    my( ct = 0 );
    for (c1 = -n, n,
    for (c2 = -n, n,
    for (c3 = -n, n,
        if ( ! polisirreducible( Pol([1,c1,c2,c3]) ), ct += 1 );
    ); ); );
    return( ct );
}
vector(12, n, a(n) ) \\ Joerg Arndt, Dec 12 2022

Dubickas (2014) shows that a(n) ~ 2(1+(2/3)Pi^2)n^2 = 15.1598... n^2 for large n.

a(26)-a(45) from Hugo Pfoertner, Nov 27 2022

A358399 a(n) is the number of reducible monic quartic polynomials (x^4 + rx^3 + sx^2 + t*x + u) with integer coefficients bounded by naïve height n (abs(r), abs(s), abs(t), abs(u) <= n).

Original entry on oeis.org

47, 271, 810, 1849, 3395, 5832, 8915, 13242, 18465, 25267, 32874, 43023, 53662, 66957, 81770, 99374, 117564, 140303, 163048, 190757, 219702, 252465, 285820, 326853, 366732

Offset: 1

Lorenz H. Menke, Jr., Nov 13 2022

Cf. A358398, A358400, A067274.

{ a(n) = \\ A358399
    my( ct = 0 );
    for (c1 = -n, n,
    for (c2 = -n, n,
    for (c3 = -n, n,
    for (c4 = -n, n,
        if ( ! polisirreducible( Pol([1,c1,c2,c3,c4]) ), ct += 1 );
    ); ); ); );
    return( ct );
}
vector(12, n, a(n) )
\\ Joerg Arndt, Dec 05 2022

A358400 a(n) is the number of reducible monic quintic polynomials (x^5 + rx^4 + sx^3 + tx^2 + ux + v) with integer coefficients bounded by naïve height n (abs(r), abs(s), abs(t), abs(u), abs(v) <= n).

Original entry on oeis.org

139, 1313, 5359, 15365, 34229, 68385, 120421, 200839, 312057, 468827, 669591, 943175, 1274089, 1701441, 2216841, 2856379, 3594651, 4510437, 5541135, 6788823, 8195941, 9845089, 11670925, 13842429, 16191555

Offset: 1

Lorenz H. Menke, Jr., Nov 13 2022

Cf. A358398, A358399, A067274.

{ a(n) = \\ A358400
    my( ct = 0 );
    for (c1 = -n, n,
    for (c2 = -n, n,
    for (c3 = -n, n,
    for (c4 = -n, n,
    for (c5 = -n, n,
        if ( ! polisirreducible( Pol([1,c1,c2,c3,c4,c5]) ), ct += 1 );
    ); ); ); ); );
    return( ct );
}
vector(7, n, a(n) )
\\ Joerg Arndt, Dec 05 2022

A270383 Number of ordered pairs (i,j) with i >= j, |i|, |j| <= n, and |i * j| <= n.

Original entry on oeis.org

1, 6, 12, 18, 27, 33, 43, 49, 59, 68, 78, 84, 98, 104, 114, 124, 137, 143, 157, 163, 177, 187, 197, 203, 221, 230, 240, 250, 264, 270, 288, 294, 308, 318, 328, 338, 359, 365, 375, 385, 403, 409, 427, 433, 447, 461, 471, 477, 499, 508, 522

Offset: 0

Lorenz H. Menke, Jr., Mar 15 2016

nonn

			For n = 2 the a(2) = 12 pairs are (2,1), (2,0), (2,-1), (1,1), (1,0), (1,-1), (1,-2), (0,0), (0,-1), (0,-2), (-1,-1), and (-1,-2). - _Danny Rorabaugh_, Apr 05 2016

David A. Corneth, Table of n, a(n) for n = 0..9999

Cf. A000005, A067274, A226355.

a[n_]:=2Sum[Length[Divisors[k]],{k,1,n}]+Floor[Sqrt[n]]+2n+1

a(n) = 2*sum(k=1, n, numdiv(k)) + sqrtint(n) + 2*n + 1; \\ Michel Marcus, Apr 05 2016

a(n) = 2*(Sum_{k=1..n} tau(k)) + floor(sqrt(n)) + 2*n + 1, where tau(k) = A000005(k) is number of divisors of k.

a(n) = A067274(n) + 2 for n >= 1.

cp's OEIS Frontend

A069274 13-almost primes (generalization of semiprimes).

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A091627 Number of ordered integer pairs (b,c) with 1 <= b,c <= n such that both roots of x^2+bx+c=0 are integers.

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A091626 Number of ordered integer pairs (b,c) with 0 <= b, c <= n such that both roots of x^2+bx+c=0 are integers.

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A379597 a(n) is the number of distinct solution sets to the quadratic equations u*x^2 + v*x + w = 0 with integer coefficients u, v, w, abs(u) + abs(v) + abs(w) <= n having a nonnegative discriminant.

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A381710 a(n) is the number of distinct solution sets to the quadratic equations u*x^2 + v*x + w = 0 with integer coefficients u, v, w, abs(u) + abs(v) + abs(w) <= n having a negative discriminant.

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A381711 a(n) = A379597(n) - A381710(n).

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A358398 a(n) is the number of reducible monic cubic polynomials x^3 + r*x^2 + s*x + t with integer coefficients bounded by naïve height n (abs(r), abs(s), abs(t) <= n).

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A358399 a(n) is the number of reducible monic quartic polynomials (x^4 + r*x^3 + s*x^2 + t*x + u) with integer coefficients bounded by naïve height n (abs(r), abs(s), abs(t), abs(u) <= n).

A379597 a(n) is the number of distinct solution sets to the quadratic equations ux^2 + vx + w = 0 with integer coefficients u, v, w, abs(u) + abs(v) + abs(w) <= n having a nonnegative discriminant.

A381710 a(n) is the number of distinct solution sets to the quadratic equations ux^2 + vx + w = 0 with integer coefficients u, v, w, abs(u) + abs(v) + abs(w) <= n having a negative discriminant.

A358398 a(n) is the number of reducible monic cubic polynomials x^3 + rx^2 + sx + t with integer coefficients bounded by naïve height n (abs(r), abs(s), abs(t) <= n).

A358399 a(n) is the number of reducible monic quartic polynomials (x^4 + rx^3 + sx^2 + t*x + u) with integer coefficients bounded by naïve height n (abs(r), abs(s), abs(t), abs(u) <= n).

A358400 a(n) is the number of reducible monic quintic polynomials (x^5 + rx^4 + sx^3 + tx^2 + ux + v) with integer coefficients bounded by naïve height n (abs(r), abs(s), abs(t), abs(u), abs(v) <= n).