A254571 -id:A254571 - cp's OEIS frontend

A254671 Numbers that can be represented as x*y + x + y, where x >= y > 1.

8, 11, 14, 15, 17, 19, 20, 23, 24, 26, 27, 29, 31, 32, 34, 35, 38, 39, 41, 43, 44, 47, 48, 49, 50, 51, 53, 54, 55, 56, 59, 62, 63, 64, 65, 67, 68, 69, 71, 74, 75, 76, 77, 79, 80, 83, 84, 86, 87, 89, 90, 91, 92, 94, 95, 97, 98, 99, 101, 103, 104, 107, 109, 110, 111, 113

Offset: 1

Alex Ratushnyak, Feb 04 2015

nonn

Apparently 8 and the elements of A061743. - R. J. Mathar, Feb 19 2015
This is true. For proof, see link.
As x*y + x + y = (x + 1)*(y + 1) - 1 where x >= y > 1 we have k = a(n) is in the sequence if and only if k + 1 is an odd composite or k + 1 is an even number with more than 4 divisors. - David A. Corneth, Oct 15 2024

			14 = 2*4 + 2 + 4.
15 = 3*3 + 3 + 3.
There is no way to express 16 in this form, so it is not in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000
Robert Israel, A254571 and A061743

Cf. A071904, A254636 is the complement.

filter:= proc(n) local t;
  if n::odd then numtheory:-tau(n+1) > 4 else not isprime(n+1) fi
end proc:
select(filter, [$1..200]); # Robert Israel, Nov 14 2024

sol[t_] := Solve[x >= y > 1 && x y + x + y == t, {x, y}, Integers];
Select[Range[200], sol[#] != {}&] (* Jean-François Alcover, Jul 28 2020 *)

def aupto(limit):
    cands = range(2, limit//3+1)
    nums = [x*y+x+y for i, y in enumerate(cands) for x in cands[i:]]
    return sorted(set(k for k in nums if k <= limit))
print(aupto(113)) # Michael S. Branicky, Aug 11 2021

from sympy import primepi
def A254671(n):
    def f(x): return int(n+(x>=7)+primepi(x+1)+primepi(x+1>>1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Oct 14 2024

A254572 Least multiple of n that is abundant or perfect (A023196).

Original entry on oeis.org

6, 6, 6, 12, 20, 6, 28, 24, 18, 20, 66, 12, 78, 28, 30, 48, 102, 18, 114, 20, 42, 66, 138, 24, 100, 78, 54, 28, 174, 30, 186, 96, 66, 102, 70, 36, 222, 114, 78, 40, 246, 42, 258, 88, 90, 138, 282, 48, 196, 100, 102, 104, 318, 54, 220, 56, 114, 174, 354, 60

Offset: 1

Jeppe Stig Nielsen, Feb 01 2015

nonn

See A254571(n) = a(n)/n for the multipliers.

Very often a(n) = A064162(n), but the definition of the latter requires strict abundance.

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

Cf. A023196, A064162, A254571.

f:= proc(n) local k; uses numtheory;
      for k from 1 to 4 do if sigma(k*n)>=2*k*n then return k*n fi od:
      6*n
end proc:
map(f, [$1..100]); # Robert Israel, Feb 10 2019

a[n_] := Do[If[DivisorSigma[1, k*n] >= 2*k*n, Return[k*n]], {k, {1, 2, 3, 4, 6}}];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 09 2023 *)

a(n) = forstep(m=n,6*n,n,if(sigma(m)>=2*m,return(m)))

A384665 Smallest odd multiplier k such that k*n is abundant.

Original entry on oeis.org

945, 9, 315, 3, 189, 3, 135, 3, 105, 3, 315, 1, 315, 3, 63, 3, 315, 1, 315, 1, 45, 3, 315, 1, 63, 3, 35, 3, 315, 1, 315, 3, 105, 3, 27, 1, 315, 3, 105, 1, 315, 1, 315, 3, 21, 3, 315, 1, 45, 3, 105, 3, 315, 1, 63, 1, 105, 3, 315, 1, 315, 3, 15, 3, 63, 1, 315, 3

Offset: 1

Sergio Pimentel, Jun 06 2025

nonn

a(n) <= 945 for all n. a(n) = 945 for prime numbers > 103.

The possible values appear to be: 1, 3, 5, 7, 9, 15, 21, 25, 27, 35, 45, 63, 105, 135, 189, 315, 945. - Michel Marcus, Jun 12 2025

			a(5) = 189 because 189 is the smallest odd multiplier k such that 5*k is abundant (i.e., 5*189 = 945 which is abundant).

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

Cf. A005101, A005231, A023196, A254571.

f:= proc(n) local k;
  for k from 1 by 2 do if numtheory:-sigma(n*k) > 2*n*k then return k fi od
end proc:
map(f, [$1..100]); # Robert Israel, Jun 09 2025

a[n_] := Module[{k = 1}, While[DivisorSigma[-1, k*n] <= 2, k += 2]; k]; Array[a, 100] (* Amiram Eldar, Jun 06 2025 *)

a(n) = my(k=1); while (sigma(k*n,-1)<=2, k+=2); k; \\ Michel Marcus, Jun 09 2025

cp's OEIS Frontend

A254671 Numbers that can be represented as x*y + x + y, where x >= y > 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Python

A254572 Least multiple of n that is abundant or perfect (A023196).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A384665 Smallest odd multiplier k such that k*n is abundant.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI