A061743 -id:A061743 - 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

A061752 Positive integers k such that k! is divisible by (k+1)^4.

Original entry on oeis.org

11, 23, 29, 31, 34, 35, 39, 41, 44, 47, 49, 53, 54, 55, 59, 62, 63, 64, 65, 69, 71, 74, 76, 77, 79, 80, 83, 84, 87, 89, 90, 94, 95, 97, 98, 99, 101, 103, 104, 107, 109, 111, 113, 114, 116, 118, 119, 120, 124, 125, 127, 129, 131, 132, 134, 135, 137, 139, 142, 143

Offset: 1

Robert G. Wilson v, Jun 21 2001

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A061743, A061751, A061753, A061754, A061755, A061756, A061757, A061758, A061759.

Select[Range[150], IntegerQ[ #!/(# + 1)^4] &]

A061753 Positive integers k such that k! is divisible by (k+1)^5.

Original entry on oeis.org

23, 29, 31, 35, 39, 41, 44, 47, 49, 53, 55, 59, 62, 63, 65, 69, 71, 74, 76, 77, 79, 80, 83, 87, 89, 90, 95, 97, 98, 99, 101, 103, 104, 107, 109, 111, 113, 116, 118, 119, 120, 124, 125, 127, 129, 131, 132, 134, 135, 137, 139, 142, 143, 146, 149

Offset: 1

Robert G. Wilson v, Jun 21 2001

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A061743, A061751, A061752, A061754, A061755, A061756, A061757, A061758, A061759.

Select[Range[150], IntegerQ[ #!/(# + 1)^5] &]

A061755 Positive integers k such that k! is divisible by (k+1)^7.

Original entry on oeis.org

35, 39, 44, 47, 53, 55, 59, 62, 63, 69, 71, 74, 79, 80, 83, 87, 89, 95, 97, 98, 99, 103, 104, 107, 109, 111, 116, 119, 124, 125, 127, 129, 131, 134, 135, 139, 142, 143, 146, 149, 151, 152, 153, 155, 159, 161, 164, 167, 169, 170, 174, 175, 179, 181, 183, 186

Offset: 1

Robert G. Wilson v, Jun 21 2001

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A061743, A061751, A061752, A061753, A061754, A061756, A061757, A061758, A061759.

Select[Range[200], IntegerQ[ #!/(# + 1)^7] &]

A061756 Positive integers k such that k! is divisible by (k+1)^8.

Original entry on oeis.org

39, 44, 47, 55, 59, 62, 63, 69, 71, 74, 79, 80, 83, 89, 95, 98, 99, 104, 107, 109, 111, 116, 119, 124, 125, 127, 129, 131, 134, 139, 142, 143, 146, 149, 152, 153, 155, 159, 161, 164, 167, 169, 170, 174, 175, 179, 181, 186, 188, 189, 191, 194, 195, 197, 199

Offset: 1

Robert G. Wilson v, Jun 21 2001

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A061743, A061751, A061752, A061753, A061754, A061755, A061757, A061758, A061759.

Select[Range[200], IntegerQ[ #!/(# + 1)^8] &]

A061757 Positive integers k such that k! is divisible by (k+1)^9.

Original entry on oeis.org

44, 47, 59, 62, 63, 69, 71, 79, 80, 83, 89, 95, 99, 104, 107, 109, 111, 119, 124, 125, 127, 129, 131, 134, 139, 142, 143, 146, 149, 153, 155, 159, 161, 164, 167, 169, 174, 175, 179, 181, 186, 188, 189, 191, 194, 195, 197, 199, 203, 207, 208, 209, 215, 219

Offset: 1

Robert G. Wilson v, Jun 21 2001

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A061743, A061751, A061752, A061753, A061754, A061755, A061756, A061758, A061759.

Select[Range[250], IntegerQ[ #!/(# + 1)^9] &]

A061758 Positive integers k such that k! is divisible by (k+1)^10.

Original entry on oeis.org

47, 59, 69, 71, 79, 83, 89, 95, 99, 104, 107, 111, 119, 125, 127, 131, 134, 139, 142, 143, 146, 149, 153, 155, 159, 161, 164, 167, 174, 175, 179, 181, 186, 188, 191, 194, 195, 197, 199, 203, 207, 208, 209, 215, 219, 220, 223, 224, 227, 230, 233, 237, 239

Offset: 1

Robert G. Wilson v, Jun 21 2001

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A061743, A061751, A061752, A061753, A061754, A061755, A061756, A061757, A061759.

Select[Range[250], IntegerQ[ #!/(# + 1)^10] &]

A061764 Positive integers k such that k! is divisible by (k+1)^12.

Original entry on oeis.org

59, 71, 79, 83, 89, 95, 104, 107, 111, 119, 125, 127, 131, 134, 139, 143, 149, 153, 159, 161, 164, 167, 174, 175, 179, 181, 188, 191, 194, 195, 197, 199, 207, 209, 215, 219, 220, 223, 224, 230, 233, 237, 239, 242, 244, 246, 249

Offset: 1

Robert G. Wilson v, Jun 21 2001

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A061743, A061751, A061752, A061753, A061754, A061755, A061756, A061757, A061758, A061759, A061768.

Select[Range[250], IntegerQ[ #!/(# + 1)^12] &]

A061744 a(n) = (2^A000959(n)) - 1.

Original entry on oeis.org

1, 7, 127, 511, 8191, 32767, 2097151, 33554431, 2147483647, 8589934591, 137438953471, 8796093022207, 562949953421311, 2251799813685247, 9223372036854775807, 147573952589676412927, 590295810358705651711, 9444732965739290427391, 37778931862957161709567

Offset: 1

Jason Earls, May 07 2001

nonn

Cf. A000959, A061743, A000668.

cp's OEIS Frontend

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

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A061752 Positive integers k such that k! is divisible by (k+1)^4.

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A061753 Positive integers k such that k! is divisible by (k+1)^5.

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A061755 Positive integers k such that k! is divisible by (k+1)^7.

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A061756 Positive integers k such that k! is divisible by (k+1)^8.

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A061757 Positive integers k such that k! is divisible by (k+1)^9.

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A061758 Positive integers k such that k! is divisible by (k+1)^10.

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A061764 Positive integers k such that k! is divisible by (k+1)^12.

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A061744 a(n) = (2^A000959(n)) - 1.

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